Topics for Exam III

• Material from Exams I and II
  • Interpretations of the derivative
    • slope of tangent line
    • instantaneous rate of change
  • Rules for computing derivatives

• 4.2 Monotinicity and Concavity
  • The sign of f '(x) tells you whether the graph of f is increasing or decreasing at x
  • The sign of f ''(x) tells you whether the graph of f is concave up or concave down at x
  • Additional topic not emphasized in book: Sketching a graph of f' from a given graph of f, and vice versa

• 4.3 Extrema, Inflection Points and Graphing
  • Finding extrema by checking critical points and endpoints
  • First (p. 178) and Second (p. 180) Derivative Tests for local minima and maxima
  • Finding inflection points
  • Asymptotes (but not oblique asymptotes)

• 4.4 Optimization

• 4.5 L'Hopital's Rule
  • With indeterminate forms 0/0 and infinity/infinity you can use L'Hopital's Rule directly.
  • If you have 0*infinity first rewrite the product as a quotient.
  • For infinity - infinity rewrite the difference as a product or quotient (e.g. by finding a common denominator or multiplying by the conjugate radical).
  • For 0^0, 1^(infinity), or (infinity)^0 use logarithms to rewrite as a limit involving one of the other indeterminate forms.

• 4.6 Numerical Methods
  • Discrete Dynamical Systems and Fixed Points
    • Iteration
    • Finding fixed points (equilibria)
    • Cobweb plots
    • Determining the stability of fixed points
      • If P is a fixed point and |f '(P)| < 1 then P is attracting (stable).
      • If P is a fixed point and |f '(P)| > 1 then P is repelling (unstable).
      • If P is a fixed point and |f '(P)| = 1 then the test tells us nothing.