MATH 1271 (Calculus I) INSTRUCTOR: SCOT ADAMS
(topics summaries)
Topic 0010
(Standard Notation)
Link to Topics website
- 0010(1): title slide
- 0010(2): abbreviations
- for all, for any
- there exists, there exist
- such that
- implies
- if and only if
- 0010(3): def'n: scalar
- 0010(3): the empty set, union, intersection, complement
- 0010(3): is an element of, is not an element of
- 0010(3): some important sets
- \Z={integers}
- \R={real numbers}
- \Q={rational numbers}
- \C={complex numbers}
- 0010(4): subset, superset
- 0010(5-20): intervals
- 0010(5): def'n and e.g.: interval
- 0010(5): the set (-1,1); visualization with open dots
- 0010(6): visualization with parentheses
- 0010(7): def'n: open interval
- 0010(8): the open unbounded interval (-\infty,1); visualization with arrow and parenthesis
- 0010(9): visualization with arrow and open dot
- 0010(10): the open unbounded interval (-\infty,\infty); visualization with arrows
- 0010(11): the interval [-1,1]; visualization with closed dots
- 0010(11): def'n: compact interval
- 0010(12): visualization with brackets
- 0010(13): the interval (-1,1]; visualization with open dot and closed dot
- 0010(14): visualization with parenthesis and bracket
- 0010(14): half-open interval, open on the left, closed on the right
- 0010(15): the interval [-1,1); visualization with closed dot and open dot
- 0010(16): visualization with bracket and parenthesis
- 0010(16): half-open interval, closed on the left, open on the right
- 0010(17): the interval (-\infty,1]; visualization with closed dot
- 0010(18): the interval (-\infty,1]; visualization with arrow and bracket
- 0010(18): closed means as closed as possible; compact is the same as closed and bounded
- 0010(19): the interval [-1,-\infty); visualization with bracket and arrow
- 0010(20): SKILL: graph interval
- 0010(20): SKILL: identify interval as open, closed, half-open, bounded, etc.
- 0010(21): transition slide
- 0010(22): visualizing \Z
- 0010(23): \Z meets some, but not every open interval
- 0010(23): you cannot approxmate \sqrt{2} to within 0.001 using integers
- 0010(24): visualizing \Q
- 0010(25): \Q meets every open interval
- 0010(25): def'n: dense means meets every open interval
- 0010(25): you can approxmate \sqrt{2} to within 0.001 using rational numbers
Topic 0020
(Functions and expressions)
Link to Topics website
- 0020(1): title slide
- 0020(2): def'n: function; def'n: domain; def'n: target
- 0020(2): e.g.: function
- 0020(2): notation for evaluation of a function
- 0020(3): e.g.: domain
- 0020(3): convention: if unspecified, domain and target as large as possible
- 0020(3): meaning of "undefined at"
- 0020(3): notation to indicate domain and range, e.g., f:[3,\infty)--->\R
- 0020(4): transition slide
- 0020(5): distinction between functions and expressions
- 0020(6): transition slide
- 0020(7): changing the domain changes the function
- 0020(7): e.g. and notation for restriction of a function
- 0020(8): def'n and e.g.s: image
- 0020(8): notation dom for domain, im for image
- 0020(8): avoid using "range"; seems to mean both target and image
- 0020(9): changing the target does not change the function, but the target must contain the image
- 0020(10): advantage of functions: no arbitrary choices of variables
- 0020(10): illustration via Fahrenheit vs. Celsius
- 0020(10): illustration of independent and dependent variables
- 0020(11): transition slide
- 0020(12): illustration of inverse functions via Fahrenheit and Celsius
- 0020(13-27): operations on functions
- 0020(13): operation: addition of functions; domain of sum
- 0020(14): operation: subtraction of functions; domain of difference
- 0020(15): operation: multiplication of functions; domain of product
- 0020(16): operation: division of functions; domain of quotient
- 0020(17): operation: scalar multiplication of functions; domain of scalar multiple
- 0020(17): linear operations
- 0020(17): problem involving closure under linear operations
- 0020(18): illustration of solution of that problem
- 0020(18): def'n and e.g.s: linear combination
- 0020(19): def'n: quadratic polynomial in x
- 0020(20): operation: evaluation of functions; evaluation is additive
- 0020(21): evaluation is multiplicative
- 0020(22): evaluation commutes with scalar multiplication
- 0020(23): def'n: linear operation; evaluation is linear and multiplicative
- 0020(24): most operations studied in this course will be linear, but NOT multiplicative
- 0020(25): operation: difference evaluation
- 0020(26): difference evaluation is linear
- 0020(27): difference evaluation is NOT multiplicative
- 0020(27): def'n and e.g.: the cocycle identity for difference evaluation
Topic 0030
(Polynomials and rational functions)
Link to Topics website
- 0030(1): title slide
- 0030(2-6): collect like terms
- 0030(2): collect like terms
- 0030(2): SKILL: collect like terms
- 0030(3-6): expand and collect like terms
- 0030(5): SKILL:expand and collect like terms
- 0030(6): def'n: polynomial in t
- 0030(7): def'n: polynomial; def'n: polynomial in x,u,q, etc.
- 0030(7): examples of polynomials, non-example of a polynomial
- 0030(7): SKILL: recognize polynomial
- 0030(7): SKILL: simplify polynomial
- 0030(8): transition slide
- 0030(9): def'n: rational function; def'n: rational expression in t,v,b,x, etc.
- 0030(9-11): examples of rational expressions
- 0030(11): SKILL: recognize rational function
- 0030(11): SKILL: simplify rational function
- 0030(12): def'n: degree of a polynomial
- 0030(12): e.g.: degree of a polynomial
- 0030(12): SKILL: find the degree of a polynomial
- 0030(12): def'n: degree of a polynomial
- 0030(12): def'n: constant, linear, quadratic, cubic, quartic, quintic
- 0030(13): def'n by example: constant term, linear term, quadratic term,
cubic term, quartic term, quintic term, degree six term
- 0030(13): SKILL: identify terms of a polynomial
- 0030(14): def'n by example: constant coefficient, linear coefficient, quadratic coefficient,
cubic coefficient, quartic coefficient, quintic coefficient, degree six coefficient
- 0030(14): def'n: leading coefficient
- 0030(14): SKILL: identify coefficients of a polynomial
- 0030(15): e.g.: leading coefficient
- 0030(15): SKILL: identify leading coefficient of a polynomial
- 0030(15): def'n: monic polynomial
- 0030(16): e.g.: leading term
- 0030(16): SKILL: identify leading term of a polynomial
- 0030(17-19): heierarchy of functions and expressions
- 0030(17): review definitions of polynomial, rational; examples
- 0030(17): def'n: algebraic, transcendental; examples
- 0030(17): heierarchy of polynomials: constant linear, quadratic, etc.
- 0030(17): easiest type of algebraic that is not rational: rational except for a single square root
- 0030(18): transition slide
- 0030(19): polynomial implies rational implies algebraic
- 0030(19): algebraic and transcendental are opposites
- 0030(20-30): polynomial division
- 0030(20): e.g.: polynomial division
- 0030(20): SKILL: polynomial division
- 0030(21): writing the dividend as: (divisor times quotient) plus remainder
- 0030(22-24): synthetic division for dividing x-a into a polynomial
- 0030(24): SKILL: synthetic division of x-a into polynomial
- 0030(24): dividing x-a into p(x), the remainder is p(a)
- 0030(25): transition slide
- 0030(26): [x-a divides evenly into p(x)] iff [a is a root of p]
- 0030(26-29): e.g.: repeated factoring of x-a from a polynomial
- 0030(29): SKILL: repeated factoring of x-a from a polynomial
- 0030(30): using the last remainder (which is nonzero) to evaluate the final quotient
- 0030(31): def'n: multiplicity
- 0030(31): SKILL: find the multiplicity of a root of a polynomial
- 0030(32): SKILLs and Whitman problems:
- find domain
- find domain of composite
- words to function and find domain
Topic 0040
(Miscellaneous precalculus)
Link to Topics website
- 0040(1): title slide
- 0040(2): SKILL: factoring a polynomial
- 0040(2): SKILL: intervals of positivity and negativity for a factored polynomial
- 0040(2): e.g.: factoring a polynomial and determining where it's pos and neg
- 0040(3): another example, with a root of multiplicity > 1
- 0040(4): another example with a negative leading coefficient
- 0040(5): SKILL: intervals of positivity and negativity for a factored rational function
- 0040(5): e.g.: intervals of positivity and negativity for a factored rational function
- 0040(6): start of binomial theorem on expanding (x+y)^n
- 0040(7-8): expansion without collecting terms
- 0040(9): expansion with collecting terms up to (x+y)^2
- 0040(9): careful expansion of (x+y)^3
- 0040(10): transition slide
- 0040(11): working with the coefficients, not the whole polynomial
- 0040(12): the coefficients of (x+y)^4
- 0040(13): the expansion of (x+y)^4
- 0040(14): how to work out the monomial in each term
- 0040(15): transition slide
- 0040(16): a triangle of numbers
- 0040(17): another row of the triangle, called Pascal's triangle
- 0040(17): expansion of (x+y)^5
- 0040(18): the first entry in the nth row is n
- 0040(18): NOTE: The top row is the 0th row.
- 0040(18): NOTE: The leftmost entry in any row is its 0th entry.
- 0040(19-22): rationalizing numerator and denominator
- 0040(19): SKILL: rationalize numerator and denominator
- 0040(19): e.g.: rationalize the denominator in 1 / \sqrt{2}
- 0040(19): e.g.: rationalize the denominator in 4 / \sqrt{7}
- 0040(20): e.g.: rationalize the numerator in \sqrt{3} / \sqrt{7}
- 0040(20): e.g.: rationalize the numerator in [7 - \sqrt{3}] / \sqrt{5}
- 0040(21): e.g.: rationalize the denominator in 5 / \sqrt{x}
- 0040(21): e.g.: rationalize the numerator in [7 - \sqrt{h}] / h
- 0040(22): e.g.: rationalize the numerator in [\sqrt{9+h} - 3] / h
Topic 0050
(Absolute value and distance)
Link to Topics website
- 0050(1): title slide
- 0050(2): def'n and notation: absolute value of x
- 0050(3-5): e.g.: absolute value
- 0050(5): SKILL: compute absolute value
- 0050(5): \sqrt{x^2} = |x|
- 0050(6): the distance from a to b, denoted dist(a,b) is |b-a| (which is the same as |a-b|)
- 0050(6): SKILL: compute distance
- 0050(7): |x| = dist(x,0)
- 0050(7): [ |x| <= r ] iff [ -r <= x <= r ]
- 0050(7): [ |x-a| <= r ] iff [ a-r <= x <= a+r ]
- 0050(8): similar for |x-a| < r
- 0050(9-12): SKILL: graph an absolute value inequality
- 0050(9-12): e.g.: graph an absolute value inequality
- 0050(13-19): SKILL: graph neighborhood
- 0050(13): e.g.: graph neighborhood
- 0050(14-19): SKILL: graph punctured neighborhood
- 0050(14): e.g.: graph punctured neighborhood
- 0050(15-19): more graphing of neighborhoods and punctured neighborhoods
- 0050(19): stretching a graph for better viewing
- 0050(20): distance in the plane
- 0050(21): distance in three dimensions
- 0050(21): SKILL: compute distances in higher dimensions
- 0050(22): the triangle inequality in the plane
- 0050(22-23): the triangle inequality on the line
- 0050(24-27): rewriting the triangle inequality as subadditivity of absolute value
- 0050(28-31): additivity of error proved via the triangle inequality
Topic 0060
(Elementary graphing)
Link to Topics website
- 0060(1): title slide
- 0060(2): def'n: graph of a function
- 0060(2): ordered pairs <--> points in the plan
- 0060(2): visualizing graphs of functions
- 0060(2-3): e.g.: graph of a f(x)=x^2
- 0060(4): change of units on the x-axis
- 0060(4): distortions caused by a change of units
- 0060(5): transition slide
- 0060(6): def'n: restriction of a function to a subset of its domain
- 0060(7): graph of the restricted squaring function, i.e. of squaring restricted to [0,\infty)
- 0060(7-23): translation and dilation
- 0060(8-11): effect (on graph) of changing x to x-a, in an equation of x and y
- 0060(8-10): equation of a circle
- 0060(10): translation means shifting
- 0060(12): effect (on graph) of changing y to y-a, in an equation of x and y
- 0060(13): effect (on graph) of changing x to x/a, in an equation of x and y
- 0060(13): dilation means stretching
- 0060(14): effect (on graph) of changing y to y/a, in an equation of x and y
- 0060(15-23): can change the axes, instead of moving the graph
- 0060(21,23): beware of distortions introduced by change of units on axes
- 0060(24-36): lines
- 0060(24-27): given two points on a line, find its slope
- 0060(24): SKILL: points to slope
- 0060(24): e.g.: points to slope
- 0060(24): slope measures steepness
- 0060(25): downhill lines have negative slope
- 0060(26): transition slide
- 0060(27): interchanging points doesn't change the slope
- 0060(28): given point and slope, find an equation for the line
- 0060(28): SKILL: point/slope to equation
- 0060(29-30): given an equation for a line, find the linear function
- 0060(29-30): SKILL: equation of line to linear function
- 0060(31-36): given two points, find linear function whose graph goes through them
- 0060(31): given two points, find linear function through them
- 0060(31): one approach: slope, then equation, then function
- 0060(32-36): another approach: linear combination of well chose linear expressions
- 0060(32-33): first variant
- 0060(34): describe second variant
- 0060(35-36): second variant
- 0060(37): SKILLS:
- two points to y=mx+b
- change equation to y=mx+b and find intercepts
- check whether lines are parallel
- find lines along edges of a triangle
- from words to an equation of a line
- translation and dilation of a graph
- equations of circles
- distance and slope, circle equation and line equation
- 0060(38): SKILLs and Whitman problems:
- two points to y=mx+b
- change equation to y=mx+b and find intercepts
- check whether lines are parallel
- find lines along edges of a triangle
- words to equation of line
- translate/dilated graph
- equations of circles
- distance and slope, circle equation and line equation
Topic 0070
(Summation)
Link to Topics website
- 0070(1): title slide
- 0070(2): 1^2 + \cdots + n^2 = ??
- 0070(2): def'n: \triangle a_n
- 0070(3): e.g.: \triangle 2n
- 0070(3): if the difference of a sequence is the zero sequence, then the sequence is constant
- 0070(3-4): e.g.: \triangle n^4
- 0070(4): \triangle reduces the degree of a polynomial in n by 1
- 0070(5): s_n := 1^2 + ... + n^2
- 0070(6): s_0 = 0
- 0070(6-7): \triangle s_n = (n+1)^2 = n^2+2n+1
- 0070(8): expect s_n to be a cubic in n
- 0070(8-9): \triangle n^3, \triangle n^2, \triangle n
- 0070(10): \triangle is linear (additive and commutes with scalar multplication)
- 0070(10): \triangle(n^3/3) has same leading term as \triangle s_n, namely n^2
- 0070(11): \triangle is not multiplicative
- 0070(12-14): differencing by parts, a.k.a. the product rule for differencing
- 0070(15): the leading two terms of \triangle((n^3/3)+(n^2/2)) agree with the leading terms of \triangle s_n, namely n^2+2n
- 0070(16-17): \triangle((n^3/3)+(n^2/2)+(n/6))=\triangle s_n
- 0070(18-19): (n^3/3)+(n^2/2)+(n/6) = s_n
- 0070(20): common denominator yields: 1^2 + ... + n^2 = s_n = (2n^3+3n^2+n)/6
- 0070(21): transition slide
- 0070(22): similar formulas for 1 + ... + n and 1^3 + ... + n^3
- 0070(23): a curious formula: (1 + ... + n)^2 = 1^3 + ... + n^3
- 0070(24): transition slide
- 0070(25-39): explanation of the curious formula (1 + ... + n)^2 = 1^3 + ... + n^3
Topic 0080
(The Sigma notation)
Link to Topics website
- 0080(1): title slide
- 0080(2): the Sigma (\Sum) notation, with examples
- 0080(3): transition slide
- 0080(4): \Sum_{j=1}^n j, \Sum_{j=1}^n j^2 and \Sum_{j=1}^n j^3
- 0080(5): \Sum is linear (additive and commutes with scalar multiplication)
- 0080(6): if an expression inside \Sum_j doesn't depend on j, it can be "factored out" to the outside of \Sum_j
- 0080(7): transition slide
- 0080(8): if it does depend on j, then it cannot be factored out
- 0080(9-10): beware of hidden dependencies through dependent variables
- 0080(11): \Sum is not multiplicative, but there is a sumation by parts formula
- 0080(12): if you add an integer to the index variable all through the expression inside \Sum,
and also subtract that same integer from the limits of summation, then the sum does not change
Topic 0090
(Basics of trigonometry)
Link to Topics website
- 0090(1): title slide
- 0090(2): degrees vs radians vs revolutions
- 0090(2): pictures of 90, 60, 45, 30 degrees
- 0090(2): def'n: degree; def'n: radian, def'n of \pi
- 0090(2): picture of 1 radian, which is approximately 57 degrees
- 0090(2): SKILL: radian/degree/revolution conversion
- 0090(3): pictures of \pi/2, \pi/3, \pi/4, \pi/6 radians (same as 90, 60, 45, 30 degrees)
- 0090(4): transition slide
- 0090(5): the standard orbiter
- 0090(5): radians are useful because the standard orbiter covers one radian per unit time
- 0090(6-9): some locations of the standard orbiter at various times
- 0090(10-12): at time t, the y-coordinate of the standard orbiter is \sin t;
NOTE: We always use radians unless otherwise specified, so \sin t mean the sine of t radians.
- 0090(13-15): sin(\pi/2), sin(\pi/3), sin(\pi/4), sin(\pi/6), sin(0)
- 0090(15): SKILL: sine computation
- 0090(16-17): some important points on the circle; learn:
- their y-coordinates
- the times when the standard orbiter reaches them
- 0090(18): def'n: cos t; def'n: complementary angles
- 0090(19-22): \cos t is the x-coordinate of the standard orbiter at time t
- 0090(23): [ \cos(t+\pi) = - \cos t ] AND [ \sin(t+\pi) = - \sin t ]
- 0090(24): t is a dummy variable; could use x, so
[ \cos(x+\pi) = - \cos x ] AND [ \sin(x+\pi) = - \sin x ]
- 0090(25): learn the x-coordinates of those important points on the circle
- 0090(26): \sin^2 + \cos^2 = 1
- 0090(26): def'n: \tan\theta; def'n: \sec\theta
- 0090(26): the geometry of \sin, \cos, \tan and \sec
- 0090(27): 1 + \tan^2 = \sec^2
- 0090(28): transition slide
- 0090(29): def'n: \cot\theta; def'n: \csc\theta
- 0090(29): some formulas
- 1 + \cot^2 = \csc^2
- \tan = \sin / \cos
- \cot = \cos / \sin
- \sec = 1 / \cos
- \csc = 1 / \sin
- 0090(30-33): the geometry of \sin, \cos, \tan and \cot
- 0090(34): SKILLs and Whitman problems:
- preimage under trig functions
- image under trig functions
- trig identities
- trig graphing
- solve trig equations
Topic 0100
(Sum of angles forumlas in trigonometry)
Link to Topics website
- 0100(1): title slide
- 0100(2): recall the standard orbiter
- 0100(3): def'n: R_t(x,y)
- 0100(3): e.g.: R_{\pi/2}(1,0), R_{\pi/2}(0,1)
- 0100(4): e.g.: R_{\pi/4}(1,0), R_{\pi/4}(0,1)
- 0100(5): R_t(1,0), R_t(0,1)
- 0100(6): transition slide
- 0100(7): R_t(5,0)
- 0100(7): the arrow from (0,0) to (5,0) has components <5,0>
- 0100(7): the arrow from (0,0) to R_t(5,0) has components <5\cos t,5\sin t>
- 0100(8): R_t(0,3)
- 0100(8): the arrow from (0,0) to R_t(0,3) has components <-3\sin t,3\cos t>
- 0100(9): to get to (5,3), start at (0,0) and follow arrows with components <5,0> and then <0,3>
- 0100(9): to get to R_t(5,3), start at (0,0) and follow arrows with components <5\cos t,5\sin t> and then <-3\sin t,3\cos t>
- 0100(9): R_t(5,3) = ...
- 0100(10): R_t(x,y) = ...
- 0100(11): transition slide
- 0100(12): R_t is linear
- 0100(12): (x,y) is a linear combination of (1,0) and (0,1) with coefficients x and y
- 0100(12): R_t(x,y) is a linear combination of R_t(1,0) and R_t(0,1) with coefficients x and y
- 0100(12): this gives another way of computing R_t(x,y)
- 0100(13-14): computing R_u(R_t(1,0)) in two different ways
- 0100(15): equating those two computations
- 0100(16): formulas for cos(t+u) and \sin(t+u)
- 0100(17): changing t,u to \alpha,\beta
- 0100(17-21): computing the "double-angle" formulas for \sin(2\alpha) and \cos(2\alpha)
- 0100(22): transition slide
- 0100(23): the sum of angles formulas and the double-angle formulas
Topic 0110
(Inverse functions)
Link to Topics website
- 0110(1): title slide
- 0110(2): def'n: one-to-one (or 1-1)
- 0110(2): horizontal line test
- 0110(2): SKILL: recognize 1-1 function
- 0110(3): def'n/notation: image
- 0110(3): def'n: onto
- 0110(3): def'n/notation: inverse
- 0110(4): transition slide
- 0110(5): f^n(x)=[f(x)]^n, *except* for n=-1
- 0110(6): restricted squaring is 1-1
- 0110(7): graphing the inverse of restricted squaring, i.e., graphing the square root function
- 0110(8): \sin, \Sin and \arcsin
- 0110(9-10): \cos, \Cos and \arccos
- 0110(11-12): \tan, \Tan and \arctan
- 0110(13-14): \cot, \Cot and \arccot
- 0110(14): \arcsec and \arcsec are not defined in this course
- 0110(14): SKILL: graphs of trig and inverse trig functions
- 0110(15): SKILL: computation of values of inverse trig functions
- 0110(15-16): some values of \sin, \Sin and \arcsin
- 0110(16): exercise: similar tables for \arccos, \arctan and \arccot
- 0110(17): visualizing \cos, \sin and \tan
- 0110(18-19): visualizing \Cos, \Sin and \Tan
- 0110(20): visualizing \Cot
- 0110(20): domains and images of \Sin, \Cos, \Tan and \Cot
- 0110(20): why "arc" is in the names of the inverse trig functions
- 0110(21): domains and images of inverse trig functions
- 0110(21): SKILL: domains and images of restricted trig functions and inverse trig functions
- 0110(22): \arcsin + \arccos = \pi/2 and \arctan + \arccot = \pi/2
- 0110(22-23): proof that \arcsin + \arccos = \pi/2
- 0110(24): exercise to show \arctan + \arccot = \pi/2
- 0110(25-29): comparing the graph of (\pi/2) - \arcsin with the graph of \arccos
- 0110(30-32): computing "inverse trig, then trig"
- 0110(30): \tan(\arccos\theta) and \cos(\arcsin\theta)
- 0110(31): \cos(\arctan\theta) and \sec(\arccos\theta)
- 0110(32): \sin(\arctan\theta) and \csc(\arctan\theta)
- 0110(32): SKILL: write inverse tri9g then trig as a no-trig function
Topic 0120
(Speed of a freely falling body)
Link to Topics website
- 0120(1): title slide
- 0120(2): one second of free fall
- 0120(3): two and three seconds of free fall
- 0120(4): first approximation to instantaneous velocity at 2 second mark, using average velocity between time 2 and time 3
- 0120(5): second approximation to instantaneous velocity at 2 second mark, using average velocity between time 2 and time 2.5
- 0120(6): third approximation to instantaneous velocity at 2 second mark, using average velocity between time 2 and time 2.1
- 0120(7): a table of average velocities
- 0120(7): using a variable h for the length of the time interval
- 0120(7): calculating the average velocity between time 2 and time 2+h
- 0120(8): simplifying when h is not 0
- 0120(9): putting result in table
- 0120(10): more average velocities, using the result for h
- between time 2 and 2.001
- between time 2 and 2.0001
- between time 2 and 2.00001
- 0120(11): some of the approximate answers, and the exact answer (4 rods per second)
- 0120(12): writing the instantaneous velocity as a limit of average velocities,
as the length of the time interval approaches 0 (without equaling 0)
- 0120(13): SKILLs and Whitman problems:
- average an instantaneous velocities
Topic 0130
(Rates of change and slopes of lines)
Link to Topics website
- 0130(1): title slide
- 0130(2): goal: equation of tangent line to y=x^2 at (1,1)
- 0130(2): m := slope of that tangent line
- 0130(2): a secant line between (1,1) and (s,s^2), with s=2
- 0130(3): slope of that secant line
- 0130(4): changing s to 1.5
- 0130(4): exercise to recalculate with s set to 1.1, 1.01, 1.001.
- 0130(5): m as a limit of slopes of secant lines
- 0130(6): replacing s by 1+h
- 0130(7): m = ... = 2
- 0130(8): the equation of the tangent line to y=x^2 at (1,1)
- 0130(9): some terminology
- difference quotient
- secant line
- tangent line
- 0130(10): given two points one a line, the slope is a difference quotient
- 0130(10): slopes of secant lines are difference quotients
- 0130(10): slopes of tangent lines are limits of difference quotients
- 0130(11): slopes are difference quotients
- rise is change in y-coordinate, which is a difference
- run is change in x-coordinate, which is a difference
- slope, calculated as rise/run, is a difference quotient
- 0130(11): for a moving particle on a line, average velocity is a difference quotient
- change in position is a difference
- change in time is a difference
- average velocity is a differnce quotient
- 0130(11): for a moving particle on a line, instantaneous velocity is a limit of difference quotients
- 0130(11): for any two quantities related by a formula, average rate of change is a difference quotient
- change in input is a difference
- change in output is a difference
- average rate of change is a differnce quotient
- 0130(11): for any two quantities related by a formula, instantaneous rate of change is a limit of difference quotients
- 0130(11): this is the theme of differential calculus
- 0130(12): often the input quantity is time, but not always, e.g.:
- Celsius temperature from Fahrenheit temperature
- tax from adjusted gross income
- 0130(13): given a graph of a function relating two quantities,
- average rates of change are slopes of secant lines
- instantaneous rates of change are slopes of tangent lines
- 0130(14-15): relating slopes to velocities for a falling body
- 0130(16): another example ...
- 0130(17-20): my trip to Chicago
- 0130(20): SKILL: compute average velocity
- 0130(21-28): example problem calculating velocities (rock on Mars)
- 0130(29-38): example problem calculating velocities (invoving sine and cosine)
- 0130(36): IOU: limits of (sin x)/x and (1-(cos x))/x, as x ---> 0
- 0130(38): SKILLS and Whitman problems
- slope of secant and tangent lines
- difference quotient and derivative
- slope positive or negative
Topic 0140
(Limits)
Link to Topics website
- 0140(1): title slide
- 0140(2): an example of a limit of a quadratic f(x) as x-->2
- 0140(3-4): an example where the function g is not defined at 2
- 0140(5-6): an example h(x), which is equal to f(x) except at 2, where it has a different value
- 0140(7): an example u(x) which is constant except at 2
- 0140(8-9): an example where computing values to guess the limit can be misleading
- 0140(10): the limit of (\sin x)/x, as x-->0
- 0140(11): the tangent line to y = \sin x, at (0,0)
- 0140(11): the tangent line to y = \sin x, at (0,0), crosses the graph at (0,0)
- 0140(12): the vertical line test
- 0140(12): a tangent line that crosses at the point of tangency, and at other points
- 0140(12): a graph of a 1-1 function, with a tangent line that crosses at the point of tangency, and at other points.
Topic 0150
(The limit game and the exact definition of a limit)
Link to Topics website
- 0150(1): title slide
- 0150(2): the limit game
- 0150(3-5): toward a rigorous definition of a limit
- 0150(6): rigorous definition, but atypically phrased
- 0150(7): toward the typical phrasing
- 0150(8): the usual definition of a limit
- 0150(9): transition slide
- 0150(10-25): limits: notation, terminology, intuitive def'n, rigorous def'n, alternative notation
- 0150(10): limit, as x ---> a, of f(x) is L
- 0150(11): limit, as x ---> a^-, of f(x) is L
- 0150(12): limit, as x ---> a^+, of f(x) is L
- 0150(12): the two-sided limit is L iff both one-sided limits are
- 0150(13): limit, as x ---> a^+, of f(x) is \infty; vertical asymptote
- 0150(14): limit, as x ---> a^+, of f(x) is -\infty; vertical asymptote
- 0150(15): limit, as x ---> a, of f(x) is -\infty; vertical asymptote
- 0150(16): limit, as x ---> a, of f(x) is \infty; vertical asymptote
- 0150(17): limit, as x ---> a^-, of f(x) is \infty; vertical asymptote
- 0150(17): the two-sided limit is \infty iff both one-sided limits are
- 0150(18-19): limit, as x ---> a^-, of f(x) is -\infty; vertical asymptote
- 0150(18-19): the two-sided limit is -\infty iff both one-sided limits are
- 0150(20): limit, as x ---> -\infty, of f(x) is -\infty
- 0150(21): limit, as x ---> -\infty, of f(x) is \infty
- 0150(22): limit, as x ---> \infty, of f(x) is \infty
- 0150(23): limit, as x ---> \infty, of f(x) is -\infty
- 0150(24): limit, as x ---> \infty, of f(x) is L; horizontal asymptote
- 0150(25): limit, as x ---> -\infty, of f(x) is L; horizontal asymptote
Topic 0160
(Sequences)
Link to Topics website
- 0160(1): title slide
- 0160(2): def'n and e.g.: sequence
- 0160(2): visualizing a sequence
- 0160(3-5): limits: notation, terminology, intuitive def'n, rigorous def'n, alternative notation
- 0160(3): limit, as n ---> -\infty, of f(x) is L
- 0160(4): limit, as n ---> -\infty, of f(x) is \infty
- 0160(5): limit, as n ---> -\infty, of f(x) is -\infty
- 0160(6-27): description of the number e
- 0160(6): problem: 6% compounded annually
- 0160(7): transition slide
- 0160(8): problem: 6% compounded semi-annually
- 0160(9): transition slide
- 0160(10): problem: 6% compounded bimonthly
- 0160(11): transition slide
- 0160(12): problem: 100% compounded annually
- 0160(13): transition slide
- 0160(14): problem: 100% compounded semi-annually
- 0160(14): problem: 100% compounded quarterly
- 0160(14): problem: 100% compounded monthly
- 0160(15): transition slide
- 0160(16): problem: 100% compounded daily
- 0160(17): transition slide
- 0160(18): problem: 100% compounded n times per year
- 0160(19): start of problem: 100% compounded continuously
- 0160(20): transition slide
- 0160(21): def'n: e
- 0160(21): from limits of functions to limits of sequences
- 0160(22): end of problem: 100% compounded continuously
- 0160(22): approximate value of e
- 0160(23-26): problem: 4% compounded continuously
Topic 0170
(Simple limit problems)
Link to Topics website
- 0170(1): title slide
- 0170(2-12): SKILL: evaluation of limits and values of a function from its graph
- 0170(13): SKILL: drawing a graph of a function with certain specificed limits and values
- 0170(14-17): SKILL: limits of rational functions, when the denominator tends to 0, but the numerator does not
- 0170(18): transition slide
- 0170(19): e.g.: a graph of a rational function which does not have a two-sided limit at -5
Topic 0180
(Additivity of limit)
Link to Topics website
- 0180(1): title slide
- 0180(2): review definitions of continuity
- 0180(2): fact: x^2 and cos x are both continuous at x = \pi / 4
- 0180(2): restatement in terms of limits
- \lim_{x\to\pi/4} x^2 = \pi^2 / 16
- \lim_{x\to\pi/4} \cos x = \sqrt{2} / 2
- 0180(3): transition slide
- 0180(4): \epsilon-\delta definition of
\lim_{x\to\pi/4} x^2 = \pi^2 / 16, thought of as a game
- 0180(5): change \delta to \alpha
- 0180(5): Alice chooses \alpha, for \epsilon = 0.1
- 0180(6): \epsilon-\delta definition of
\lim_{x\to\pi/4} \cos x = \sqrt{2} / 2, thought of as a game
- 0180(7): change \delta to \beta
- 0180(7): Ben chooses \beta, for \epsilon = 0.07
- 0180(8): goal is to show that
\lim_{x\to\pi/4} (x^2 + \cos x) = (\pi^2 / 16) + (\sqrt{2} / 2)
- 0180(9): \epsilon-\delta definition of
\lim_{x\to\pi/4} (x^2 + \cos x) = (\pi^2 / 16) + (\sqrt{2} / 2), thought of as a game
- 0180(10-17): example: \epsilon = 0.008, thought of as a game
- 0180(10): changing \epsilon to 0.008
- 0180(11): transition slide
- 0180(12): cut \epsilon in half, getting 0.004
- 0180(12): Alice chooses \alpha for 0.004
- 0180(12): Ben chooses \beta for 0.004
- 0180(12): We make \delta the minimum of \alpha and \beta
- 0180(12): Our opponent chooses x
- 0180(13): goal is to show that x^2 + \cos x is 0.008-close to (\pi^2 / 16) + (\sqrt{2} / 2)
- 0180(14): x^2 is 0.004-close to \pi^2 / 16
- 0180(15): \cos x is 0.004-close to \sqrt{2} / 2
- 0180(16-17): additivity of error comletes the proof
- 0180(17): theorem: limit of the sum is equal to the sum of the limits
- 0180(18-20): proof of theorem
- 0180(18): given \epsilon, want \delta
- 0180(18): choice of \alpha and \beta for \epislon / 2
- 0180(18): choice of \delta as the minimum of \alpha and \beta
- 0180(18): choice of x
- 0180(18): want (f(x)) + (g(x)) is \epsilon-close to L + M
- 0180(19): f(x) is (\epsilon / 2)-close to L
- 0180(19): g(x) is (\epsilon / 2)-close to M
- 0180(20): additivity of error completes the proof
Topic 0190
(Limit laws)
Link to Topics website
- 0190(1): title slide
- 0190(2): limit is additive and commutes with scalar multiplication
- 0190(3): limit is linear and is multiplicative and distributes over division
- 0190(4): transition slide
- 0190(5): limit respects linear combination
- 0190(6): example linear combination limit problem
- 0190(7): limit distributes over subtraction
- 0190(7): limit commutes with positive integer powers
- 0190(8): def'n: continuous at a; def'n: the identity function
- 0190(8): constants and the identity are continuous
- 0190(9): the limit, as x --> \infty, of x is \infty
- 0190(9): the limit, as x --> -\infty, of x is -\infty
- 0190(10): transition slide
- 0190(11): positive integer powers of a continuous function are again continuous
- 0190(12): positive integer powers are continuous
- 0190(13): functions commute with limits at points of continuity
- 0190(13): example problem: the limit, as x --> 7, of \sin(x^2)
- 0190(14): positive integer roots are continuous (for even roots, only true at positive numbers)
- 0190(15): transition slide
- 0190(16-18): recall definitions and e.g.s of polynomials and rational functions
- 0190(19): polynomials are continuous at all real numbers
- 0190(19): a rational function is continuous at any number in its domain
- 0190(19): polynomials and rational functions are continous
- 0190(20): example problem: the limit, as x --> \infty, of 1/x
- 0190(20): answer to example problem: because 1 / (very positive) = (very close to zero), the limit is equal to 0
- 0190(20): the form of the answer is 1 / \infty; "1 / \infty = 0"
- 0190(20): meaning of "1/\infty = 0" is
if a limit of f(x) is \infty, then the same limit of 1 / (f(x)) is zero.
- 0190(20): the form of the problem, which is 1 / \infty, determines the answer, which is 0
- 0190(20): 1 / \infty is an example of a "determinate form"
- 0190(20): Because 1 / (very negative) = (very close to zero), we see that the form 1 / (-\infty) is also determinate,
and, specifically that "1 / (-\infty) = 0"
- 0190(21): WARNING: 1 / 0 is not always \infty; in fact, 1 / 0 is slightly indeterminate;
1 / 0 might be \infty or -\infty or DNE; examples of each of these three
- 0190(21): "1 / (0^+) = \infty" and "1 / (0^-) = -\infty"
- 0190(22): exact meaning of "1 / (0^+) = \infty"
- 0190(22): exact meaning of "1 / (0^-) = -\infty"
- 0190(23): example problem: the limit, as x --> 0 of x^2 has the form "1 / 0^+", and so, is equal to 0
- 0190(24-25): some determinate and indeterminate forms
- 0190(25): explanation of how indeterminate forms often show a competition between two expressions
- 0190(25): speficially, 0 . \infty is a competition between 0 and \infty;
sometimes 0 wins, sometimes they tie, sometimes \infty wins
- 0190(25): speficially, 0 . (-\infty) is a competition between 0 and -\infty;
sometimes 0 wins, sometimes they tie, sometimes -\infty wins
- 0190(26-29): some determinate and indeterminate forms
- 0190(30-32): left monotonicity of limit
- 0190(32): exercise: work out the statement for right monotonicity of limit
- 0190(33): two-sided monotonicity of limit
- 0190(34-35): the (two-sided) squeeze theorem
- 0190(35): exercise: the left and right squeeze theorems
(also the squeeze theorem at -\infty, and, also, the squeeze theorem at \infty)
- 0190(36-37): if two functions agree on a punctured neighborhood of a and one has a limit at a,
then the other has the same limit at a
- 0190(38): transition slide
- 0190(39-40): a limit of a difference quotient, done with care
- 0190(41): SKILLS and Whitman problems:
- compute limits
- oscillatory limit
- find \delta
- limits from graph
- calculator estimation of limits
Topic 0200
(Limit problems)
Link to Topics website
- 0200(1): title slide
- 0200(2): example limit of a polynomial
- 0200(2): def'n: continuous
- 0200(2): rational functions (and, in particular, polynomials) are continuous
- 0200(2): example limit of a rational function at a number in its domain
- 0200(2): SKILL: limit of a polynomial
- 0200(3): def'n: continuous in x, continuous in s, etc.
- 0200(4): fact: square root of a nonnegative continuous function is a continuous function
- 0200(5): absolute value is continuous
- 0200(6): transition slide
- 0200(7-8): example limit with an absolute value
- 0200(7): composition of continuous function with a continuous expression is a continuous expression
- 0200(8): a linear combination of continuous expressions is continuous
- 0200(8): solution to the limit with an absolute value
- 0200(8): SKILL: limit with absolute value
- 0200(9): example limit of a rational expression at a number in the domain
- 0200(9): SKILL: limit of a rational expression
- 0200(10-15): example limit of a rational expression at a number NOT in the domain
- 0200(10): using synthetic division for evaluation of the denominator
- 0200(10): strategy: factor and cancel
- 0200(11): repeated synthetic division to factor the denominator
- 0200(12): factoring the numerator
- 0200(13): rewriting the limit problem
- 0200(14): transition slide
- 0200(15): calcellation and completing the limit
- 0200(15): SKILL: limit of a rational expression
- 0200(16): example limit of a rational expression at a number NOT in the domain
- 0200(16): SKILL: limit of a rational expression
- 0200(17): example limit of a rational expression at a number NOT in the domain
- 0200(17): SKILL: limit of a rational expression
- 0200(18): example limit of a rational expression at a number NOT in the domain
- 0200(18): SKILL: limit of a rational expression
- 0200(19-21): general procedure for limits (at finite numbers) of rational expressions
- 0200(22-23): limits of rational expressions at 0
- 0200(22): e.g.: easy to factor out x - 0, which is equal to x
- 0200(22): e.g.; easy to evaluate at x = 0
- 0200(23): e.g.: easy to compute limit at 0
- 0200(23): SKILL: limit of a rational expression
- 0200(24-25): example limit of a rational function at a number not in its domain;
in this case, the function requires simplification
- 0200(25): SKILL: limit of a rational function
- 0200(26): example limit of a nearly rational function (rational, but for a square root)
- 0200(26): SKILL: limit of a nearly rational function (rational, but for a square root)
- 0200(27-29): example limit of a nearly rational function (rational, but for a square root);
in this case, the function requires simplification
- 0200(29): SKILL: limit of a nearly rational function (rational, but for a square root)
- 0200(30-33): example limit of an oscillatory function, via the Squeeze Theorem
- 0200(34): visualization of that last oscillatory limit
- 0200(34): SKILL: oscillatory limit
- 0200(35-36): limit, as x-->0, of [ (x^4 if x \notin \Q) and (0 if x \in \Q) ], via the Squeeze Theorem
- 0200(37-38): visualization of that last limit
- 0200(39): SKILLS and Whitman problems:
- compute limits
- oscillatory limit
- find \delta
- limits from graph
- calculator estimation of limits
Topic 0210
(Continuity)
Link to Topics website
- 0210(1): title slide
- 0210(2): recall definition of continuity at a number
- 0210(2): visualizing a limit at a number; need defined on a punctured neighborhood
- 0210(2): visualizing continuity at a number; need defined on a neighborhood
- 0210(3): transition slide
- 0210(4): def'n: discontinuous at a number
- 0210(4): visualizing continuity and discontinuity at a number; sometimes neither
- 0210(5): def'n: continuous from the left at a number
- 0210(5): def'n: continuous from the right at a number
- 0210(5): visualizing continuity from left and right at a number; sometimes neither
- 0210(5): SKILL: recognize continuity visually
- 0210(6): def'n: infinite discontinuity
- 0210(7): def'n: jump discontinuity
- 0210(8): def'n: removable discontinuity
- 0210(9): recall definition of continuous at a number
- 0210(9): some properties of continuity (linear combinations, products, quotients)
- 0210(10): same properties for continuity from the left
- 0210(11): same properties for continuity from the right
- 0210(12): def'n: continuous on (a,b)
- 0210(12): def'n: continuous on [a,b)
- 0210(12): def'n: continuous on (a,b]
- 0210(13): def'n: continuous on [a,b]
- 0210(13): recall definition of continuous
- 0210(14): e.g.: continuous
- 0210(15): e.g.: not continuous
- 0210(16): classes of functions that are continuous
- 0210(16): classes of functions that are not continuous, but are continuous on their domains
- 0210(17): the Intermediate Value Theorem (IVT)
- 0210(17): visualization of the IVT
- 0210(18): IVT requires continuity
- 0210(19): finding points of disconinuity, stating whether they are points of right or left continuity
- 0210(20): finding maximal intervals of continuity
- 0210(20): classifying types of discontinuity
- 0210(20): SKILL: find maximal intervals of continuity
- 0210(20): SKILL: recognized types of discontinuities visually
- 0210(21): explaining the word maximal (some maximal intervals of continuity are shorter than others)
- 0210(22): sketch a graph with specified continuity properties
- 0210(22): SKILL: sketch types of discontinuities
- 0210(23): consequences of discontinuities in the tax code
- 0210(24): another discontinuity horror: the dorm shower control
- 0210(25-26): finding the continuous function within a two-parameter ("a" and "b") class of piecewise-defined functions
- 0210(25-26): SKILL: force continuity
- 0210(27-30): the horse-goes-up-and-down-the-hill problem
Topic 0220
(Limits of power functions)
Link to Topics website
Topic 0230
(Trigonometric limits)
Link to Topics website
- 0230(1): title slide
- 0230(2-10): the area inside a circle of radius r
- 0230(2-5): the strategy
- for each integer n >= 1, inscribe a regular n-gon inside the circle
- compute the area enclosed in each of these n-gons
- let n go to \infty
- 0230(6): the inscribed regular hexagon (n=6); six isosceles triangles
- 0230(7): height is h_6, base is b_6
- 0230(8): area of one triangle
- 0230(9): area inside hexagon
- 0230(10): area inside the 20-gon
- 0230(10): area inside the n-gon
- 0230(10): limit of the area inside the n-gon, as n ---> \infty
- 0230(11): area in a sector of angle between 0 and \pi/2
- 0230(12): area in an inscribed triangle in the sector with chord as side
- 0230(13): area in a triangle larger than the sector
- 0230(14-15): sin\theta < \theta < tan\theta, for \theta between 0 and \pi/2
- 0230(16): (\sin\theta)/\theta is between 1 and \cos\theta, for 0 < \theta < \pi/2
- 0230(17): the limit, as \theta ---> 0^+, of (\sin\theta)/\theta, is 1
- 0230(18): the limit, as \theta ---> 0^-, of (\sin\theta)/\theta, is 1
- 0230(19): the limit, as \theta ---> 0, of (\sin\theta)/\theta, is 1
- 0230(19): start of limit, as \theta ---> 0, of (1-\cos\theta)/\theta
- 0230(20): "sine-izing" the numerator
- 0230(20): def'n: asymptotic
- 0230(20): [ (f asymptotic to g) and (g approaches L) ] implies [ f approaches L ]
- 0230(20): sometimes a transcendental can be asymptotic to a polynomial,
e.g., \sin\theta and \theta, as \theta ---> 0
- 0230(20): completion of limit, as \theta ---> 0, of (1-\cos\theta)/\theta
- 0230(21): transition slide
- 0230(22): references for limits of (\sin\theta)/\theta and (1-\cos\theta)/\theta
- 0230(22): start of asymptotic alert: can multiply and divide asymptotics
- 0230(23): an example limit done by asymptotics
- 0230(23): completion of asymptotic alert: cannot add or subtract asymptotics
- 0230(24): an example problem where we would like to subtract asymptotics
- 0230(25): can add asymptotics when there are square roots, but cannot subtract
- 0230(26): transition slide
- 0230(27): in that example problem
- rationalize the numerator so that the difference of square roots becomes a sum
- use the addition of asymptotics when there are square roots
- 0230(28): completion of that example problem via asymptotics
Topic 0240
(Bounded functions and horizontal asymptotes)
Link to Topics website
- 0240(1): title slide
- 0240(2): recall the definition of image
- 0240(2): def'n of bounded
- 0240(2): example of a bounded function: \cos
- 0240(3): non-example: \tan is not bounded
- 0240(4-6): recall the definition and graph of arctan
- 0240(7): vertical asymptotes (limits that are equal to \infty or -\infty)
- 0240(8): inverse interchanges vertical and horizontal asymptotes
- 0240(9-10): horizontal asymptotes (limits at \infty or -\infty)
- 0240(11): def'n: horizontal asymptote
- 0240(12-14): the Joy of Asymptotics
- 0240(12): asymptotics are preserved under taking powers
- 0240(12): using the asymptotics alert (from earlier topic) to solve a limit problem
- 0240(13): addition causes problems (and so does subtraction)
- 0240(14): addition of square roots does not (but subtraction does)
- 0240(14): can sometimes resolve subtraction of square root by rationalizing the numerator
- 0240(15): for limits of rational functions, we seek asymptotics of polynomials
- 0240(15): first asymtotics at 0, then at \infty and -\infty
- 0240(16): a polynomial is asymptotic at 0 to its lowest order term
- 0240(17): a polynomial is asymptotic at \infty to its leading term
- 0240(18): a polynomial is asymptotic at -\infty to its leading term
- 0240(19): same degree in numerator and denominator implies: limit (at \infty or -\infty) is
the quotient of the leading coefficients
- 0240(20-21): larger degree in denominator implies: limit (at \infty or -\infty) is 0
- 0240(22-25): larger degree in numerator implies: limit at -\infty is either \infty or -\infty; how to figure out which
- 0240(26): the limit at \infty of a particular rational function
- the computation
- the tabulation of values
- the graph
- 0240(27-28): the limit at -\infty of a particular rational function
- the computation
- the tabulation of values
- the graph
- 0240(29-33): e.g.: all asymptotes (vertical and horizontal) for a function f that is nearly rational
- 0240(34): an example of an \infty - \infty indeterminate form which is nearly rational
- rationalize the numerator to change the difference to a sum
Topic 0250
(Problems involving horizontal asymptotes)
Link to Topics website
- 0250(1): title slide
- 0250(2-4): SKILL: limits from the graph
- 0250(2-4): e.g.: finding limits from a graph
- 0250(5): SKILL: limits of rational functions
- 0250(5): e.g.: limits of rational functions
- 0250(6-26): SKILL limits of nearly rational functions (rational, but for a square root)
- 0250(6): e.g.s: limit of a nearly rational function (rational, but for a square root)
- 0250(7): e.g.: a limit that is a determinate form; no tricks needed
- 0250(8-26): e.g.: a limit that is a -\infty + \infty indeterminate form
- 0250(8): the example problem
- 0250(9): the difficulty with subtracting asymptotics
- 0250(10): situations where you can add asymptotics
- 0250(11): an example of adding asymptotics
- 0250(12): transition to a different addition of asymptotics
- 0250(13): another situation where you can add asymptotics
- 0250(14): another example of adding asymptotics
- 0250(15): rationalize the numerator in the example problem to change subtraction to addition
- 0250(16): transition slide
- 0250(17): completion of the example problem via asymptotics
- 0250(18): not tradition in freshman calculus to teach asymptotic methods; instead ...
- 0250(19): in the example problem, go back to the point just before the use of asymptotics
- 0250(20-24): completion of the example problem by traditional freshman calculus methods; this is harder
- 0250(25): SKILL: limits of rational functions
- 0250(25): e.g.: limits of rational functions
- 0250(26): SKILL: limit at infinity
- 0250(26): e.g.: limit at infinity
- 0250(27): e.g.: limit at infinity
- 0250(27): care is required with asymptotics, and sometimes easier NOT to use them
- 0250(28): SKILL: oscillatory limit
- 0250(28): e.g.: oscillatory limit; use of the Squeeze Theorem at \infty
- 0250(29): an oscillatory function with no limit; Squeeze Theorem gives no information
- 0250(30): e.g.: oscillatory limit; use of the Squeeze Theorem at -\infty
- 0250(31): SKILL: finding the input specification, given the output specification, for a limit problem
- 0250(31): e.g.: finding the input specification, given the output specification, for a limit problem
Topic 0260
(Definition of logarithm)
Link to Topics website
- 0260(1): title slide
- 0260(2): description of various exponential functions
- 0260(2): contrasting x^2 and 2^x
- 0260(2): description of 1^x and 0^x
- 0260(3): graph of y = 10^x; 1-1; increasing
- 0260(4): graph of y = 100^x; 1-1; increasing
- 0260(5): graph of y = 2^x; 1-1; increasing
- 0260(6): graph of y = e^x; 1-1; increasing
- 0260(7): backward graph of y = (1/10)^x
- 0260(8): graph of y = (1/10)^x; 1-1; decreasing
- 0260(9): def'n: exponential function; exponential increase and decay
- 0260(9): def'n: \log_b : (0,\infty) ---> \R, for b \in (0,\infty)\backslash\{1\}
- 0260(9): def'n: \ln : (0,\infty) ---> \R
- 0260(9): def'n: logarithmic function; logarithmic increase and decay
- 0260(10): transition slide
- 0260(11): \log_b(b^x)=x and b^{\log_b(x)}=x
- 0260(11): \ln(e^x)=x and e^{\ln(x)}=x
- 0260(11): SKILL: domain and image of exponential and logarithmic functions
- 0260(12): recall graph of y = (1/10)^x
- 0260(13): adjust units
- 0260(14): graph of y = \log_{1/10}(x)
- 0260(15): recall graph of y = e^x
- 0260(16): adjust units
- 0260(17): transition slide
- 0260(18): graph of y = \ln x
- 0260(18): SKILL: graphs of exponential and logarithmic functions
- 0260(19): problem: use Intermediate Value Theorem to prove existence of solution of a transcendental equation (involving exp and log)
- 0260(19): SKILL: root of an equation via the IVT
Topic 0270
(Derivatives and rates of change)
Link to Topics website
- 0270(1): title slide
- 0270(2): average rate of change in output per unit rate of change in input (between u and a) = ... = slope of secant line from (u,f(u)) to (a,f(a))
- 0270(3-5): take limit as u ---> a
- 0270(5): limit gives slope of tangent line
- 0270(6-8): changing u to a+h and letting h ---> 0
- 0270(8): def'n: derivative of f at a; notation: f'(a)
- 0270(9-11): e.g.: computation of a derivative of a polynomial at a number
- 0270(12-13): e.g.: computation of a derivative of a polynomial at a variable
- 0270(14): e.g.: finding an equation of a tangent line
- 0270(15): common error: using the derivative at a variable, rather than at a number
- 0270(15): SKILL: point and slope to equation of line
- 0270(15): note that the function whose graph is the line is on the RHS of y = ... and NOT on the RHS of y - y_0 = ...
- 0270(16-21): e.g.: computation of a derivative of a rational function at a variable
- 0270(21): SKILL: differentiate rational function
- 0270(22): e.g.: going from a limit of a difference quotient to the underlying function and number
- 0270(22): SKILL: recognize derivative
Topic 0280
(The derivative of a function is a function)
Link to Topics website
- 0280(1): title slide
- 0280(2): def'n: derivative of a function
- 0280(2): derivative has all slopes of all tangent lines; to get one slope, plug in a number
- 0280(3): SKILL: graph f' from the graph of f
- 0280(3): the domain of f' is contained in the domain of f
- 0280(4): def'n: differentiable at a; def'n: differentiable
- 0280(5): e.g.: the derivative of x^2 w.r.t. x
- 0280(5): need to deal with expressions of x and h, but final answer is an expression of x alone
- 0280(5-6): replacing h by \triangle x
- 0280(7): notation
- for derivative of a function (prime)
- for derivative of an expression of x w.r.t. x (d/dx)
- 0280(8-10): differentiation of a dependent variable w.r.t. a dependent variable
- 0280(8): def'n of \triangle y, where y depends on x
- 0280(9): \triangle y does NOT mean \triangle times y
- 0280(9): calculation of (\triangle y)/(\triangle x)
- 0280(9): \triangle x does NOT mean \triangle times x
- 0280(9): no cancellation of (\triangle)s
- 0280(10): conclusion of the differentiation
- 0280(10): (\triangle y)/(\triangle x) depends on x and \triangle x
- 0280(10): dy/dx depends only on x
- 0280(11-14): the general setup
- 0280(11): y depends on x
- 0280(11): \triangle y depends on x and \triangle x, through a formula
- 0280(11): \triangle x is a single variable, represented by two symbols
- 0280(12): the derivative is the limit of (\triangle y)/(\triangle x) as \triangle x ---> 0
- 0280(13): same idea, but for expressions y of t
- 0280(14): same idea, but for expressions r of t
- 0280(15): e.g.: the derivative of r = \sqrt{t} w.r.t. t
- 0280(15): SKILL: Find the derivative from the definition
- 0280(16): e.g.: the derivative of y = \sqrt{x} w.r.t. x
- 0280(16): SKILL: Find the derivative from the definition
- 0280(17): e.g.: the derivative of y = 1/x w.r.t. x
- 0280(17): SKILL: Find the derivative from the definition
- 0280(17-18): e.g.: the equation of the tangent line to y = 1/x at (2,1/2)
- 0280(17): be careful to find ONE slope of ONE tangent line; don't use ALL slopes of ALL tangent lines simultaneously
- 0280(17): SKILL: find slope of tangent line (by evaluating the derivative at the x-coordinate of the point of tangency)
- 0280(18): conclusion of the problem
- 0280(18): SKILL: point/slope to equation
- 0280(18): SKILL: find equation of a tangent line
- 0280(19): various notations for the derivative, recall two definitions of the derivative
- 0280(20): recall the definition of differentiable at a
- 0280(20): def'n: differentiable on an open interval
- 0280(20-28): e.g.: where is f(x) = |x| differentiable
- 0280(20-21): the graph of y = |x|
- 0280(21-22): computation of f'(3)
- 0280(23): f'(x) for any x>0; f is differentiable on (0,\infty)
- 0280(24): f'(x) for any x<0; f is differentiable on (-\infty,0)
- 0280(25-28): f'(0) does not exist; f is not differentiable at 0
- 0280(28): (continuous at a) does not imply (differentiable at a)
- 0280(29-33): (differentiable at a) implies (continuous at a)
- 0280(34-35): recognizing, from the graph, where a function is not differentiable
- 0280(34): (not defined at a) implies (not continuous at a)
- 0280(34): (not continuous at a) implies (not differentiable at a)
- 0280(35): picture intuition behind defined, continuous and differentiable
- defined: there's a dot there
- continuous: there's no break there
- differentiable: there's neither a break nor a sudden change of direction there
Topic 0290
(Intervals of increase/decrease and intervals of continuity)
Link to Topics website
- 0290(1): title slide
- 0290(2): def'n and visualization: increasing
- 0290(2): non-example (not increasing)
- 0290(3-4): maximal interval of increase
- 0290(5): transition slide
- 0290(6): def'n: decreasing
- 0290(6): def'n: nondecreasing (or semi-increasing)
- 0290(6): def'n: nonincreasing (or semi-decreasing)
- 0290(7): def'n and visualization: concave up on I
- the secant line segment ... lies above the graph
- focus on the word "segment"
- 0290(8): sometimes called "convex"
- 0290(8): should be "strictly concave up"
- 0290(8): algebraic form of the definition
- 0290(9): back to the geometric form of the definition
- 0290(10): should be "lies strictly above"
- 0290(11): should be "open secant line segment"
- 0290(12): making the interval as large as possible
- 0290(13): for differentiable functions: concave up on I iff graph lies above its tangents on I
- 0290(14): def'n: concave down
- 0290(14): similar remarks as were made for "concave up"
- 0290(15): (increasing/decreasing for f) corresponds to (positive/negative for f')
- 0290(16): (concave up/concave down for f) corresponds to (increasing/decreasing for f')
- 0290(17): points of inflection
Topic 0300
(Differentiation problems without techniques of differentiation)
Link to Topics website
- 0300(1): title slide
- 0300(2-3): derivative of a polynomial; domain of function and derivative
- 0300(2): SKILL: find derivative from the definition
- 0300(4-6): derivative of a polynomial and equation of a tangent line
- 0300(6): SKILL: find slope of a tangent line
- 0300(6): SKILL: find equation of a tangent line
- 0300(7): derivative of a rational function
- first: at a specific number
- then: in general
- 0300(8-13): derivative of a rational function
- 0300(14): derivative of a nearly rational function (except for a square root);
domain of function and derivative
- 0300(14): SKILL: find derivative from the definition
- 0300(15): SKILL: sketch graph of f', given the graph of f
- 0300(16): draw conclusions from a graph of temperature (vs. time) and its derivative
- 0300(16): SKILL: sketch graph of f', given the graph of f
- 0300(17): SKILL: sorting out f, f' and f'', from the graphs
- 0300(18): SKILLS and suggested problems:
- find derivative from definition
- graph f' from graph of f
- sketch graph with specifications
- recognize and bound a bounded function
- check continuity
- find a root
Topic 0310
(The power rule)
Link to Topics website
- 0310(1): title slide
- 0310(2): derivative of 1
- 0310(3): derivative of a constant
- 0310(4): derivative of x (w.r.t. x)
- 0310(5-6): derivative of x^4 (w.r.t. x)
- 0310(7-8): derivative of x^5 (w.r.t. x)
- 0310(9): transition slide
- 0310(10): the power rule (for positive integer powers)
- 0310(10): power rule is slightly incorrect for 1 power
- 0310(10): power rule works even for 1/2 and -1 powers
- 0310(11): transition slide
- 0310(12): power rule works even for -1/2
- 0310(13): transition slide
- 0310(14): the general version of the power rule
- 0310(15): derivative of \sqrt{13}
- 0310(15): SKILL: derivative of a constant
- 0310(15): differentiation and evaluation don't commute
- 0310(16): SKILL: derivative of a power function
- 0310(16): SKILL: equation of a tangent line
- 0310(16-17): equation of a tangent line to y = x^{1/5}
- 0310(18): equation of a tangent line to y = x^{4/3}
- 0310(19): SKILLs and Whitman problems:
- derivative of powers
- word problems that lead to derivatives
Topic 0320
(Linearity of the derivative and derivatives of polynomials)
Link to Topics website
- 0320(1): title slide
- 0320(2): derivative commutes with scalar multiplication
- 0320(3): derivative is additive
- 0320(4): transition slide
- 0320(5): linearity of differentiation; differentiation respects linear combinations
- 0320(6): SKILL: differentiation of a linear combination of power functions
- 0320(6): e.g.: differentiation of a linear combination of power functions
- 0320(6-7): SKILL: differentiation of polynomials
- 0320(6-7): e.g.: differentiation of polynomials
- 0320(8-10): e.g.: differentiation of a scalar multiple of a power function
- 0320(8-10): SKILL: differentiation of a linear combination of power functions
- 0320(10): in context of one of the problems, "k" is a constant, not a dependent variable
- 0320(11-13): derivative of a cube of a linear combination of power functions
- 0320(11): expand the cube to get a linear combination of power functions
- 0320(12): SKILL: expand to a linear combination of power functions
- 0320(12): transition slide
- 0320(13): differentiate the linear combination of power functions
- 0320(14): SKILL: find horizontal tangent lines
- 0320(14): e.g.: find horizontal tangent lines
- 0320(15): motion along a line
- 0320(15): derivatives and higher derivatives of position (w.r.t. time)
- velocity
- acceleration
- jerk
- snap
- crackle
- pop
- 0320(15): example: falling body
- position = t^2 rods
- velocity = 2t rods/sec
- acceleration = 2 rods/sec/sec
- jerk, snap, crackle, pop are all 0
- 0320(16): example (computing the velocity, acceleration, jerk, snap crackle and pop of a train, given its position)
- 0320(17): SKILL: find polynomial with a given jet
- 0320(17): e.g.: find polynomial with a given jet
- 0320(18): the maximal distance apart occurs when the two cars have the same velocity
- 0320(19): SKILLs and Whitman problems:
- use linearity to find derivatives
- equation of a tangent line
- word problems about going from position to acceleration
- graph of scalar multiple and derivative of scalar multiple
- difference is differentiable; derivative of difference
- derivative of a general polynomial
- find a polynomial with given properties
Topic 0330
(Derivatives of exponential functions)
Link to Topics website
- 0330(1): title slide
- 0330(2): try to find derivative of g(x)=10^x, at x=7, via the definition of derivative
- 0330(3-5): the limit of (e^h-1)/h, as h ---> 0
- 0330(5): better to change to from base 10 to base e
- 0330(6): the derivative of f(x)=e^x, at x=7, via the definition of derivative
- 0330(7): answer is e^7; change 7 to x
- 0330(8): (d/dx)e^x=e^x
- 0330(9-10): example problem: a linear combination of exponentials (with base e) and power functions
- 0330(10): SKILL: derivative of an expression involving an exponential function
Topic 0340
(The product rule)
Link to Topics website
- 0340(1): title slide
- 0340(2): goal: (d/dt)((t^5)(e^t))
- 0340(2): setting up notation and rectangle (base u=t^5, height v=e^t) at time 1
- 0340(3): rectangle at time 1.03
- 0340(4): change to base
- 0340(5): change to height
- 0340(6): change to area, in terms of change to base and height
- 0340(6): a formula for \triangle(uv)
- 0340(7): another proof of that formula for \triangle(uv)
- 0340(8): formula for (d/dt)(uv)
- 0340(9-10): working out the answer in the specific case of u=t^5 and v=e^t
- 0340(11): the formula for (d/dt)(uv) is called the product rule
- 0340(12): transition slide
- 0340(13): other ways of writing the product rule
- 0340(14): transition slide
- 0340(15): the product rule is called differentiation by parts
- 0340(15): a phrase to summarize the product rule
- 0340(16): the product rule for functions (as opposed to expressions)
- 0340(17): an example: (d/du)((u^2)(e^u))
- 0340(20): SKILL: product rule
- 0340(18): another way of organizing the answer to (d/du)((u^2)(e^u))
- 0340(18): because multiplcation and addition are commutative, the product rule takes several forms
- 0340(19): an example
- 0340(20): an example with three factors
- 0340(20): SKILL: many factor product rule
- 0340(21): SKILLs and Whitman problems:
- product rule
- product rule, sketch, equation of tangent line
- many factor product rule
Topic 0350
(The quotient rule)
Link to Topics website
- 0350(1): title slide
- 0350(2): a formula for \triangle(u/v)
- 0350(3): transition slide
- 0350(4): a formula for (d/dt)(u/v)
- 0350(5): transition slide
- 0350(6): the formula for (d/dt)(u/v) is called the quotient rule
- 0350(6): the case where u=t^5 and v=e^t
- 0350(7): various forms of the quotient rule
- 0350(8): transition slide
- 0350(9): a phrase to summarize the quotient rule
- 0350(10): an example, and an exercise
- 0350(10): SKILL: the quotient rule
- 0350(11-13): an example
- 0350(11): a. worked by the quotient rule
- 0350(11): b. worked by simplification, the power law and linearity of the derivative
- 0350(12): transition slide
- 0350(13): c. show the two solutions are the same
- 0350(13): d. which solution is better?
- 0350(14-17): more examples
- 0350(18): SKILLs and Whitman problems:
- quotient rule
- equation of a tangent line
- derivative of a rational function
- from (the 1-jets of the dividend and divisor) to (the 1-jet of the quotient)
Topic 0360
(Derivatives of trigonometric functions)
Link to Topics website
- 0360(1): title slide
- 0360(2): recall two basic trig limits
- 0360(3): change \theta to h and mutiply one of the two limits by -1
- 0360(4): transition slide
- 0360(5-9): the derivative of sine
- 0360(10-16): the derivative of cosine
- 0360(17-24): remembering the derivatives of sine and cosine by the geometry of the standard orbiter
- 0360(24-29): the derivative of tangent
- 0360(30): recall the derivatives of sine, cosine and tangent
- 0360(31): the derivative of secant
- 0360(32): transition slide
- 0360(33): the derivatives of sine, cosine, tangent, cotangent, secant and cosecant
- 0360(34): transition slide
- 0360(35): the derivatives of trig functions occur in complementary pairs
- 0360(36): example problem (derivative of product of algebraic function and trig function)
- 0360(37): example problem (derivative of function involving exponential and trig functions)
- 0360(38): example problem (a quotient rule problem involving trig functions)
- 0360(39): example problem (a triple product rule problem involving trig functions)
- 0360(40): example problem (equation of a tangent line to a curve whose definition involves a trig function)
- 0360(41-43): example problem (evaluating a second derivative of a function involving trig functions)
- 0360(44): example problem (high derivatives of \sin and \cos)
- 0360(45): example problem (finding all horizontal tangent lines for a rational function of trig functions)
- 0360(46): SKILLs and Whitman problems:
- trig limits
- squeeze theorem problems (involving trig functions)
- trig derviatives
- horizontal tangent line (involving trig functions)
- equation of a tangent line (involving trig functions)
- geometric limit via trigonometry
Topic 0370
(The chain rule)
Link to Topics website
- 0370(1): title slide
- 0370(2-12): (d/dx)[sin(x^3)]
- 0370(2): NOT the product rule
- 0370(2): setting up notation: y=f(x)=x^3, g=\sin, z=g(y)
- 0370(2): (\triangle z)/(\triangle x) broken down
- 0370(2): f is 1-1, so \triangle x nonzero implies \triangle y nonzero
- 0370(3): letting \triangle x ---> 0
- 0370(4): get dz/dx and dy/dx, but
limit of (\triangle z)/(\triangle y) is unclear
- 0370(4): z is not an expression of y
- 0370(5): z is an expression of x
- 0370(6-8): formula for \triangle z in terms of y and \triangle y
- 0370(9): formula for (\triangle z)/(\triangle y) in terms of y and \triangle y
- 0370(10-11): the limit is g'(y)
- 0370(12): completion of the problem
- 0370(13-17): (d/dx)[sin(x^4)]
- 0370(13): setting up notation: y=f(x)=x^4, g=\sin, z=g(y)
- 0370(13): problems because f is NOT 1-1
- 0370(14): def'n of (\box z)/(\box y)
- 0370(14): replacing (\triangle z)/(\triangle y) with (\box z)/(\box y)
- 0370(14): problem becomes the limit of (\box z)/(\box y)
- 0370(15-16): the limit is g'(y)
- 0370(16): get dz/dx and dy/dx, as before
- 0370(17): completion of the problem
- 0370(18): transition slide
- 0370(19): a general formula for dz/dx when z=g(y) and y=f(x)
- 0370(19): dz/dy, even though y is a dependent variable (sloppy)
- 0370(19): one version of the chain rule: dz/dx = (dz/dy) (dy/dx)
- 0370(20): another version: (d/dx)(g(f(x)) =
[g'(f(x))] [(d/dx)(f(x))]
- 0370(21): buzz phrase:
- given an expression
- plugged into a function,
- to differentiate the result
- take the derivative of the function
- plug in the expression
- and then multiply by
- the derivative of the expression
- 0370(21): example problem: (d/dx)(\sin(\cot x))
- 0370(22): example problem: (d/dx)(e^{\tan x})
- 0370(23): example problem: (d/dx)(cos^3(x))
- 0370(24): the chain rule for expressions of t
- 0370(24): the chain rule for expressions of s
- 0370(24): the chain rule for functions: (f \circ g)' =
[g' \circ f] \cdot f'
- 0370(25): triple chain rule example problem: (d/dx)(\sin(\tan(x^7)))
- 0370(26): complicated problem involving:
- quotient rule
- product rule
- triple chain rule
- 0370(27): complementing (d/dx)(\sin x) = \cos x;
minus sign from the Chain Rule
- 0370(28): complementing (d/dx)(\tan x) = \sec^2 x;
minus sign from the Chain Rule
- 0370(29): complementing (d/dx)(\sec x) = (\sec x)(\tan x);
minus sign from the Chain Rule
Topic 0380
(Chain rule problems)
Link to Topics website
- 0380(1): title slide
- 0380(2): chain rule problem (difference between \cos^3(x) and \cos(x^3))
- 0380(3-4): chain rule problem (expand is inefficient, better to use chain rule)
- 0380(5-6): chain rule problem via: dy/dx = (dy/du) (du/dx)
- 0380(7): chain rule problem (square root is 1/2 power)
- 0380(8): chain rule problem (seventh root in numerator is 1/7 power)
- 0380(9): chain rule problem (cube root in denominator is -1/3 power)
- 0380(10): chain rule problem (square root of a quotient)
- 0380(11-12): chain rule problem (triple composite)
- 0380(13): equation of a tangent line problem (remember to evaluate)
- 0380(14): search for horizontal tangent lines problem
- 0380(15-20): given G(x) = f( x f( x f(x) ) ), and given 1-jet on f at 2 and 12, find G'(2)
- 0380(21): SKILLs and Whitman problems:
- misc differentiation
- equation of a tangent line
Topic 0390
(Derivatives of logarithmic functions)
Link to Topics website
- 0390(1): title slide
- 0390(2): computation of (d/dx)(\ln x), for x>0
- 0390(3): transition slide
- 0390(4): the real answer is 1/x,x>0, i.e., ln' = (1/\bullet) | (0,\infty)
- 0390(4): graph of 1/\bullet, i.e, of y = 1/x
- 0390(5): graph of (1/\bullet) | (0,\infty), i.e., of y = 1/x, x>0
- 0390(6): (d/dx)(\ln(-x)) = 1/x,x<0
- 0390(6): (d/dx)(\ln|x|) = 1/x
- 0390(6): def'n of \la x
- 0390(7-10): recall the graph of \ln
- 0390(11): transition slide
- 0390(12): graphs of several equations
- y = \ln x, i.e., of y = \ln |x|,x>0
- y = \ln(-x), i.e., of y = \ln |x|,x<0
- y = 1/x, x>0
- y = 1/x, x<0
- 0390(13): graphs of y = \ln|x| = \la x and of y = 1/x
- 0390(14): an example problem: [d/dx][cube root of (\ln x)]
- 0390(15): an example problem: [d/dx][\ln(a polynomial)]
- 0390(15): [d/dx][\ln(f(x))] and [d/dx][\la(f(x))] in general
- 0390(16): an example problem: [d/dx][\ln(a polynomial)]
- 0390(17): an example problem: [d/dx][\ln(a polynomial)] and [d/dx][\la(a polynomial)]
- 0390(18-19): an example problem: [d/dx][\ln(an expression of x)]
- 0390(20): an example problem: first and second derivative of an expression involving \ln x
Topic 0400
(Logarithmic differentiation)
Link to Topics website
- 0400(1): title slide
- 0400(2): recall \la and \la'
- 0400(2-3): def'n of logarithmic derivative
- 0400(2-3): the principle of logarithmic differentiation
- to compute the derivative of an expression, multiply the expression by its logarithmic derivative
- 0400(4-5): example: (d/dx)(x^x)
- 0400(6): properties of \la
- 0400(7): the power law: (d/dx)(x^n) = n x^{n-1}, for all n
- 0400(7-8): occasional exceptions when x=0
- 0400(9): example: (d/dx)(10^x)
- 0400(9): general: (d/dx)(b^x), when b>0
- 0400(10): example: (d/dx)(\log_2(x))
- 0400(10): general: (d/dx)(\log_b(x)), when b>0 and b \ne 1
- 0400(11): writing \log_b(x) in terms of \ln, and then differentiating
- 0400(12): examples of logarithmic derivatives
- 0400(13): properties of logarithmic derivative
- 0400(13): using the properties of logarithmic derivative
- 0400(14): transition slide
- 0400(15): an example of logarithmic differentiation (computing a derivative via a logarithmic derivative)
- 0400(15): exceptions
- 0400(16): a logarithmic derivative followed by a derivative; think about possible exceptions
- 0400(17): an example of a derivative involving \ln; what happens with \la
- 0400(18): an example of logarithmic differentiation; think about possible exceptions
- 0400(19-20): an example of logarithmic differentiation; think about possible exceptions
- 0400(21): my way or the slow way of logarithmic differentiation
- 0400(22): an example of logarithmic differentiation (an x-in-the-exponent problem); think about exceptions
- 0400(23): an example of logarithmic differentiation (an x-in-the-exponent problem); think about exceptions
- 0400(24-25): an example of logarithmic differentiation (an x-in-the-exponent problem); think about exceptions
- 0400(26): SKILLs and Whitman problems:
- log properties
- graphs of logarithmic expressions
- derivative of logarithmic functions
- solve logarithmic expressions
- logarithmic differentiation
- derivatives of expressions with exponential functions
- derivatives of expressions with logarithmic functions
- tangent line problems
Topic 0410
(l'Hôpital's rule)
Link to Topics website
- 0410(1): title slide
- 0410(2): some determinate forms: 8/2 and 56/7
- 0410(2): recall the definition of determinate form
- 0410(3): sometimes the form can be both 1/0 and 1/(0^+)
- 0410(4): another determinate form: 1/(0^+) = \infty
- 0410(5): another determinate form: 1/(0^-) = -\infty
- 0410(5): the slightly indeterminate form 1/0; answer could be
- \infty, or -\infty, or "does not exist"
- 0410(6): transition slide
- 0410(7): recall the definition of indeterminate form
- 0410(7): another determinate form: (0^+)^\infty = 0
- 0410(7): SKILL: (0^+)^\infty
- 0410(8): 0/0 is VERY indeterminate
- 0410(8-15): a 0/0 indeterminate form problem via difference quotients and differentiation
- 0410(16): summary of that last problem: the limit of the quotient is the limit of the quotient of the derivatives
- 0410(17): doesn't work except for 0/0, \infty/\infty, (-\infty)/\infty, \infty/(-\infty), (-\infty)/(-\infty)
- 0410(18): the limit of the quotient is the limit of the quotient of the derivatives
- 0410(18): if the limit of the quotient of the derivatives doesn't exist, the original limit might nevertheless exist
- 0410(19): two-sided l'Hôpital's rule
- 0410(20): example where the limit of the quotient of the derivatives doesn't exist, but the original limit does
- 0410(21): l'Hôpital's rule from the right
- 0410(22): l'Hôpital's rule from the left
- 0410(23): l'Hôpital's rule at infinity
- 0410(24): example where the limit of the quotient of the derivatives doesn't exist, but the original limit does
- 0410(25): l'Hôpital's rule at minus infinity
- 0410(26): an example problem with \infty/\infty indeterminate form
- 0410(26): SKILL: l'Hôpital's rule
- 0410(27): fact: bdd / \infty = 0
- 0410(28-32): example where the limit of the quotient of the derivatives doesn't exist, but ...
- 0410(33): ... the original limit is equal to 0
- 0410(33): another example where the limit of the quotient of the derivatives doesn't exist, but ...
- 0410(33): ... the original limit is equal to \sqrt{3}
- 0410(33): SKILL: 0/0 indeterminate forms
- 0410(34): a determinate form example
- 0410(34): an example where l'Hôpital's rule must be used repeatedly to get to the answer
- 0410(34): if a factor in the numerator or denominator tends to 1, then it can be eliminated without changing the problem
Topic 0420
(Indeterminate forms)
Link to Topics website
- 0420(1): title slide
- 0420(2): table of determinate and indeterminate forms
- 0420(3): transition slide
- 0420(4): four types of indeterminate forms
- l'Hôpital indeterminate forms: 0/0 , (\pm\infty) / (\pm\infty)
- indeterminate producs: 0 \cdot \infty and 0 \cdot (-\infty)
- indeterminate differences: \infty - \infty, and similar forms
- indeterminate powers: 1^(\pm\infty), (0^+)^0, \infty^0
- 0420(5-7): indeterminate products
- 0420(5-6): the general method
- 0420(7): example problem
- 0420(7): SKILL: (0)(plus or minus \infty)
- 0420(8-9): indeterminate differences
- 0420(8): the general method
- 0420(9): example problem
- 0420(9): SKILL: \infty - \infty
- 0420(10-14): indeterminate powers
- 0420(10): types of indeterminate powers
- 0420(11): transition slide
- 0420(12): the general method
- 0420(13): example problem by l'Hopital
- 0420(13): SKILL: indeterminate power
- 0420(14-16): redo the problem by asymptotics, esp. asymptotics of ln [ 1 + (f(x)) ]
- 0420(16): SKILL: indeterminate power
- 0420(17): example problem
- 0420(17): SKILL: indeterminate power
- 0420(18-29): SKILL: general limits
- 0420(18-29): miscellaneous determinate and indeterminate forms problems
- 0420(18-19): 0/0, with trig functions
- 0420(18): by l'Hopital
- 0420(19): redo the problem by asymptotics
- 0420(20): determinate form, with trig functions
- 0420(21-23): 0/0, with log and trig functions
- 0420(21): by l'Hopital
- 0420(22-23): redo the problem by asymptotics
- 0420(24): (-\infty)(0), with polynomials and exponentials
- 0420(25): determinate form, with trig functions
- 0420(26): (1/0)-(1/0), with trig functions
- 0420(27): \infty^0, with log functions
- 0420(28): \infty^0, with exponential functions and rational functions
- 0420(29): 1^\infty, with trig and algebraic functions
- 0420(30): SKILLs and Whitman problems:
- general limits
- horizontal asymptotes
Topic 0430
(Implicit differentiation)
Link to Topics website
- 0430(1): title slide
- 0430(2): first goal is to find dy/dx, where x^2 + y^2 = 25
- 0430(2): x^2 + y^2 = 25 is not the graph of a function, but breaks up into two semicircles,
each of which is the graph of a function
- 0430(3): peeking ahead, we see that we want the upper semicircle, which we call f(x)
- 0430(4): even without peeking ahead, we can still get that the derivative is f'(x) = -x/y
- 0430(4): -x/y works for either semicircle
- 0430(5): finding the equation of the tangent line to the circle x^2 + y^2 = 25 at (3,4)
- 0430(6): reviewing the calculation of the derivative -x/y
- 0430(6): abbreviations: y for f(x), and then y' for f'(x)
- 0430(7): the same calculation using abbreviations
- 0430(7): comment on the chain rule (d/dy)(y^2)=2y, but ...
- 0430(8): (d/dx)(y^2)=2yy'
- 0430(9): the equation 2x + 2yy' = 0 is linear in y'
- 0430(10): solving a linear equation in one unknown is not hard
- 0430(11): how to solve a linear equation in one unknown
- 0430(12-16): an implicit differentiation problem (find y', find a tangent line, find points with horizontal tangent)
- 0430(17): an implicit differentiation problem (find y')
- 0430(18): an implicit differentiation problem (find y', solve for y and find y', check that the two answers are the same)
- 0430(19): an implicit differentiation problem (find y', solve for y and find y', check that the two answers are the same)
- 0430(20): an implicit differentiation prolbem (find y')
- 0430(21): recall how to solve one equation in one unknown
- 0430(22): an alternate bookkeeping system for solving a linear equation in one unknown
- 0430(23-24): applying that alternate bookkeeping system to implicit differentiation
- 0430(25): an implicit differenitation problem (find g'(0) given an implicit formula for g(x))
- 0430(26): an implicit differentiation problem (equation of a tangent line)
- 0430(27): an alternate system for finding equations of tangent lines, when y is given implicitly
- 0430(28-29): an implicit differentiation problem (equation of a tangent line, using the alternate system)
- 0430(30): an implicit differentiation problem (equation of a tangent line, using the alternate system, where there are
parameters in the implicit formula for y and where the point of tangency is variable)
- 0430(31-44): find the length of the shadow of a specified figure (an ellipse) with a specific light source
- 0430(31-40): finding the coordinates of the two points of tangency
- 0430(31): for one of the points: setting up two equations in the two unknown coordinates
- 0430(32): for the other point: setting up two equations in the two unknown coordinates
- 0430(32): focus on the first point
- 0430(33): seeing that we have two quadratics in two unknowns
- 0430(34): seeing that the quadratic parts cancel, giving a linear and a quadratic
- 0430(34-35): the plan: solve the linear for one of the variables, then plug into the quadratic,
obtaining one quadratic in one variable
- 0430(35): solving for the linear and plugging in
- 0430(36-38): expanding and simplifying
- 0430(39): transition slide
- 0430(40): setting up the same computations for the other point, and solving for both coordinates of both points
- 0430(41): finding the slopes of the two blue tangent lines
- 0430(42-43): finding the endpoints of the shadow and the length of the shadow
- 0430(44): inverse problem: finding the position of the light source, given the length of the shadow
- 0430(45): SKILLs and Whitman problems:
Topic 0440
(Derivatives of inverse functions (The Inverse Function Theorem))
Link to Topics website
- 0440(1): title slide
- 0440(2-5): geometric: derivative of the inverse squaring function, evaluated at 4/25
- 0440(6-8): symbolic: derivative of the inverse squaring function, evaluated at 4/25
- 0440(9): statement of the Inverse Function Theorem (IFT)
- 0440(10): quick derivation of the IFT formula
- 0440(11): transition slide
- 0440(12): derivative of \arcsin, first formula
- 0440(13): transition slide
- 0440(14): derivative of \arcsin, second formula
- 0440(14): second formula is preferred because it's algebraic
- 0440(15-18): derivative of \arccos, preferred formula (algebraic)
- 0440(19): comparison of derivatives of \arcsin and \arccos
explanation of why one is the negative of the other
- 0440(20-22): derivative of \arctan, preferred formula (rational)
- 0440(22): exercise to compute derivative of \arccot
- 0440(23): comparison of derivatives of \arctan and \arccot
explanation of why one is the negative of the other
- 0440(24): transition slide
- 0440(25): all four inverse trigonometric derivatives; only two to memorize
- 0440(25): we don't define \arcsec and \arccsc in this course
- 0440(26): example problem: chain rule and inverse trig function
- 0440(27): example problem: chain rule and inverse trig function and trig function
- 0440(28): example problem: chain rule and inverse trig function and rational function
- 0440(29): SKILLs and Whitman problems:
- inverse trigonometric differentiation
Topic 0450
(Maxima and minima)
Link to Topics website
- 0450(1): title slide
- 0450(2): def'ns
- global maximum = absolute maximum
- global minimum = absolute minimum
- extremum
- 0450(2): plurals (minima, maxima, extrema)
- 0450(3): def'n and visualizations: local maximum = relative maximum
- 0450(3): must have a full neighborhood of the point in the domain
- 0450(4): def'n and visualizations: local minimum = relative minimum
- 0450(5): def'n: critical point = critical number
- 0450(5): visualization of maxima, minima and critical numbers
- 0450(6): global maxima need not occur at a unique number, but the max value is unique
- 0450(7): five questions
- are global maxima guaranteed?
- is every endpoint a critical number?
- is every global max a local max?
- is every local extremum a critical number?
- is every global extremum a critical number?
- 0450(7): is every endpoint a critical number? NO
- 0450(8): are global maxima guaranteed? NO
- 0450(9): points of discontinuity (in the domain of the function) are always critical numbers
- 0450(10): points not in the domain of the funcion are never critical numbers
- 0450(11): points of discontinuity (in the domain of the function) are always critical numbers
- 0450(12): is every global max a local max? NO
- 0450(13): Fermat's Theorem for local extrema
- 0450(13): is every local extremum a critical number? YES
- 0450(13): Fermat's Theorem for global extrema
- 0450(13): is every global extremum a critical number? YES
- 0450(14-15): even for bounded functions, global maxima are not guaranteed
- 0450(15): Extreme Value Theorem (EVT)
- 0450(15): visualization of the EVT
- 0450(16): recall the meaning of continuity on a compact interval
- continuity from the right at the left endpoint
- continuity from the left at the right endpoint
- two-sided continuity on the "interior"
- 0450(17-18): continuity is needed in the EVT
- 0450(19): interval must be closed in the EVT
- 0450(20): interval must be bounded in the EVT
- 0450(21): algorithm for finding global max and min for a continuous function on a compact interval
- 0450(21): example finding global max and min for a continuous function on a compact interval
- 0450(21): SKILL: global max-min
- 0450(22): SKILL: global max-min
- 0450(22): example: find global max min of a polynomial on a compact interval
- 0450(23): SKILL: global max-min
- 0450(23): example: find global max min of a rational function on a compact interval on which it's continuous
- 0450(24): SKILL: global max-min
- 0450(24): problem makes no sense unless the interval is contained in the domain
- 0450(24): if you try to extremize on a set that is not an interval, there may be no answer
- 0450(24): SKILL: global max-min
- 0450(25): example: find global max min of a rational function on a compact interval on which it's continuous
- 0450(26-27): SKILL: global max-min;
- 0450(26-27): example in which a critical number occurs because f' does not exist
- 0450(26): pictorial solution
- 0450(26): find the derivative
- 0450(27): symbolic solution
- 0450(28): SKILL: global max-min
- 0450(28): example: find global max min of a transcendental function on a compact interval on which it's continuous
- 0450(29-31): SKILL: applied max-min
- 0450(29-31): example finding global max and min for a continuous function on a compact interval (acceleration of spacecraft)
- 0450(32): SKILL: critical numbers
- 0450(32): example: finding critical numbers of a polynomial
- 0450(32): sometimes there are no numberse points
- 0450(33-34): SKILL: critical numbers
- 0450(33-34): example: finding critical numbers of a transcendental function
- 0450(35): SKILL: max-min from graph; example: max-min from graph
- 0450(36): SKILL: max-min from graph; example: max-min from graph
- 0450(37): SKILLs and Whitman problems:
- local extrema
- miscellaneous local extrema and critical numbers
- local extrema and critical numbers for families of functions
- find global extrema and global extremum values
Topic 0460
(The Mean Value Theorem)
Link to Topics website
- 0460(1): title slide
- 0460(2): preliminaries
- 0460(2): the tangent line to a line is just the line itself
- 0460(2): the slope of the tangent line to a line is just the slope of the line itself
- 0460(3-5): my trip to Chicago, revisited
- 0460(5): instantaneous velocity matches average velocity
- 0460(6): if you end where you start and achieve positive displacement, then you must have stopped
- 0460(7): the tame hypotheses are needed
- 0460(8): if you end where you start and achieve negative displacement, then you must have stopped
- 0460(9): Is hypothesis on displacement needed?
- 0460(10): Start of proof that it's not needed
- 0460(11): End of proof that it's not needed
- 0460(12): tame hypotheses in the Mean Value Theorem (MVT)
- 0460(13): conclusion of the MVT, with visualization
- 0460(14): there may be choices for c
- 0460(15-17): proof of a special case of the MVT, when the function goes through (3,1) and (9,13)
- 0460(18): proof of the MVT in general
- 0460(19-30): consequences to the MVT
- 0460(19): Rolle's Theorem (horizontal secant line implies horizontal tangent line)
- 0460(20): the one-to-one test (no horizontal tangent line implies no horizontal secant line)
- 0460(21): the constant test (all tangent lines horizontal implies all secant lines horizontal)
- 0460(22): if two fuctions have the same derivative, then they differ by a constant
- 0460(23): the increasing test (all tangent lines uphill implies all secant lines uphill)
- 0460(24): the decreasing test (all tangent lines downhill implies all secant lines downhill)
- 0460(25): the nonincreasing test (no tangent lines uphill implies no secant lines uphill)
- 0460(26): the nondecreasing test (no tangent lines downhill implies no secant lines downhill)
- 0460(26): converse for the nondecreasing test (no secant lines downhill implies no tangent lines downhill)
- 0460(27): converse for the nonincreasing test (no secant lines uphill implies no tangent lines uphill)
- 0460(28): recall the increasing test
- 0460(29): there is no perfect converse for the increasing test
- 0460(30): there is no perfect converse for the decreasing test
- 0460(31): example: if you start at -3, and the speed limit is 5, then, two hours later, you can't get beyond 7
- 0460(31): SKILL: maximize value of function from an upper bound on its derivative
- 0460(32): from [ (d/dx)((\arctan x)+(\arccot x)) = 0 ] to [ (\arctan x)+(\arccot x) = \pi/2 ]
- 0460(32): SKILL: use calculus to prove algebra or trig identity
- 0460(33-35): verify hypothesis of Rolle's Theorem and then find solutions
- 0460(33-35): SKILL: Rolle's Theorem
- 0460(36): show, by example, that the tame hypotheses are actually needed in Rolle's Thorem
- 0460(36): SKILL: Rolle's Theorem
- 0460(37): show, by example, that the tame hypotheses are actually needed in the MVT
- 0460(37): SKILL: Mean Value Theorem
- 0460(38): show that a certain family of quartics all have at most one real root in a certain interval
- 0460(38): SKILL: Mean Value Theorem
- 0460(39-45): example: prove a trig idenity via calculus
- 0460(45): SKILL: use calculus to prove algebra or trig identity
- 0460(46): SKILLs and Whitman problems:
Topic 0470
(Derivative tests and graphing)
Link to Topics website
- 0470(1): title slide
- 0470(2): recall definitions of increasing, decreasing, nondecreasing and nonincreasing
- 0470(3): illustration by a graph that intervals of increase/decrease for f correspond to
intervals of positivity/negativity for f'
- 0470(4): increasing/decreasing test
- 0470(5): improved increasing/decreasing test
- 0470(5-8): example of the improved increasing/decreasing test
- 0470(9): another example of the improved increasing/decreasing test, in which
the derivative has a root of multiplicty two
- 0470(10): another example of the improved increasing/decreasing test, in which
the derivative has a root of multiplicty two
- 0470(11): graph of the preceding example
- 0470(12-14): the first derivative test
- 0470(15): example of the first derivative test
- 0470(16): SKILL: local extrema via the first derivative test
- 0470(16): another example of the first derivative test
- 0470(17-19): recall the definitions of concave up and concave down
- 0470(20): illustration by a graph that intervals of concave up/down for f correspond to
intervals of positivity/negativity for f''
- 0470(21): concavity test
- 0470(21): def'n and e.g.s of interiors of intervals
- 0470(21): improved concavity test
- 0470(22): example of the use of the improved concavity test
- 0470(23): example of sketching a graph of a function with various requirements specified
- 0470(24): illustration by a graph of points of inflection, a.k.a. inflection points
- 0470(25): def'n of inflection point (a.k.a. point of inflection, a.k.a. flex point)
- 0470(26): non-example for inflection point because no tangent line (even vertical)
- 0470(27): example of an inflection point with a vertical tangent line
- 0470(28): non-example for inflection point because not continuous
- 0470(29-30): example of finding concavity intervals and points of inflection and local extrema for a polynomial,
then sketching the graph
- 0470(29-30): SKILL: sketching
- 0470(31): the second derivative test
- 0470(32): example of finding intervals of increase/decrease and local extrema,
along with concavity intervals and points of inflection,
for a rational function
- 0470(32): SKILL: intervals, local max/min
- 0470(33): example of finding local max/min values using the first derivative test
- 0470(34): same example, but use the second derivative test and state preference between the two derivative tests
- 0470(34): SKILL: derivative tests
- 0470(35-38): example of finding intervals of increase/decrease, local extrema, concavity intervals and then sketch
- 0470(38): SKILL: intervals, local max/min
- 0470(39): example of going from an inequality on the derivative of a function to an inequality on the function itself
- 0470(39): SKILL: inequalites from increasing/decreasing
- 0470(40): every cubic has exactly one point of inflection
- 0470(41): when the cubic has three real roots, the point of inflection is at their average
- 0470(42): SKILLs and Whitman problems:
- first derivative test
- second derivative test
- concavity, inflection points
Topic 0480
(Graphing problems)
Link to Topics website
- 0480(1): title slide
- 0480(2): graphing checklist
- symmetry
- intervals of positivity and negativity
- domain
- x- and y-intercepts
- vertical and horizontal asymptotes
- intervals of increase and decrease
- concavity and points of inflection
- 0480(3-11): example: sketching the a graph of a rational function
- 0480(3): symmetry: even
- 0480(4): domain
- 0480(4): intervals of positivity and negativity
- 0480(4): intercepts and asymptotes
- 0480(5): intervals of increase and decrease
- 0480(6-8): concavity and points of inflection
- 0480(9): transition slide
- 0480(10): sketching the graph over [0,\infty)
- 0480(11): reflecting through x=0
- 0480(12-23): example: sketching the a graph of a function that has a square root
- 0480(12): no symmetry
- 0480(13): domain
- 0480(13): intervals of positivity and negativity
- 0480(13): intercepts and asymptotes
- 0480(14-16): intervals of increase and decrease
- 0480(17-20): concavity and points of inflection
- 0480(21): transition slide
- 0480(22-23): sketching the graph
- 0480(24-29): example: sketching the a graph of a transcendental function (with an expoential function in it)
- 0480(24): no symmetry
- 0480(25): domain
- 0480(25): intervals of positivity and negativity
- 0480(25): intercepts and asymptotes
- 0480(26): intervals of increase and decrease
- 0480(27): concavity and points of inflection
- 0480(28): transition slide
- 0480(29): sketching the graph
- 0480(30-43): example: sketching the a graph of a transcendental function (with trigonometric functions in it)
- 0480(30): symmetry: 2\pi periodic and odd
- 0480(31): domain
- 0480(31): intervals of positivity and negativity
- 0480(31): intercepts and asymptotes
- 0480(32-34): intervals of increase and decrease
- 0480(35-38): concavity and points of inflection
- 0480(39): transition slide
- 0480(40): sketching the graph over [0,\pi]
- 0480(41): reflecting trhough 0 to get the graph over [-\pi,\pi]
- 0480(42-43): repeating to get the full graph
Topic 0490
(More graphing problems)
Link to Topics website
Topic 0500
(Even more graphing problems)
Link to Topics website
Topic 0510
(Optimization)
Link to Topics website
- 0510(1): title slide
- 0510(2): Steps in solving optimization problems:
- 1. Understand the problem (and identify the important quantities).
- 2. Draw a diagram.
- 3. Introduce notation.
- 4. Express the quantity to be extremized as a function (of possibly more than one variable);
express the constraints.
- 5. Express the quantity to be extremized as a function of one variable.
- 6. Use the methods fo Chapter 5 to maximize or minimize the function
- 0510(2-4): example (fixed amount of fencing; pen along a river; max area)
- 0510(4): SKILL: max-min
- 0510(5-8): example (can with fixed volume; minimize surface area)
- 0510(8): SKILL: max-min
- 0510(9-15): example (maximize area of printed material on poster, fixed margins, fixed total area of poster)
- 0510(15): SKILL: max-min
- 0510(16-21): example (minimize distance from a point to a certain parabola)
- 0510(21): SKILL: max-min
- 0510(22-31): example (minimize travel time to point on other side of river)
- 0510(31): SKILL: max-min
- 0510(32): example (maximize revenue based on rebate)
- 0510(32): SKILL: max-min
- 0510(33): example (minimize product, given that difference is 10)
- 0510(33): SKILL: max-min
- 0510(34): example (minimize area of rectangle with a fixed perimeter)
- 0510(34): SKILL: max-min
- 0510(35-36): example (minimize area of open topped box with a fixed volume)
- 0510(36): SKILL: max-min
- 0510(37-38): example (minimize distance from a point to a line)
- 0510(38): SKILL: max-min
- 0510(39-46): example (maximize length of ladder going around a corner)
- 0510(46): SKILL: max-min
- 0510(47): SKILLs and Whitman problems:
Topic 0520
(Related rates)
Link to Topics website
- 0520(1): title slide
- 0520(2): sample related rates problem with a growing cube
- 0520(2): 12 step program for solving related rates problems
- 1. Admit you have a problem.
- 2. Read the problem.
- 3. Draw a snapshot.
- 4. Identify the important quantities.
- 5. Notate the drawing.
- 6. Notate the requested rate.
- 7. Notate other information given in the problem.
- 8. RELATE the quantities
- 9. DIFFERENTIATE (with respect to time), RELATING THE RATES.
- 10. Plug in the information given in the problem.
- 11. Solve for the requested rate.
- 12. Celebrate! (Responsibly!)
- 0520(2): SKILL: related rates
- 0520(3): example related rates problem (air pumped into spherical balloon)
- 0520(3): SKILL: related rates
- 0520(4): example related rates problem (ladder sliding out from wall)
- 0520(4): SKILL: related rates
- 0520(5): example related rates problem (growing rectangle)
- 0520(5): SKILL: related rates
- 0520(6): example related rates problem (water pumped into conical tank)
- 0520(6): SKILL: related rates
- 0520(7-11): example related rates problem (distance between moving ships)
- 0520(11): SKILL: related rates
- 0520(12-18): example related rates problem (carts connected by rope over pulley)
- 0520(18): SKILL: related rates
- 0520(19): SKILLs and Whitman problems:
Topic 0530
(Newton's method)
Link to Topics website
- 0530(1): title slide
- 0530(2): finding a solution to f(x)=0
- 0530(2): visulization of Newton's method, via x_1,x_2,x_3,...
- 0530(3): finding the formula for x_2 in terms of x_1
- 0530(4): finding the formula for x_3 in terms of x_2
- 0530(5): finding the formula for x_4 in terms of x_3
- 0530(5): finding the formula for x_{n+1} in terms of x_n
- 0530(6-8): what can go wrong with Newton's method
- 0530(6): sometimes the denominator is zero
- 0530(7): may repeat
- 0530(8): may not have a limit
- 0530(9): usually Newton's method works, and sometimes incredibly well
- 0530(10): for square roots, the accuracy doubles with each iteration
- 0530(11): how to remember the formula for Newton's method for f(x)=0
- 0530(11): to get x_{n+1}
- take the logarithmic derivative of f(x) with respect to x
- reciprocate
- change every x to x_n
- subtract from x_n
- 0530(12-14): example (solving a cubic)
- 0530(15-18): example (finding \sqrt{7})
- 0530(15-16): formula says to average x_n with 7/x_n to get x_{n+1}
- 0530(17-18): stability: if x_n=\sqrt{7} then x_{n+1}=\sqrt{7}
- 0530(18): number of decimals of accuracy doubles with each iteration
- 0530(19): example (finding the fifth root of 3)
- 0530(20): example (solving a transcendental equation)
- 0530(21): example (solving a quintic)
- 0530(22): example (finding a 100th root)
- 0530(22): sometimes with a bad initial guess, the method can be *very* slow
- 0530(22): starting with another initial guess, it's much better
- 0530(23): example to show that Newton's method can fail
- 0530(24): SKILLs and Whitman problems:
Topic 0540
(Linear approx)
Link to Topics website
- 0540(1): title slide
- 0540(2-5): preliminaries
- 0540(2): recall the definition of \triangle y (called the "difference of y")
- 0540(2): e.g. of \triangle y, and of an evaluation of \triangle y
- 0540(2): def'n of dy (called the "differential of y")
- 0540(2): e.g. of dy, and of an evaluation of dy
- 0540(3): def'n of linearization
- 0540(3-5): e.g. of linearization
- 0540(5): general formula for linearization
- 0540(6): guessing the temperature in the future, 0th order approximation
- 0540(7): guessing the temperature in the future, 1st order approximation = linear approximation = approximation by differentials
- 0540(7): reference to higher order approximations
- 0540(8): the temperature curve, guessing the temperature in the future, linear approximation
- 0540(8): the tangent line hugs the curve, for a little while
- 0540(9): the general picture of linear approximation
- 0540(9): first formula describing the principle of linear approximation
- 0540(10): second formula describing the principle of linear approximation, via linearization
- 0540(11-14): third formula describing the principle of linear approximation, via differentials and differences
- 0540(11): the run is \triangle x
- 0540(12): the curve rise is \triangle y
- 0540(13): the run is dx and the line rise is dy
- 0540(14): \triangle y is approximately equal to dy, provided \triangle x = dx is small
- 0540(15): transition slide
- 0540(19-23): example (linearization, then estimation, then overestimate or underestimate?)
- 0540(19): graphing the function
- 0540(20): slope of the tangent line
- 0540(21): equation of the tangent line and doing the estimation
- 0540(22-23): the estimation by differentials
- 0540(23): overestimate because graph is concave down
- 0540(23): SKILL: linear approximation
- 0540(24): example: radius accurate up to \pm 0.07, estimate error in volume in the sphere
- 0540(24): SKILL: linear approximation
- 0540(25): example: estimate 1.001^7
- 0540(25): SKILL: linear approximation
- 0540(26): example: estimate e^{-0.032}
- 0540(26): SKILL: linear approximation
- 0540(27): SKILLs and Whitman problems:
Topic 0550
(Antidifferentiation)
Link to Topics website
- 0550(1): title slide
- 0550(2-5): differentiation = splitting apart; integraion = putting them back together
- 0550(6): def'n: antiderivative
- 0550(7-8): example: antiderivatives of x^2 w.r.t. x
- 0550(9): d/dx is not "1-1", so is not invertible
- 0550(10): recall that if two functions have the same derivative on an interval,
then they differ by a constant on that interval
- 0550(10): the set of all antiderivatives of x^2 w.r.t x
- 0550(11): notation: \int; example: int x^2 dx
- 0550(11): traditional not to use set notation
- 0550(12): transition slide
- 0550(13): example: antiderivative of 1/(\sqrt{1-x^2}), careful answer
- 0550(14): def'n: antiderivative of f on (a,b)
- 0550(14): e.g.: antiderivative of f(x) on a < x < b
- 0550(15): def'n: antiderivative of f on [a,b]
- 0550(16): e.g.: antiderivative of f on a <= x <= b
- 0550(17): Recall: definition and e.g. for \int f(x) dx
- 0550(18): \int x^n dx, for n positive
- 0550(18): \int x^n dx, for n = -1/2, careful answer
- 0550(19-27): \int x^n dx, for n = -1, careful answer and sloppy answer
- 0550(19): x^{-1} = 1/x
- 0550(20-21): \ln(|x|) is an antiderivative of 1/x w.r.t. x
- 0550(22): \ln(|x|) + C is not right
- 0550(23-25): \ln_{AB}
- 0550(26): \int 1/x dx, careful answer is too complicated
- 0550(27): \int 1/x dx, sloppy answer
- 0550(28-30): \int x^n dx, sloppy answer
- 0550(28): examples of sloppiness
- 0550(29): some sloppiness is acceptable
- 0550(30): some is not
- don't forget + C
- x^{n+1}/(n+1) is not correct for n=-1
- in \ln(|x|), don't forget the absolute value bars; \int 1/x dx = [\ln(x)] + C is not acceptable
- 0550(31-33): table of particular antiderivatives
- c(f(x))
- (f(x))+(g(x))
- x^n
- 1/x
- e^x
- \cos x
- \sin x
- \sec^2 x
- (\sec x)(\tan x)
- 1/\sqrt{1-x^2}
- 1/(1+x^2)
- 0550(34-35): example: all antiderivatives of a linear combination of \cos x and powers of x
- 0550(36-37): example: particular antiderivative of a linear combination of e^{2x} and 1/\sqrt{1-x^2} with initial value condition
- 0550(36): antiderivative of e^{2x} w.r.t. x
- 0550(37): finding all antiderivatives
- 0550(37): using the initial value condition to find the right one
- 0550(38): antiderivatives of f(ax+b), where F'=f
- general result: [F(ax+b)]/a
- examples
- 0550(39): no similar result for f(ax^2+bx+c)
- 0550(39): no elementary antiderivative of e^{x^2}
- 0550(39): Fundamental Theorem of Calculus yields a non-elementary antiderivative of e^{x^2}
- 0550(40): recall motion along a line
- 0550(40): derivatives and higher derivatives of position (w.r.t. time)
- velocity
- acceleration
- jerk
- snap
- crackle
- pop
- 0550(40): antiderivatives and higher antiderivatives of pop
- crackle
- snap
- jerk
- acceleration
- velocity
- position
- 0550(41): transition slide
- 0550(42): acceleration (on Earth) = 2 rods/second; going from that to velocity, then position
- 0550(43): example (motion along a line)
- 0550(43): SKILL: motion on a line
- 0550(44): example (motion on a vertical line, due to gravity)
- 0550(44): SKILL: motion on a line
Topic 0560
(Antidifferentiation problems)
nk to Topics website
- 0560(1): title slide
- 0560(2): example (all antiderivatives of a polynomial)
- 0560(2): SKILL: all antierivatives
- 0560(3-4): checking the answer by differentiation
- 0560(5-7): example (all antiderivatives of a linear combination of powers of x)
- 0560(5): need to compare domains of problem and answer
- 0560(6): use \ln(|x|), not \ln x, as an antiderivative of 1/x
- 0560(6): the technically correct answer is "too good"
- 0560(7): SKILL: all antierivatives
- 0560(8-9): example (rational function with denominator 1+x^2)
- 0560(8): doing the division
- 0560(9): completing the problem
- 0560(9): checking the answer by differentiation
- 0560(9): SKILL: all antierivatives
- 0560(10): example: sketching the graph of the derivative, given the graph of the function and an initial value condition
- 0560(10): SKILL: antiderivative from graph
- 0560(11): example: sketching the graph of the derivative, given the graph of the function and an initial value condition,
phrased in terms of velocity and position
- 0560(11): SKILL: antiderivative from graph
- 0560(12): antiderivative of 5+2(1+x^2)^{-1}, with an initial value condtion
- 0560(12): finding the answer, which is 5x + 2 \arctan x - 6 - (\pi/2)
- 0560(12-16): checking the answer by comparing graphs
- 0560(12-15): hints toward the graph of 5+2(1+x^2)^{-1}
- 0560(16): hings toward the graph of 5x + 2 \arctan x - 6 - (\pi/2)
- 0560(16): SKILL: graphing
- 0560(16): SKILL: one antiderivative
- 0560(17): example (finding all double antiderivatives)
- 0560(17): SKILL: all double antiderivatives
- 0560(18): example (finding all double antiderivatives)
- 0560(18): SKILL: all double antiderivatives
- 0560(19): example (finding one double antiderivatives, with initial conditions)
- 0560(19): SKILL: one double antiderivatives
- 0560(20-22): example (finding one double antiderivatives, with initial conditions)
- 0560(20-22): SKILL: one double antiderivatives
Topic 0570
(Indefinite integration)
Link to Topics website
- 0570(1): title slide
- 0570(2): recall the notation \int f(x) dx, which is
read "the indefinite integral of f(x) with respect to x"
- 0570(2-9): properties and examples of indefinite integration
- 0570(2-4): linearity of indefinite integration
- 0570(5): integrating a constant
- 0570(5): integrating a power of x (caveat for x^0 and doesn't work for x^{-1})
- 0570(5): integrating a x^{-1} = 1/x
- 0570(5): integrating e^x and a^x
- 0570(6-8): dx sometimes moves into the numerator to save space
- 0570(9): integrating \cos x, \sin x, \sec^2(x), \csc^2(x), (\sec x)(\tan x) and (\csc x)(\cot x)
- 0570(10): example (verifying an indefinite integration result by differentiation)
- 0570(10): SKILL: verify indefinite integral
- 0570(11): example (verifying an indefinite integration result by differentiation)
- 0570(11): SKILL: verify indefinite integral
- 0570(12): example (indefinite integral of a polynomial)
- 0570(12): SKILL: find indefinite integral
- 0570(13): example (indefinite integral of a certain polynomial in trig functions)
- 0570(13): SKILL: find indefinite integral
- 0570(14): example (indefinite integral of a certain polynomial in trig functions)
- 0570(14): SKILL: find indefinite integral
- 0570(15): example (indefinite integral, then graph several of the antiderivatives)
- 0570(15): SKILL: find and graph antiderivatives
Topic0580
(Riemann sums and the definition of the definite integral)
Link to Topics website
- 0580(1): title slide
- 0580(2): setup of problem to compute area under y=x^2 from x=0 to x=1
- 0580(3): an approximation by 8 rectangles
- 0580(4): the area of the union of the 8 rectangles
- 0580(5): an approximation by 40 rectangles and the area of their union
- 0580(6): the area under the curve, as a limit
- 0580(7-8): computing that limit
- 0580(9): the Fundamental Theorem of Calculus (FTC) gives an easier way
- 0580(10): the general setup of area under a function f between a and b
- 0580(10): def'n h_3
- 0580(10-14): 3rd partition: three subintervals, left endpoints, midtpoints and right endpoints
- 0580(13): stepping from a to b by three steps
- 0580(13): the "1 convention"
- 0580(13): the left endpoint of the third subinterval in the third partition, using the 1 convention
- 0580(15-17): change to and examine the 10th partition
- 0580(16): stepping from a to b by ten steps
- 0580(16): def'n: h_{10}
- 0580(16): def'n: h_n
- 0580(16): WARNING: h is for "horizontal" not "height"
- 0580(16-17): h_n is the common *width* (NOT height) of the rectangles in the nth partition
- 0580(17): Alternate notation: \triangle x, instead of h_n
- 0580(18): back to the 3rd partition
- 0580(19): formula for the right 3rd Riemann sum from a to b of f, denoted R_3 S_a^b f
- 0580(20): h_n can be factored out of the sum
- 0580(21-22): formula for the midpoint 3rd Riemann sum from a to b of f, denoted M_3 S_a^b f
- 0580(23-24): formula for the left 3rd Riemann sum from a to b of f, denoted L_3 S_a^b f
- 0580(25): change to the picture of the rectangles for the left 60th Riemann sum
- 0580(26): goal is to take the limit of the nth Riemann sums as n ---> \infty, but which one (left, right or midpoint)?
- 0580(27): theorem: they come out the same in the limit
- 0580(27): true even with finitely many jump discontinuities
- 0580(28): def'n of definite integtral; notation: \int_a^b f(x) dx
- 0580(29): represents area
- 0580(29): don't need to use x; can use *any* independent variable, or even use none at all (via function notation)
- 0580(30): distinguishing between definite and indefinite integrals
- 0580(30): integral from c to c
- 0580(30): integral from b to a, when a < b
- 0580(31-34): algebra yields alternate versions of the formulas
- 0580(34): the "0 convention"
- 0580(34): various formulas for various Riemann sums
- 0580(35): transition slide
- 0580(36-38): reworking \int_0^1 x^2 dx, using right Riemann sums again, but with summation notation this time
- 0580(38): the Fundamental Theorem of Calculus (FTC) gives an easier way
Topic0590
(Definite integration and Riemann sum problems)
Link to Topics website
- 0590(1): title slide
- 0590(2-5): example: estimating the area under a graph using Riemann sums
- 0590(5): SKILL: Riemann sums from graph
- 0590(6): example: computing area under piecewise linear graph
- 0590(6): SKILL: integral as area
- 0590(7): example: computing the area under a semicircular graph
- 0590(7): SKILL: definite integral
- 0590(7): integrating from 3 to -3, instead of from -3 to 3
- 0590(8-10): example: computing Riemann sums given a table with values of the function
- 0590(10): SKILL: Riemann sum from table
- 0590(11): example: Riemann sum of a polynomial
- 0590(11): SKILL: compute Riemann sum
- 0590(12): example: Riemann sum of a transcendental function
- 0590(12): SKILL: compute Riemann sum
- 0590(13): Review of formulas for \sum j^k
- 0590(14-16): example: showing that \int_0^1 x^2 dx is 1/3 by left and midpoint Riemann sums
- 0590(16): SKILL: limit of Riemann sums
- 0590(16): the Fundamental Theorem of Calculus (FTC) gives an easier way
- 0590(17-25): example: compute a limit of Riemann sums of a polynomial
- 0590(25): SKILL: limit of Riemann sums
- 0590(26): example: express an integral as a limit of right-endpoint Riemann sums
- 0590(26): steps in the process
- For each n = 1,2,3,...
- find width of rectangles (h_n)
- find right-endpoint of jth subinterval
- find height of jth rectangle
- find area of jth rectangle
- add over j = 1,...,n
- Then take limit as n ---> \infty
- 0590(26): SKILL: write integral as a limit of Riemann sums
- 0590(27): change to left-endpoint Riemann sums
- 0590(28): change to midpoint Riemann sums
- 0590(29): example: express limit of Riemann sums as an integral
- 0590(29): SKILL: interpret Riemann sums
- 0590(30): SKILLs and Whitman problems:
Topic 0600
(Variations on the definition of the definite integral)
Link to Topics website
Topic 0610
(The Fund Th'ms of Calc, statements and motivations)
Link to Topics website
- 0610(1): title slide
- 0610(2): recall the definition of the indefinite integral
- 0610(2): goal: connecting antidifferentiation to area
- 0610(2): specific goal: connecting position (antiderivative of velocity) to the area under velocity
- 0610(3-14): example using motion along a line: given velocity v(t), find change in position p(t) between t=5 and t=11
- 0610(3): subintervals, midpoints
- 0610(4): estimate (using midpoint velocity) of changes in position (t=5 to t=7, t=7 to t=9 and t=9 to t=11)
- 0610(5): transition slide
- 0610(6): adding to get an estimate of the change in position from t=5 to t=11, i.e., of [p(11)] - [p(5)]
- 0610(7): transition slide
- 0610(8-12): computation of M_3 S_5^{11} v yields that same estimate
- 0610(13): limit as n ---> \infty of M_n S_5^{11} v is equal to [p(11)] - [p(5)]
- 0610(14): \int_5^{11} v(t) dt = [p(11)] - [p(5)]
- 0610(14): using \int_5^{11} v(t) dt = [p(11)] - [p(5)] and antidifferentiation to calculate \int_5^{11} v(t) dt
- 0610(15): transition slide
- 0610(16): key idea: to compute a definite integral
- find an antiderivative
- evaluate at the limits of integration
- subtract
- 0610(17): the Fundamental Theorem of Calculus (FTC) for the integral of the derivative
- 0610(18): another version involving the derivative of the integral
- 0610(19): calculating \int_5^x t^2 dt and then differentiating that w.r.t. x
- 0610(20): the Fundamental Theorem of Calculus (FTC) for the derivative of the integral
- 0610(21): the domain of the integral and the domain of its derivative
- 0610(22-23): slight change of notation
- 0610(23): avoid using the variable of integration as a limit of integration
- 0610(24): rephrasing the two FTCs using functions rather than expressions
- 0610(24): integration and differentation are opposites, seen in three separate formulas
- 0610(25): the definite integral as an antiderivative
- 0610(26): ends up being an antiderivative on a compact interval
- 0610(27): if a function is continuous on a compact interval, then it has an antiderivative on that interval
- 0610(28): can replace the lower limit of integration by any number in the compact interval
- 0610(29-33): loose ends from previous topics
- 0610(29-31): an antiderivative (not elementary) of e^{x^2}
- 0610(32): easier way (via the FTC) to compute \int_0^1 x^2 dx
- 0610(32): SKILL: definite integration
- 0610(33): easier way (via the FTC) to compute \int_2^7 3x^2+4x^3 dx
- 0610(33): SKILL: definite integration
- 0610(34-41): example (a Riemann sum and a definite integral)
- 0610(34-36): the computation of the Riemann sum
- 0610(36): SKILL: Riemann sums
- 0610(37): transition slide
- 0610(38): the corresponding rectangles
- 0610(38): rectangles below the x-axis count negative
- 0610(39): analyzing the definite integral
- 0610(39): area below the x-axis counts negative; expect a negative answer
- 0610(40): transition slide
- 0610(41): computation of the integral
- 0610(41): evaluation is linear
- 0610(41): SKILL: definite integration
- 0610(42): backing up a few steps
- 0610(43): interpreting the evaluations as integrals
- 0610(44): transition slide
- 0610(45): definite integration is linear; more on that in a later topic
Topic 0620
(The Fundamental Theorems of Calculus, problems)
Link to Topics website
- 0620(1): title slide
- 0620(2): example (the derivative of an integral from expression to expression)
- 0620(2): SKILL: differentiate an integral that goes from a constant to a variable
- 0620(3): example (the derivative of an integral from expression to expression)
- 0620(3): SKILL: differentiate an integral that goes from a constant to a variable
- 0620(4): example (definite integral of a polynomial)
- 0620(4): SKILL: evaluation of a definite integral
- 0620(5-7): example (integration of an exponential function)
- 0620(7): SKILL: evaluation of a definite integral
- 0620(8): example (definite integral of a polynomial)
- 0620(8): SKILL: evaluate a definite integral
- 0620(9): example (definite integral of a polynomial)
- 0620(9): SKILL: evaluate a definite integral
- 0620(10): example (definite integral of a product of two polynomials)
- 0620(10): SKILL: evaluate a definite integral
- 0620(10): WARNING: the integral of the product is not the product of the integrals
- 0620(11): example (definite integral of a linear combination of powers)
- 0620(11): SKILL: evaluate a definite integral
- 0620(12-14): example (definite integral of a quotient of a polynomial by a power)
- 0620(14): SKILL: evaluate a definite integral
- 0620(15): example (definite integral of a quotient of (a linear combination of powers) by (a power))
- 0620(15): SKILL: evaluate a definite integral
- 0620(16): example (definite integral w.r.t. t of a rational function with 1+t^2 in the denominator)
- 0620(16): SKILL: evaluate a definite integral
- 0620(17): example (definite integral w.r.t. x of a rational function with 1+x^2 in the denominator)
- 0620(17): example (interpret a definite integral as an area)
- 0620(17): SKILL: evaluate a definite integral
- 0620(17): fat tails and thin tails, and finance
- 0620(18): example (definite integral involving trig functions)
- 0620(18): SKILL: evaluate a definite integral
- 0620(19): example (definite integral of dx/x between various limits)
- 0620(19): SKILL: evaluate a definite integral
- 0620(19): no integration across an infinite discontinuity in this class
- 0620(20): technically \int (1/x) dx = [\ln(|x|)] + C is wrong, because we should have two degrees of freedom;
however, for definite integrals of dx/x,
because \ln(|x|) is an antiderivative of (1/x) w.r.t. x,
this inaccuracy becomse unimportant
- 0620(21): example (the derivative of an integral from expression to expression)
- 0620(21): SKILL: differentiate an integral that goes from an expression to an expression
- 0620(22): example (the derivative of an integral from expression to expression)
- 0620(22): SKILL: differentiate an integral that goes from an expression to an expression
- 0620(23): example (the derivative of an integral from expression to expression)
- 0620(23): SKILL: differentiate an integral that goes from an expression to an expression
- 0620(24-29): differentiate an integral that goes from an expression to an expression, in two different ways:
- 0620(24-27): evaluate the antiderivative between two expressions and differentiate
- 0620(28-29): example (the derivative of an integral from expression to expression)
- 0620(28-29): SKILL: differentiate an integral that goes from an expression to an expression
- 0620(30): example (the double derivative of a double integral from expression to expression)
- 0620(30): SKILL: differentiate^2 an integral^2 that goes from an expression to an expression
- 0620(31): example (interpretation of an integral of a physical quantity)
- 0620(31): SKILL: interpret definite integral
- 0620(32-33): example (integration of a graph)
- 0620(32-33): SKILL: integrate a graph
- 0620(34): SKILLs and Whitman problems:
- fundamental theorem of calculus
Topic 0630
(Properties of the definite integral)
Link to Topics website
- 0630(1): title slide
- 0630(2-11): properties of the definite integral
- 0630(2-3): linearity of definite integral
- 0630(2): commutes with scalar multiplication
- 0630(3): distributes over addition (i.e., is additive)
- 0630(4): integrating a constant
- 0630(5): distributes over subtraction
- 0630(6): nonnegative functions have nonnegative definite integrals
- 0630(7-8): monotonicity of definite integration
- 0630(9): upper and lower bounds on a function give upper and lower bounds on its definite integral
- 0630(10-11): the cocycle identity
- 0630(11): recall linearity of the definite integral
- 0630(12): many operations are linear but not multiplicative
- differentiation
- summation
- indefinite integration
- definite integration
- change between two evaluations
- difference
- 0630(12): there are replacements for multiplicativity, though:
- product rule (differentiation by parts)
- summation by parts
- indefinite integration by parts
- definite integration by parts
- product rule for change in evaluation (change in evaluation by parts)
- product rule for differencing (differencing by parts)
- 0630(13): some operations are both linear and multiplicative
- 0630(14-15): example (definite integral of a polynomial, using properties of integration)
- 0630(15): SKILL: Definite integration
- 0630(16): example (using the cocycle identity)
- 0630(16): SKILL: Properties of integration
- 0630(17-21): example (integrating the absolute value of a function with known antiderivative)
- 0630(17-18): figure out where the function is positive and negative
- 0630(18): use the cocycle identity to split the integral into integrals where the function doesn't change sign
- 0630(19): take out the absolute value bars, putting in negative signs when necessary
- 0630(20): transition slide
- 0630(21): finish off the integral by antidifferentiating and evaluating
- 0630(21): SKILL: find definite integral
- 0630(22-29): example (displacement and distance traveled by a particle moving on a line)
- 0630(22): difference between displacement and distance moved, explained via another particle
- 0630(23): displacement
- 0630(24): transition slide
- 0630(25): explain what displacement -9 means
- 0630(25): discuss the travel path of the particle and why distance traveled will be greater than 9
- 0630(25): figure out where the velocity is positive and negative
- 0630(26): split into integrals where the velocity doesn't change sign
- 0630(26-29): antidifferentate and evaluate
- 0630(29): SKILL: compute displacement and distance traveled
- 0630(30-32): example (integral of a negative function from left to right and from right to left)
- 0630(30): from left to right
- 0630(31): transition slide
- 0630(32): from right to left
- 0630(32): SKILL: Properties of integration
- 0630(33): SKILLs and Whitman problems:
- properties of integration
Topic 0640
(The Integral Mean Value Theorem)
Link to Topics website
- 0640(1): title slide
- 0640(2): calculation of the mean (a.k.a. average) of four numbers, and visualization
- 0640(2): the mean is between the max and the min
- 0640(2): the mean may not be attained as one of the four numbers
- 0640(3): calculation of the mean of x^2 on 1 <= x <= 3
- 0640(3): the mean is between the max and the min
- 0640(3): the mean is attained as a value of the function
- 0640(4): a general function
- 0640(5): def'n: average or mean of a function; notation
- 0640(6): the mean is between the max and the min
- 0640(6): the mean is attained as a value of the function
- 0640(7): the Integral Mean Value Theorem (IMVT) a.k.a. the Average Mean Value Theorem
- 0640(8): notation of averaging varies with the independent variable
- 0640(8): notation of averaging for functions (with no independent variable)
- 0640(8): averaging is linear, but not multiplicative
- 0640(8): no cocycle identity for averaging
- 0640(9): change of notation in the IMVT (v for the function, t for the independent variable)
- 0640(10): interpretation of the IMVT in terms of average velocity being attained
- 0640(10): my trip to Chicago and the IMVT
- 0640(11-12): my trip to Chicago and the MVT
- 0640(13): my velocity function on the way to Chicago and the IMVT
- 0640(14): example (average a polynomial)
- 0640(14): SKILL: find average value
- 0640(15): transition slide
- 0640(16): finding where the average is attained
- 0640(16): SKILL: find where average is attained
- 0640(17-19): example (finding where the average is attained)
- 0640(17): computing the average
- 0640(18): setting up the equation
- 0640(19): solving the equation
- 0640(19): SKILL: find where average is attained
- 0640(20): example (finding where the average is attained)
- 0640(20): SKILL: find where average is attained
- 0640(21): example (average a trig function)
- 0640(21): SKILL: find average value
- 0640(21): If you average an odd function across an interval with 0 as its midpoint, you get 0.
- 0640(22): example (average a trig function)
- 0640(22): SKILL: find average value
- 0640(23): example (average a trig function)
- 0640(23): SKILL: find average value
- 0640(24): example (average a trig function)
- 0640(24): SKILL: find average value
- 0640(25): example (average a trig function)
- 0640(25): SKILL: find average value
- 0640(26): example (average a rational function)
- 0640(26): SKILL: find average value
- 0640(27): example (average a power function, find where the average is attained, sketch with averaging rectangle)
- 0640(27): SKILL: find average value
- 0640(27): SKILL: find where average attained
- 0640(27): SKILL: show averaging rectangle
- 0640(28): example (average a rational function, find where the average is attained, sketch with averaging rectangle)
- 0640(28): SKILL: find average value
- 0640(28): SKILL: find where average attained
- 0640(28): SKILL: show averaging rectangle
- 0640(29): SKILLs and Whitman problems:
Topic 0650
(The Fundamental Theorems of Calculus, proofs)
Link to Topics website
- 0650(1): title slide
- 0650(2-7): proof of the Fundamental Theorem of Calculus (FTC) for the derivative of the integral
- 0650(2): recall the statement of the FTC for the derivative of the integral
- 0650(2): interpreting the derivative as a limit of difference quotients
- 0650(2): using the cocycle identity to simplify the numerator to a single integral
- 0650(3): transition slide
- 0650(4): want: (the two-sided limit) of (the difference quotient) is f(x)
- 0650(4): interpreting the difference quotient as an average value, when h>0
- 0650(5): using the Integral Mean Value Theorem (IMVT), when h>0
- 0650(6): proof that (the limit from the right) of (the difference quotient) is f(x)
- 0650(7): proof that (the limit from the left) of (the difference quotient) is f(x)
- 0650(8-12): proof of the Fundamental Theorem of Calculus (FTC) for the integral of the derivative
- 0650(8): recall the statement of the FTC for the integral of the derivative
- 0650(8): interchange t and x in the FTC for the derivative of the integral
- 0650(9): transition slide
- 0650(10): computing the constant of integration
- 0650(11): transition slide
- 0650(12): finishing the proof
Topic 0660
(Integration by substitution)
Link to Topics website
- 0660(1): title slide
- 0660(2-12): indefinite integration by substitution
- 0660(2-3): example problem (polynomial substitution)
- 0660(4-6): reorganization to make it simpler to remember and recognize
- 0660(7): general statement about integrating F'(u) du, where u is a dependent variable
- 0660(7-8): proof of this general statement
- 0660(8): example problem (polynomial substitution)
- 0660(8): SKILL: integration by substitution
- 0660(9): example problem (trig substitution to integrate \tan x dx)
- 0660(9): SKILL: integration by substitution
- 0660(10-12): indicating and correcting the sloppiness in the answer to \int \tan x dx
- 0660(13): transition slide
- 0660(14-18): definite integration by substitution
- 0660(14): an example wokred via indefinite integration by substitution
- 0660(14): rewriting the answer as an integral
- 0660(15): transition slide
- 0660(16): showing the effect on the limits of integration
- 0660(16): the general rule (including changing the limits of integration)
- 0660(17): double use of a variable as both dependent and independent
- 0660(18): example (polynomial substitution)
- 0660(19): can change from one independent variable to another
- 0660(20-27): integrating symmetric functions
- 0660(20): statement of results (even and odd cases)
- 0660(20-23): proof of the even case
- 0660(24-25): proof of the odd case
- 0660(26): example in the even case
- 0660(26): SKILL: integration using symmetry
- 0660(27): example in the odd case
- 0660(27): SKILL: integration using symmetry
Topic 0670
(Integration by substitution, problems)
Link to Topics website
- 0670(1): title slide
- 0670(2): example (indefinite integral: 100th power of a quartic, times x^3)
- 0670(2): SKILL: integration by substitution
- 0670(3): example (indefinite integral: trig composed with x^{-3}, times x^{-4})
- 0670(3): SKILL: integration by substitution
- 0670(4-7): example (indefinite integral: square root of a quadratic, times an odd power of x)
- 0670(8): example (indefinite integral: 100th power of a cubic, times x^2)
- 0670(7): SKILL: integration by substitution
- 0670(8): SKILL: integration by substitution
- 0670(9): example (indefinite integral: cos(e^x), times e^x)
- 0670(9): SKILL: integration by substitution
- 0670(10): example (indefinite integral: reciprocal of a cubic, times x^2)
- 0670(10): SKILL: integration by substitution
- 0670(11): example (indefinite integral: e^{e^t}, times e^t)
- 0670(11): SKILL: integration by substitution
- 0670(12): example (indefinite integral involving trig functions)
- 0670(12): SKILL: integration by substitution
- 0670(13): example (definite integraal: square root of a quadratic, times x)
- 0670(13): SKILL: integration by substitution
- 0670(14): example (definite integral: exponential of a cubic, times x^2)
- 0670(14): SKILL: integration by substitution
- 0670(15-17): example (definite integral: reciprocal of square root of linear, times x)
- 0670(17): SKILL: integration by substitution
- 0670(18-20): example (definite integral in two parts, one done geometrically, the other by substitution)
- 0670(20): SKILL: integration by substitution
- 0670(21): example (definite integral involving an unspecificed function, and integral information about that function)
- 0670(21): SKILL: integration by substitution
- 0670(22): SKILLs and Whitman problems:
- integration by substitution
Topic 0680
(Area between curves)
Link to Topics website
- 0680(1): title slide
- 0680(2-8): area between two curves that don't cross on a certain interval
- 0680(2): statement of problem
- 0680(3): visualization
- 0680(4): area under the bottom curve
- 0680(5): area under the top curve
- 0680(6): both areas
- 0680(7): difference is the area we seek
- 0680(8): formula
- 0680(9-10): area between two curves that may cross on a certain interval
- 0680(9): visualization and max-min formula
- 0680(10): absolute value formula
- 0680(11): example (area between two polynomials)
- 0680(11): SKILL: area between curves
- 0680(12-14): example (area between two polynomials that cross more than one)
- 0680(14): SKILL: area between curves
- 0680(15-17): example (area between two polynomials that cross in y)
- 0680(17): SKILL: area between curves
- 0680(18): can interchange x and y, if it's less confusing that way
- 0680(19-29): area enclosed in a circle
- 0680(19): graphing a circle
- 0680(20): semicircles and seting up the integral
- 0680(21): transition slide
- 0680(22): using symmetry
- 0680(23-25): trig change of variables
- 0680(25): reduction to an integral of cos^2
- 0680(25-26): formula for cos^2 in terms of cos
- 0680(27): completion of the definite integral
- 0680(28): transition slide
- 0680(29): simplification of the answer
- 0680(29): SKILL: area by integration
Topic 0690
(Area between curves, problems)
Link to Topics website
- 0690(1): title slide
- 0690(2-11): example (interpret and estimate area between curves)
- 0690(2-6): interpret area between velocity curves
- 0690(6-11): estimate area between curves by midpoints
- 0690(11): SKILL: interpret and estimate area between curves
- 0690(12-14): example (area between two parabolas)
- 0690(14): SKILL: area between curves
- 0690(15): example (area between two curves)
- 0690(16): SKILL: area between curves
- 0690(16): example (area between two curves)
- 0690(16): SKILL: area between curves
- 0690(17): example (area bewteen two curves, one involving absolute value)
- 0690(17): SKILL: area between curves
- 0690(18): example (area between two curves that cross three times)
- 0690(18): SKILL: area between curves
- 0690(19): example (area inside a triangle)
- 0690(20): SKILL: area of triangle from vertices
- 0690(20): example (estimate area inside a closed curve, given cross sectional lengths)
- 0690(20): SKILL: estimate area, given cross sections
- 0690(21-25): example (area inside an algebraic loop)
- 0690(21): graph the loop
- 0690(22): break the graph into upper and lower curves
- 0690(22): set up the integral
- 0690(23-25): evaluate the integral
- 0690(25): SKILL: area between curves
- 0690(26): SKILLs and Whitman problems:
Topic 0700
(Volume by slices)
Link to Topics website
- 0700(1): title slide
- 0700(2-5): defining and computing the volume inside a solid
- 0700(2): add a number line into three-space
- 0700(3): compute area of cross section
- 0700(3): compute volume of (infinitesimally) widened cross section
- 0700(4): add up (i.e., integrate) those infinitesimal volumes to get the volume of the solid
- 0700(5): use that integral as the definition of the volume of the solid
- 0700(6): the number line need not be horizontal, and need not point to the right
- 0700(6): the variable need not be u
- 0700(7-8): Cavalieri's principle
- 0700(9-13): in volume, cone + hemisphere = cylinder
- 0700(14-15): volume of a generalized cylinder
- 0700(16-17): volume of a generalized cone
- 0700(18-20): comparing cones to cylinders
- 0700(20): in n dimensions:
- (volume of a generalized cone) is (1/n) x (volume of the corresponding generalized cylinder)
Topic 0710
(The disk and washer methods)
Link to Topics website
- 0710(1): title slide
- 0710(2-13): volume in a sphere of radius 7
- 0710(2-4): first method
- 0710(2): recall: in volume, cone + hemisphere = cylinder
- 0710(3): volume of cylinder and then cone
- 0710(4): volume of (solid) hemisphere, then volume in sphere
- 0710(5-13): second method
- 0710(5): comparison of (2-D pictures with revoluation) and (3-D pictures)
- 0710(6): def'n: solid of revolution
- 0710(6): radius and area of disk in both 2-D picture and 3-D picture
- 0710(6): writing the volume as a definite integral
- 0710(7-12): the computation of that integral
- 0710(12): SKILL: volume of a solid
- 0710(13): vertical disks
- 0710(14): the disk method with vertical disks
- 0710(15): can factor out \pi from the integral
- 0710(16): the disk method with horizontal disks
- 0710(17): some comments
- axis might be neither vertical nor horizontal; it might point left down
- variable might be neither x nor y, though x and y are tranditional for a horizontal or vertical axis of revolution
- required for the disk method: all cross sections are disks
- 0710(18-21): example problem (region under algebraic function revolved about the x-axis)
- 0710(18): picture an approximating cylinder
- 0710(19): find the volume of an approximating cylinder; add them up by summation
- 0710(20): switch to infinitesimals and integrals
- 0710(21): compute the integral
- 0710(21): SKILL: disk method
- 0710(22): recall the disk method
- 0710(23): revolving the region between two functions
- 0710(23): the washer method with vertical washers
- 0710(24): the washer method with horizontal washers
- 0710(24): some comments
- axis might be neither vertical nor horizontal; it might point left down
- variable might be neither x nor y, though x and y are tranditional for a horizontal or vertical axis of revolution
- required for the washer method: all cross sections are washers
- 0710(25-28): example (region cut out by lines and a parabola, revolved about the y-axis)
- 0710(25): finding the area of the washer and the volume of the thickened washer
- 0710(25): setting up the definite integral for calculating the volume of the solid
- 0710(26-28): computing that integral
- 0710(28): SKILL: washer method
Topic 0720
(Volume by slices and the disk and washer methods, problems)
Link to Topics website
- 0720(1): title slide
- 0720(2-10): example (wedge shape, volume by slices)
- 0720(2-6): display the solid
- 0720(7): display a slice, in the shape of a rectangular slice
- 0720(8): calculate the base and height of the rectangular slice
- 0720(9): set up the definite integral that integrates the area of the rectangular slice
- 0720(10): compute the definite integral
- 0720(10): SKILL: volume of a solid
- 0720(11): example (solid of revolution about x-axis, disk method)
- 0720(11): SKILL: disk method
- 0720(12-14): example (solid of revolution about x-axis, washer method)
- 0720(14): SKILL: washer method
- 0720(15): example (solid of revolution about x-axis, washer method)
- 0720(15): SKILL: washer method
- 0720(16): example (solid of revolution about a vertical line, washer method)
- 0720(16): SKILL: washer method
- 0720(17): example (solid of revolution about a vertical line, washer method)
- 0720(17): SKILL: washer method
- 0720(18): example (describe a solid of revolution from its integral)
- 0720(18): SKILL: region from integral
- 0720(19): example (describe a solid of revolution from its integral)
- 0720(19): SKILL: region from integral
- 0720(20): example (describe a solid of revolution from its integral)
- 0720(20): SKILL: region from integral
- 0720(21): example (describe a solid of revolution from its integral)
- 0720(21): SKILL: region from integral
Topic 0730
(Volume by cylindrical shells)
Link to Topics website
- 0730(1): title slide
- 0730(2-3): create a shell
- 0730(4-6): find its area
- 0730(4): find its radius and circumference
- 0730(4): find its height
- 0730(6): cut it open and find its area
- 0730(5): generally: the area of a shell is: (its circumference) x (its height)
- 0730(7-10): example where the washer method is hard, but the shell method is easy
- 0730(7): describe the solid of revolution
- 0730(7): try the washer method and observe the difficulties
- 0730(8): make a shell
- 0730(9): find its volume
- 0730(10): integrate to get the volume of the solid
- 0730(10): SKILL: shell method
- 0730(11-16): general shell method, in various cases
- 0730(11): revolving region under a y=f(x) about the y-axis
- 0730(12): revolving region between y=f(x) and y=g(x) about the y-axis
- 0730(13): revolving region between y=f(x) and y=g(x) about a vertical line
- 0730(14): revolving region to the left of x=f(y) about the x-axis
- 0730(15: revolving region to the between x=f(y) and x=g(y) about the x-axis
- 0730(16): revolving region to the between x=f(y) and x=g(y) about a horizontal line
Topic 0740
(Volume by cylindrical shells, problems)
Link to Topics website
- 0740(1): title slide
- 0740(2-3): example (both shell and disk methods, region under a power function, about the x-axis)
- 0740(3): SKILL: disk and shell methods
- 0740(4): example (shell method, region between parabola and line, about x-axis)
- 0740(4): SKILL: shell method
- 0740(5): example (shell method, region bounded by hyperbola and lines, about y-axis)
- 0740(5): SKILL: shell method
- 0740(6-8): example (shell method, region between parabola and line, about y-axis)
- 0740(8): SKILL: shell method
- 0740(9): example (shell method, region between parabola and line, about a vertical line)
- 0740(9): SKILL: shell method
- 0740(10-14): example (shell method, region between parabola and line, about x-axis)
- 0740(10): set up the integral
- 0740(11-14): compute the integral
- 0740(14): SKILL: shell method
- 0740(15-16): example (shell method, region bounded by power function and lines, about a vertical line)
- 0740(16): SKILL: shell method
- 0740(17): example (shell method, regtion between y=x^6 and x=y^6, about a horizontal line)
- 0740(17): SKILL: shell method
- 0740(18-19): example (shell method, region between parabola and line, about a vertical line)
- 0740(19): SKILL: shell method
Topic 0750
(Volume by cylindrical shells, more problems)
Link to Topics website