Unit 5 Summary (College Board AP® Calculus AB): Study Guide

Analytical applications of differentiation summary

Key definitions

• An extremum (plural extrema) is a maximum or minimum point of a function

  • A global extremum is the max/min over the function's whole domain

  • A local (relative) extremum is the max/min over some open interval around the point

• If at a point, then is increasing at that point

• If at a point, then is decreasing at that point

• A critical point of a function is a point where or does not exist (provided is defined there)

• If at a point, then is concave up at that point

• If at a point, then is concave down at that point

• A point of inflection is a point where the concavity of the graph changes

Key theorems

• The mean value theorem states that if is continuous on and differentiable on , then there exists a in such that

  • 

• Rolle's theorem is the special case of MVT when , guaranteeing a with

• The extreme value theorem states that if is continuous on , then has at least one global maximum and one global minimum on

Key facts

• The first derivative test classifies a critical point (where ) by checking the sign of on either side:

  • positive → negative: local maximum

  • negative → positive: local minimum

  • no sign change: point of inflection

• The second derivative test classifies a critical point (where ) by checking the sign of

  • → local minimum

  • → local maximum

  • → test inconclusive (use first derivative test)

• The candidates test finds global extrema on a closed interval by comparing the values of at all critical points and at the endpoints

  • The largest is the global max

  • The smallest is the global min

• All local extrema occur at critical points, but not all critical points are local extrema (some are points of inflection)

• For an implicitly-defined curve, the tangent is

  • horizontal where

  • vertical where