AP Calculus AB Syllabus
Course Description:
This is a college level course covering derivatives, integrals, limits, applications and
modeling of these topics.
Course Objectives:
The objectives of this course are:
To work with functions represented in a variety of ways: graphically, analytically,
numerically. They should understand the relationship of each way.
To understand the meaning of the derivative in terms of a rate of change and local
linear approximation and should be able to use derivatives to solve several problems
To understand the meaning of a definite integral as both a limit of a Riemann
sums and the net accumulation of a rate of change and use integrals to solve several
problems
To know the relationship between a derivative and a definite integral dealing with
the Fundamental Theorem of Calculus.
To be able to use a graphing calculator to plot a graph of a function within an
arbitrary viewing window, find zeros of a function, numerical calculate the derivative of
a function, and numerically calculate the value of a definite integral.
Course Expectations:
To inquire about material and concepts that may not be clear
To strive to apply knowledge from previous courses in a new way
To conceptualize the calculus curriculum and apply knowledge
To communicate mathematics both orally and in well written sentences.
Teaching Strategies Some examples are imbedded in the curriculum using parenthesis.
Students must write out solutions to application problems with logically explanations to
this. They must interpret graphs and tables. Use mirroring techniques that are given in the
AP released calculus exams to show how to communicate mathematically and what an
acceptable answer looks like. Test and quizzes are given over skills and applications.
Students are assigned problems in their text, pre-made handouts, calculator labs, board
problems, and released items. Overheard projector attached to a computer is used to show
graphical representation, especially in area of revolution to create volume and solids of a
known cross section.
Course Outline:
I.
Functions and Graphs: A Precalculus Review (approximately 3 weeks)
a. Functions & function notation
i. Definition of function
ii. Vertical line test
iii. Function notation
iv. Finding input and output values
v. Domain and range
b. Absolute Value & Piecewise Defined Functions
i. Piecewise defined functions
ii. The absolute value function
iii. Solving equations involving absolute value
c. Inequalities
i. Solving inequalities of all types
ii. Writing situations and applying inequalities
d. Composition & Combination of Functions
i. Composition of functions
ii. Inverse functions
iii. Non-invertible functions
iv. Combinations of functions
e. Transformation of Functions
i. Vertical and horizontal shifts
ii. Reflections and symmetry
iii. Vertical stretches and compressions
iv. Horizontal stretches and compressions
v. Combining transformations
f. Trigonometric Functions
i. The graph of cosx, sinx, tranx, cotx, secx, cscx
ii. Shifts and distortions of trigonometric functions
iii. Trigonometric identities
iv. Inverse trigonometric functions
v. Solving trigonometric equations
g. Culminating Presentation from Precalculus review
II.
Limits and Continuity (approximately 3 weeks)
a. Intuitive definition of a limit
i. Definition
ii. Using table and graphs to find limits (students use a table in their
calculator to approach a limit numerically)
b. Algebraic techniques for finding limits
i. Calculating limits using limit laws
ii. Direct substitution properly
iii. Indeterminate forms
c. One-Sided limits
i. Definition
ii. Finding one-sided limits
iii. Existence of limits
d. Infinite limits
i. Definition
ii. Vertical asymptotes
e. Limits at Infinity
i. End-behavior of functions and horizontal asymptotes
ii. Limit laws for limits at infinity
iii. Oblique asymptotes
f. Limits of special trigonometric functions
i. Special limits involving the sine function ( evaluating sine x as is it
approaches zero graphically and comparing it to y=x at that time
students can see the limit)
ii. Special limits involving the cosine function
g. Continuity
i. Continuity at a point
ii. Continuity on a closed interval
iii. Continuity on an open interval
iv. Discontinuous functions
a. removable discontinuity
b. Jump discontinuity
c. Infinite discontinuity
v. Intermediate Value Theorem
III.
Derivatives (approximately 5 weeks)
a. Definition of the Derivative
i. The derivative as the slope of a tangent (limit of difference
quotient)
ii. The derivative as the rate of change
iii. The derivative as a function
b. Differentiation Rules
i. Constant rule, constant multiple rule
ii. Power rule
iii. Sum rule, difference rule
iv. Product rule
v. Quotient rule
vi. Trigonometric rules
vii. Chain rule
viii. Higher order derivatives
c. Differentiability & Continuity
i. Differentiability implies continuity
ii. Non-differentiable functions
iii. Local linearity
iv. Numerical derivatives with a calculator
d. Application as applied to certain concepts
i. Position function
ii. Velocity function
iii. Acceleration function
e. rectilinear motion
i. position, velocity, acceleration (analyzing signs of velocity and
acceleration)
ii. turning points, coasting, speeding up slowing down
iii. graphing in parametric mode
IV.
Application of the Derivative (approximately 5 weeks)
a. Tangent and Normal Lines
b. Related Rates
i. Sample Problems
c. Relative Extrema and the First Derivative Test
i. Theorem on increasing and decreasing
ii. First derivative test for extrema: critical points
d. Concavity and the Second Derivative Test
i. Definition of concavity
ii. Theorem: test for concavity
iii. Definition of point of reflection
iv. Second derivative test for extrema
v. Curve sketching ( students apply what they know and then use
written sentences to explain how to back them selves up that their
curve is correct, students also present this to the class verbally,
they evaluate the functions analytically, demonstrate them
graphically and test some components numerically)
e. Absolute Extrema and Optimization
i. Extreme value theorem
ii. Sample problems with applications (students form conclusions and
use support such as sign change with derivatives)
f. Rolle’s and the Mean Value Theorems
i. Rolle’s Theorem
ii. Mean Value Theorem
g. Differentials
V.
Antidervatives and Definite Integrals (approximately 5 weeks)
a. Differential equations and slope fields ( students discover the parent
functions and see the beginnings of an antiderivative.
b. Antiderivaties
i. Definitions of an antiderivative
ii. Theorem on antiderivative
iii. Notation
iv. Vocabulary
v. Constant multiple rule
vi. Sum and difference rule
vii. Trigonometric rules
viii. Power rule
c. Substitution of Variable
i. Simple substitution
ii. Trigonometric integrals
d. Antiderivatives on Inverse Trigonometric Functions
e. Trigonometric Substitutions
VI.
Application of Antderivatives and Definite Integrals
a. Definitions of the definite integral
i. Riemann sums: left, right and midpoint
ii. Definition of the definite integral; notation
iii. Theorem: Trapezoidal Rule
iv. Theorem: Area between curves ( the students explain what this
represents both verbally and written form)
b. Fundamental theorem of calculus
i. Integral defined functions
ii. Four properties of the definite integral
iii. Property of even and odd functions
iv. Variable bounds property
v. Average-value property
vi. Integrating rate of change
(use a calculator to find a definite integral, getting the number of
people in an amusement park at a given time, or amount of sand on
a beach, here students use written sentences to explain the solution.
They also use their calculator to get their values in which to
integrate from.
c. Analyzing curves with Antiderivatives
i. Position, velocity and acceleration
d. Volume
i. Vertical axis of rotation (disc/washer)
ii. Horizontal axis of rotation (disc/washer)
iii. Solids with known cross sections
(students find volumes of irregular shapes both analytically and with
a calculator)
VII.
Exponential and Logarithmic Functions (approximately 5 weeks)
a. The family of exponential functions
b. The number e
c. Logarithmic differentiation
d. Antiderivatives and the Natural log
e. Antiderivative of other Exponentials and Logs
f. Differential equations with dy/dt=ky
VIII.
IX.
X.
AP exam review (use released items and ABCD cd ROM)
Final project ( take different vases, plot curvature, store in calculator, use
regression, find volume, fill vase with water, measure volume, find percent
error, analyze error, and write a conclusion)
Teacher resourses
a. primary text book - Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards.
Calculus I with Precalculus. Boston: Houghton Mifflin.
b. Graphing calculator labs for students ---Benita Albert and Phyllis Hillis
c. AP Calculus Resource Notebook from AP Calculus Summer Institute – with Phyllis Hillis
d. Web sites-- AP Central, Visual Calculus, and Calculus graphics
e. ABCD Calculus Review CD ROM
f. The College Board AP advanced placement program course description –
Calculus AB, Calculus BC
g. Released exams 1998 AP Calculus AB and Calculus BC
h. All released exams on AP Central Calculus