Topics for Exam I

• 1.3 Graphing
  • Transformations of functions
    • vertical shift
    • horizontal shift
    • vertical stretch or shrink
    • reflection across x- or y-axis
  • Log-log and log-linear plots: read, draw, and find the relation between y and x

• 2.1 Limits
  • Informal definition
  • Finding limits from a graph or table
  • When limits don't exist
    • function values approach positive or negative infinity
    • left- and right-hand limits don't agree
    • function oscillates, for example y = Sin[1/x] near x = 0
  • Vertical and horizontal asymptotes
  • Limit laws

• 2.2 The Sandwich Theorem and Some Trigonometric Limits
  • using the Sandwich Theorem
  • Limit[Sin[x]/x, x -> 0] = 1
  • Limit[(1 - Cos[x])/x, x -> 0] = 0

• 2.3 Limits at Infinity
  • Limits of rational functions
  • Limit[e^(-x), x -> infinity] = 0
  • Arithmetic with infinity

• 2.4 Continuity
  • Informal idea: no holes, jumps, or vertical asymptotes
  • Formal definition: Limit[f(x), x -> c] = f(c)
  • Combining continuous functions
  • Classes of functions that are continuous where defined
  • Finding the domain of a function

• 2.5 Theorems about Continuous Functions
  • Extreme Value Theorem
  • Intermediate Value Theorem
  • Using the Bisection Method to find a root of an equation

• 3.1 Formal Definition of the Derivative
  • Informal idea:
    • slope of tangent line
    • instantaneous rate of change
  • Estimating derivatives using a graph
  • Estimating derivatives using the difference quotient
  • Formal definition
  • Computing derivatives using the formal definition
  • When the derivative doesn't exist at a point
    • function not continuous
    • function has a cusp (i.e. corner)
    • function has a vertical tangent
  • The derivative as a function

• 3.2 The Power Rule, the Basic Rules of Differentiation, and the Derivatives of Polynomials
  • The Power Rule: (d/dx)(x^n) = nx^(n-1)
  • (d/dx)(c) = 0 (i.e. the derivative of a constant function is 0)
  • Constant Multiple Rule: (d/dx)(cf(x)) = cf'(x)
  • Sum Rule: (d/dx)(f(x) + g(x)) = f'(x) + g'(x)
  • Derivatives of polynomials

• Some standard models (IMPORTANT NOTE: you do not need to memorize anything about these models - if any model appears on the exam you will be given the specifics)
  • Monod growth
  • Exponential growth and decay
  • Logistic growth
  • Reaction rates