12/15/05, 1:30 to 4:30 p.m., Blegen Hall 5, 10, or Willey Hall 125,
depending on your discussion section: The Final Exam.
12/15/05, 1:30 to 4:30 p.m., Blegen Hall 5, 10, or Willey Hall 125,
depending on your discussion section: The Final Exam.
12/14/05: Discussion of our sample final ends (Problems 19 through
21). The answer key to the sample final from Michigan is posted on our
class web page.
12/12/05: Discussion of the sample final continues (Problems 14
through 18).
12/09/05: Discussion of the sample final continues (Problems 8 or 9 to
13). Filling out evaluation forms.
12/07/05: Discussion of the sample final begins (Problems 1-7 or
8).
12/05/05: The average value of a function. Example: the average
velocity is the same as the average value of the velocity
function. Examples: a problem similar to Problem 13; compute the
average values of \sqrt{4-x^2} on [-2,2] and -|x| on [-1,1]. [Section
6.5]
12/02/05: The shell method of computing volumes of solids of
revolution. Examples: Problems 12 and 7. [Section 6.3]
11/30/05: Volumes and their computation as integrals of the
cross-sectional area. The disk and washer methods for solids of
revolution. Examples: computing volumes of a pyramid, a paraboloid, a
sphere, and a funnel obtained by revolving the region bounded by y =
x^2 + 1 and y = 3-x around the x-axis. [Section 6.2]
11/28/05: Solutions to Midterm 3. Areas between curves. Example: the
area between y = 2-x^2 and y = - x. Study compound regions (such as in
Example 5 in Section 6.1) and integration with respect to y to find
areas (such as Example 6) on your own. [Section 6.1]
11/23/05: Movie time! Two prize-winning math videos.
11/22/05, Tuesday: Third midterm exam in your discussion sections.
11/21/05: Review: solutions to the sample test. Solutions to Problems
5 and 6 are posted on the web. [Sections 4.2-5, 4.7, 4.10, and
5.1-5]
11/18/05: The substitution rule. Using it to compute indefinite and
definite integrals. Using symmetry arguments in computing
integrals. Examples: Problems 4, \int cot x dx, 18, 34, 50,
56. [Section 5.5]
11/16/05: Indefinite integrals and net change. Examples: Problems 4,
12, 26, 34, and 46. [Section 5.4]
11/14/05: The Fundamental Theorem of Calculus. Examples:
differentiating various integrals with varying limits, evaluating
definite integrals (and areas) by figuring out
antiderivatives. [Section 5.3]
11/11/05: The definite integral. Net area. The midpoint rule. Riemann
sums. Problems 12, 20, 30, and 36. [Section 5.2]
11/09/05: Areas: rectangle approximation, sigma notation. Examples:
Problem 4 (and 18 in the morning class). Study distances on your
own. [Section 5.1]
11/07/05: Antiderivatives. Examples: x^5 and Problems 12, 24
(modified), and 41. [Section 4.10 (skip The Geometry of
Antiderivatives and Rectilinear Motion)]
11/04/05: Optimization problems. Examples: Problems 4, 6, and
10. [Section 4.7]
11/02/05: Curve sketching. Strategy. Example: graphing
(1+x)^2/(1+x^2). [Section 4.5 (skip Slant Asymptotes)]
10/31/05 [Halloween]: Indeterminate forms of type 0/0 and infty/infty
and l'Hospital's rule. Examples: Problems 5, 12, 26, 38, 48,
58. [Section 4.4]
10/28/05: The first two derivatives of f and the shape of the graph
(increasing, decreasing, concave up or down). The first and the second
derivative tests for local extrema. Example: Problem 16. [Section
4.3]
10/26/05: Discussion of the second midterm. Rolle's theorem. Example:
Problem 19. The Mean Value theorem (study on your own from the
text). [Section 4.2]
10/25/05, Tuesday: Second midterm exam in your discussion
sections.
10/24/05: Review: solutions to the sample test. [Sections 2.9, 3.1-2,
3.4-8, 3.10-11, and 4.1]
10/21/05: Absolute and local extrema and extreme values. The Extreme
Value theorem. Fermat's theorem. Critical numbers. Finding absolute
extrema on closed intervals. Examples: y = x^2 on different intervals;
y = x^{2/3} (afternoon class) or exp(x) - 3 exp(-x) - 4x (morning
class). [Section 4.1]
10/19/05: Linear approximation and differentials. Examples: Problems
32, 34, 38, and computing cos(0.1) using linear
approximation. [Section 3.11 (skip Applications to Physics and
Examples 1 and 3)]
10/17/05: Related rates. Examples: the heated plate problem from the
first sample midterm and the inverse problem. Problems 6 and
10. Example 3. [Section 3.10]
10/14/05: Derivatives of logarithms. Examples: Problems 4, 8, 12,
18. Logarithmic differentiation. Examples: Problems 36 and
40. [Section 3.8 (skip The Number e as a Limit)]
10/12/05: Higher derivatives, acceleration and jerk. Finding formulas
for nth derivatives. Examples: the 135th derivative of sin x; the
higher derivatives of x^3; Problems 34 and 30. [Section 3.7]
10/10/05: Implicit differentiation. Derivatives of inverse
trigonometric functions. Examples: problems 18 and 50. [Section 3.6
(skip Orthogonal Trajectories)]
10/07/05: The chain rule. Examples with exponentials and
powers. [Section 3.5]
10/05/05: Derivatives of trigonometric functions: basic rules and
examples. Computation of the derivative of sine from scratch. The
limit sin h/h as h --> 0. Examples of trig limits. [Section 3.4]
10/03/05: The product and quotient rules for derivatives. Further
computations of derivatives: examples. Proofs of the sum and the
product rules. [Section 3.2]
09/30/05: Computing the derivatives of polynomial and exponential
functions. First differentiation rules: the derivatives of sums,
differences, constant multiples, powers. Examples: derivatives of
polynomials, tangent lines to polynomial and exponential
functions. [Section 3.1]
09/28/05: Discussion of the first midterm. The derivative f'(x) as a
function of x. Differentiability and continuity. [Section 2.9]
09/27/05: Midterm test I. [Sections 1.3 and 2.1-8]
09/26/05: Review: solutions to the sample test. [Sections 1.3 and
2.1-8]
09/23/05: Derivatives: definition as the limit of the difference
ratio. Velocity, slope, and other rates of change as examples of a
derivative. Examples: Exercises 8, 18, and 29 from the text. [Section
2.8]
09/21/05: Tangents (Example: Problem 10), velocities (Example: Problem
19), and other rates of change (to study on your own). [Section
2.7]
09/19/05: The Intermediate Value Theorem. Limits at
infinity. Horizontal asymptotes. Skip the precise definitions (last
section of 2.6). [Sections 2.5 and 2.6]
09/16/05: The direct substitution property for some limits. An example
with the Squeeze Theorem. Continuity at a point and on an
interval. [Section 2.5]
09/14/05 (I am out of town, lecture given by another instructor): The
precise definition of a limit via epsilon-delta. Skip the precise
definitions of infinite limits. [Section 2.4]
09/12/05 (I am out of town, lecture given by another instructor):
Calculating limits. The limit laws. The Squeeze Theorem. [Section
2.3]
09/09/05: Limits of functions. Examples: 2^x, |x|+1, sin(pi/x)
(Example 4 on p. 96). Study on your own: One-sided limits. Infinite
limits. Vertical asymptotes. [Section 2.2]
09/07/05: Introduction. Shifting, stretching, and reflecting graphs of
functions. Example: 1 - 2 sqrt (x+3). Composition of functions (study
on your own). Velocity. Eaxmple: #39 on p. 178. [Sections 1.3 and
2.1]