Unit 3: The Derivative

AP Calculus BC Course Outline
Unit 1: Precalculus Review
Unit 2: Limits and Continuity
Unit 3: The Derivative
A. Definition of the derivative
B. Relationships between the graphs of f and f’
C. Differentiability
1. Local linearity
2. How f’ might fail to exist
3. Numeric derivatives using the calculator
4. Differentiability and continuity
D. Derivatives of algebraic functions
E. Velocity and other rates of change
F. Derivative rules when combining functions
G. Applications to velocity and acceleration
H. Derivatives of trigonometric functions
I. The chain rule
J. Implicit derivatives
1. Differential method
2. y' method
K. Derivatives of inverse trigonometric functions
L. Derivatives of logarithmic and exponential functions
Unit 4: Applications of the Derivative
A. Extreme values
1. Local (relative) extrema
2. Global (absolute) extrema
B. Using the derivative
1. Mean value theorem
2. Increasing and decreasing functions
3. Antiderivatives
C. Analysis of graphs using the first and second derivatives
1. Critical values
2. First derivative test for extrema
3. Concavity and points of inflection
4. Second derivative test for extrema
5. Graphing f given f’ and/or f”
D. Optimization problems
E. Linearization models
1. linear approximations
2. Newton’s method (By hand and through the use of calculator program)
3. Differentials
F. Related rates
Unit 5: The Definite Integral
A. Approximating areas
1. Riemann sums
2. Rectangular Approximation Methods (By hand and through the use of a calculator
3. Trapezoidal rule (By hand and through the use of a calculator program)
B. Definite integrals and antiderivatives
1. Average Value Theorem
2. Mean Value Theorem
3. Definite integrals using the calculator
C. The Fundamental Theorem of Calculus (part 1)
D. The Fundamental Theorem of Calculus (part 2)
Unit 6: Differential Equations and Mathematical Modeling
Slope field’s and Euler’s Method (By hand and through the use of a calculator program)
Differential equations
Integration using u-substitution
Integration by parts
Separable differential equations
1. Growth and decay
2. Slope fields (By hand and through the use of a calculator program)
3. General differential equations
Unit 7: Applications of Definite Integrals
A. Integral as Net Change
B. Areas in the plane
1. Area between curves
2. Area enclosed by intersecting curves
3. Areas with boundaries
4. Integrating with respect to “y”
5. Integrating using geometry formulas
C. Volumes
D. Volumes of solids with known cross sections.
E. Volumes of solids of revolution
F. Disk method
G. Shell method
Unit 8: Advanced Topics
L’Hopital’s Rule
Relative Rates of Growth
Improper Integrals
Partial Fractions and Integral Tables
Arc Length
Conic Sections
Unit 9: Series
A. sequences
B. Power series (geometric series, representing a function, differentiation and integration of
C. Taylor and MacLaurin series (general and ex, sin x and cos x)
D. Manipulation of series
E. Taylor polynomials and Lagrange error bound
F. Radius of convergence (nth term test, comparison test, ratio test)
G. Interval of convergence (integral test, p-series, alternating series)
Unit 10: Polar, Parametric and Vectors
Parametric Functions
Vectors in the Plane
Vector-valued Functions
Modeling Projectile Motion
Polar Coordinates and Polar graphs
Calculus of Polar Curves