AP CALCULUS AB OUTLINE
1. Functions (approximately 3 weeks)
a.
Analyze functions and relations with respect to intercepts, slope, and symmetry
b.
Find the domain and range of a function and write in interval notation
c.
Write functions in piecewise form
d.
Graph functions using the calculator
e.
Function Transformations
i.
Write functions as compositions
ii.
Use a calculator to discover impact of changes to a graph
iii.
Describe transformations and make graph without a calculator
f.
Evaluate inverses of polynomial and trigonometric functions
g.
Use exponential and logarithmic functions
h.
Use parametric equations and explore these using the calculator
2. Limits and Continuity (approximately 3 weeks)
a.
Demonstrate an intuitive understanding of limits
b.
Explain the concept of a limit using the definition and using the calculator with charts and graphs
c.
Evaluate finite limits
d.
Understand the relationship between one and two-sided limits
e.
Demonstrate an intuitive understanding of continuity and the relationship between continuity and limits
f.
Evaluate limits to determine asymptotes
g.
Understand asymptotes as related to graphs
h.
Evaluate infinite limits and apply the concept of end-behavior models
i.
Decide if a function is continuous at a point
j.
Understand the graphs of continuous functions as related to the Intermediate Value Theorem and
Extreme Value Theorem
k.
Understand the continuity of trigonometric and inverse trigonometric functions
l.
Obtain limits of trigonometric functions using the squeezing theorem
3. Derivatives (approximately 5 weeks)
a.
Understand the relationship between and compute the slope of the tangent line and instantaneous
speed
b.
Understand the relationship between and compute the slope of the secant line and average velocity
c.
Find derivatives of a function using the definition
d.
Understand the relationship between differentiability and continuity to decide when a function is and is
not differentiable
4.
e.
Use the definition to decide differentiability
f.
Use appropriate symbolism to express the concept of a derivative
g.
Use the graph of a function to sketch the graph of its derivative
h.
Use NDERIV function of the calculator to evaluate and graph derivatives
i.
Apply rules of differentiation to functions to find the first and second derivatives
j.
Differentiate trigonometric and inverse trigonometric functions
k.
Use implicit differentiation and logarithmic differentiation
l.
Use linearization and differentials to estimate function values
m.
Differentiate logarithms and exponentials
n.
Use L’Hopital’s Rule to determine limits of indeterminate form
Applications of Derivatives (approximately 4 weeks)
a.
Explain the Mean-Value Theorem and Rolle’s Theorem
b.
Find local maxima and minima using the first derivative
c.
Use the first derivative to find increasing and decreasing intervals
d.
Use the second derivative to find inflection points and determine concavity
e.
Use first and second derivative tests for local extrema
f.
Find the absolute extrema on closed and open intervals
g.
Use the graphing calculator to find the local and absolute extrema using the trace key, calc key and the
function’s table
h.
Use Newton’s method to estimate zeros of functions
i.
Apply first and second derivatives to find maximum and minimum values for optimization problems
j.
Use Implicit differentiation to solve related rates problems
k.
Use derivatives in applications of slope velocity, acceleration, marginal cost, etc.
l.
5.
Sketch the graph of a function with and without the calculator using the first and second derivatives
Integrations (approximately 5 weeks)
a.
Use slope fields to sketch solution curves
b.
Apply basic rules for evaluating a definite integral
c.
Apply Riemann Sums and Trapezoidal Rule to estimate area under a curve
d.
Make connection between a function and its antiderivative with respect to integrals using the
Fundamental Theorem of Calculus
6.
e.
Use the calculator to graph an integral and evaluate a definite integral
f.
Use the antidifferentiation to find derivatives of basic functions
g.
Use the substitution method to evaluate integrals involving composite functions
h.
Find the area bounded by a relation and an axis
Applications (approximately 3 weeks)
a.
Find area between two curves
b.
Find the volume of a solid using the disk/washer method
c.
Find the volume using shell method
d.
Find the length of an arc
e.
Use integration to find position and velocity
f.
Solve separable differential equations
g.
Solve various application problems by setting up an approximating Riemann Sum and using its limit as a
definite integral
7.
Techniques of integration (approximately 5 weeks)
a.
Use the substitution method to find integrals involving trigonometric functions
b.
Use integration by parts
c.
Integrate integrals involving trigonometric functions
d.
Use trigonometric substitution
e.
Use partial fractions