Practice Exercises
3 Differentiation
A. Definition of Derivative
B. Formulas
C. The Chain Rule; the Derivative of a Composite Function
D. Differentiability and Continuity
E. Estimating a Derivative
E1. Numerically
E2. Graphically
F. Derivatives of Parametrically Defined Functions BC ONLY
G. Implicit Differentiation BC ONLY
H. Derivative of the Inverse of a Function
I. The Mean Value Theorem
J. Indeterminate Forms and L’Hôpital’s Rule BC ONLY
K. Recognizing a Given Limit as a Derivative BC ONLY
Practice Exercises
4 Applications of Differential Calculus
A. Slope; Critical Points
B. Tangents and Normals
C. Increasing and Decreasing Functions
Case I. Functions with Continuous Derivatives
Case II. Functions Whose Derivatives Have Discontinuities
D. Maximum, Minimum, and Inflection Points: Definitions
E. Maximum, Minimum, and Inflection Points: Curve Sketching
Case I. Functions That Are Everywhere Differentiable
Case II. Functions Whose Derivatives May Not Exist Everywhere
F. Global Maximum or Minimum
Case I. Differentiable Functions
Case II. Functions That Are Not Everywhere Differentiable
G. Further Aids in Sketching
H. Optimization: Problems Involving Maxima and Minima
I. Relating a Function and Its Derivatives Graphically
J. Motion Along a Line
K. Motion Along a Curve: Velocity and Acceleration Vectors BC ONLY
L. Tangent-Line Approximations
M. Related Rates
N. Slope of a Polar Curve BC ONLY
Practice Exercises
