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Ms. Roeske
AP
Calculus AB
Philosophy
Students will study four
major ideas during the year: limits, graphical analysis, derivatives,
and integration. Students will accomplish maximum results when they have
achieved an understanding of the basic calculus concepts. The focus for
the course is not to master mechanical manipulation of calculus
techniques but to understand the ideas and concepts which will provide
students with the background information necessary to use calculus in
application. When students grasp the concepts for an idea or theorem,
they can apply it in application. As the concepts are developed, so are
the mechanics that go along with the topics. I believe it is important
to maintain a high level of student expectation which enables students
to rise to that level of expectation. A teacher needs to have more
confidence in the students than they have in themselves.
Teaching
Strategies
During the first few weeks,
students are familiarized with their graphing calculators and calculator
manuals. The course teaches students how to use graphing calculators to
help solve problems, experiment, interpret results, and support
conclusions.
Students are taught the rule
of three: Ideas can be investigated analytically, graphically, and
numerically. Students are expected to relate the various representations
within the concepts of calculus. The course emphasizes the connections
among functions represented graphically, numerically, analytically, and
verbally.
It is important for students
to understand that graphs and tables are not sufficient to prove an
idea. Verification always requires an analytic argument. Each chapter
exam includes questions that involve only graphs or numerical data.
Communication is important
and is utilized in a variety of methods. Students will demonstrate
understanding verbally and in demonstration as student participation
includes student presentation of problems in the classroom. Students are
expected to explain problems using proper vocabulary and terminology.
Students explain solutions on the board to their classmates which
provides clarification of understanding and will assist the teacher if
students need extra help and which topics need additional
reinforcement. The course teaches students how to communicate
mathematics and explain solutions to problems verbally, analytically,
and in written sentences.
Students will understand the
derivative in terms of a rate of change and local linearity. In
addition, the representation of a limit of Riemann sums and the definite
integral. Students will also understand the relationship between the
derivative and the integral.
Cross curricular activities
between math, science, and psychology are coordinated using lab
activities, the Texas Instruments’ Calculator, and other science
equipment which provides students with a better understanding of the
concepts of calculus when they see concrete applications.
Much of calculus depends on an
understanding of a concept taught in a previous lesson. Students are
encouraged to form study groups and tutor themselves.
Calculator Ideas
The graphing calculator is used as a tool
to help students develop an intuitive feel for concepts and illustrate
ideas and topics before they are approached through typical algebraic
techniques. The following are required functionalities of graphing
technology:
1. Sketching a function in a specified
window
2. Finding a root
3. Finding points of intersection of
functions
4. Locating maxima and minima
5. Approximating the derivative at a
point using numerical methods
6. Approximating the value of a definite
integral using numerical methods
Activities
The following sample activities
demonstrate ways to help students gain an increased understanding of
calculus.
Numerical Limits
The “table” feature will allow students
to locate a numerical value for a limit.
For example, to find.
 
We view the values of the function from
x-values from -7.5 to -6.5 with an increment step of 0.1. At x=7 the
table records “error” or “not defined”. Students should observe that the
y-values follow a pattern. We then redo the process beginning the
x-values of the function from -7.03 to -6.97 with an increment step of
0.01. At x = 7 the table records “error” or “not defined.”
Students should observe the y-values are converging to 3. The process
can be repeated with smaller and smaller steps.
The limit can also be shown visually by
graphing the function in a window that has a pixel step of 0.1. Using
the trace function will produce no value at x = 7. Trace in either
direction of x = 7 shows the corresponding x- and y-coordinates,
but at x = 7, the y-coordinate disappears. It “reappears”
when the tracing continues at x = -7.010638 or -6.989362.
Students can see graphically that the y-coordinates cluster at
about -3 as x nears 7.
For comparison, do
the same exploration with
lim 
x  2
This function is also undefined at x
= 2, but the y-values do not converge as x approaches
2. Instead, the values explode, giving students a numerical look at
asymptotic behavior.
The Derivative of the Sine Function
(This activity works well on
an overhead display.)
Graph the function y = sin x
in a standard trigonometric viewing window. Estimate the slope of
the tangent line at various x-values and plot the slope values as
a function of x on the overhead screen. (The slope values are
clearly zero at the turning points and can be estimated to be +1 or -1
at the x-intercepts. A few more estimates will enable students to
guess the curve.) Students should see that the slope curve follows the
path of the cosine function. To test this conjecture, graph the
numerical derivative of the sine in the same window. Then graph the
cosine function and note that the two graphs are superimposed. Tracing
gives the same values on both curves. From this point it is easy to
proceed to an analytic proof.
AP Calculus AB Course Outline
Chapter 1: Precalculus Review (11-13
days)
1.1 Lines
1. Slope as rate of change
2. Parallel and perpendicular lines
3. Equations of lines
4. Applications
1.2 Functions and graphs
1. Functions
2. Domain and range
3. Families of function
4. Piecewise functions
5. Absolute value functions
6. Composition of functions
1.3 Exponential functions
1. Exponential growth and decay
2. Applications
1.4 Parametric equations
1. Relations
2. Circles
3. Ellipse
4. Lines and other curves
1.5 Functions and logarithms
1. Inverses
2. Logarithmic functions
3. Properties of logarithms
4. Applications
1.6 Trigonometric functions
1. Graphs of trigonometric functions
a. domain and range
2. Transformations of trigonometric
graphs
3. Inverse trigonometric functions
a. restricting the domain
Chapter 2: Limits and Continuity (11-13
days)
2.1 Rates of change and limits
1. Average and instantaneous speed
2. Definition of a limit
3. Properties of limits
4. One-sided and two-sided limits
5. Sandwich Theorem
2.2 Limits involving infinity
1. Finite limits and horizontal
asymptotic behavior
2. Infinite limits
3. End behavior
4. Properties of limits
5. Visualizing limits to infinity
2.3 Continuity
1. Continuous functions
2. Discontinuous functions
a. Removable discontinuity
b. Jump discontinuity
c. Infinite discontinuity
2.4 Rates of change and tangent lines
1. Average rates of change
2. Tangent to a curve
3. Slope of a curve
4. Normal to a curve
5. Instantaneous rates of change
Chapter 3: Derivatives (31-33 days)
3.1 Derivative of a function
1. Definition of the derivative and
notation
2. Relationship between the graphs of
f and f′
3. Graphing the derivative from data
4. One-sided derivatives
3.2 Differentiability
1. How derivatives fail to exist
2. Local linearity
3. Numeric derivatives using the
calculator
3. Differentiability and continuity
4. Intermediate Value Theorem for
derivatives
3.3 Rules for differentiation
1. Constants
2. Sum and difference
3. Powers
2. Product
3. Quotient
4. Higher order derivatives
3.4 Velocity and other rates of change
1. Instantaneous rates of change and
velocity
2. Motion along a line; speed, velocity,
and acceleration
3. Particles in motion
3.5 Derivatives of trigonometric
functions
1. Derivatives of trigonometric functions
2. Harmonic motion
3. Jerk
3.6 Chain rule
1. Derivative of a composite function
2. Slopes of parameterized curves
3. Power chain rule
3.7 Implicit differentiation
1. Implicitly defined functions
2. Tangent and normal lines
3. Higher order derivatives
4. Rational powers of differentiable
functions
3.8 Derivatives of inverse trigonometric
functions
1. Derivatives of inverse functions
3.9 Derivatives of exponential and
logarithmic functions
1. Derivatives involving natural and
other base functions
Chapter 4: Applications of the Derivative
(25-27 days)
4.1 Extreme values of functions
1. Local (relative) extrema
2. Absolute (global) extrema
3. Finding extreme values
4.2 Mean Value Theorem
1. Physical interpretation
2. Increasing and decreasing functions
3. Velocity and position
4.3 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
4.4 Modeling and optimization
1. Examples and models
4.5 Linearization and Newton’s method
1. Linear approximations and local
linearity
2. Differentials and change
4.6 Related rates
1. Equations
2. Related motion
3. Strategies
Chapter 5: The Definite Integral (26-28
days)
5.1Estimating the finite sums
1. Distance traveled
2. Rectangular Approximations
2. Volume of a sphere
3. Cardiac output
5.2 Definite Integrals
1. Riemann sums
2. Integral notation and terminology
3. Definite integrals and area
4. Integrals on the calculator
5. Discontinuous integrable functions
5.3 Definite integrals and
antiderivatives
1. Properties of the definite integral
2. Average Value Theorem
3. Mean Value Theorem
4. Connecting differentiable and
integrable calculus
5.4 The Fundamental Theorem of Calculus
1. Fundamental Theorem, part 1
2. Graphing
3. Fundamental Theorem, part 2
4. Graphically analyzing antiderivatives
5.5 Trapezoidal rule
1. Approximations
2. Simpson’s rule
Chapter 6: Differential Equations and
Mathematical Modeling (20-22 days)
6.1 Slope fields and Euler’s method
1. Differential equations
2. Slope fields
3. Euler’s method
6.2 Antidifferentiation and substitution
1. Properties of indefinite integrals
2. Notation using u-substitution
3. Substitution
6.4 Exponential growth and decay
1. Separable differential equations
1. Growth and decay
Chapter 7: Applications of Definite
Integrals (21-23 days)
7.1 Integral as net change
1. Summing rates of change
2. Particle motion
3. Strategy
4. Consumption
5. Change
6. Work
7.2 Areas in the plane
1. Areas between curves
2. Area enclosed by intersecting curves
3. Changing functions
4. Integration with respect to y
5. Geometric formulas
7.3 Volumes
1. Volumes of solids with known cross
sections
2. Volumes of solids of revolution
a. Disk method
b. Shell method
This timeline is based on a school year
starting after Labor Day, with approximately 160 teaching days before
the Advanced Placement exam. This timeline gives approximately 15 days
to “review” the course before the exam.
The course teaches all topics associated
with Functions, Graphs, and Limits; Derivatives; and Integrals as
delineated in the Calculus AB Topic Outline in the AP Calculus Course
Description.
Major Text
Finney, Ross L., Franklin D. Demana, Bert
K. Waits, and Daniel Kennedy. Calculus—Graphical, Numerical,
Algebraic. 3rd ed. Pearson Education, Inc., Pearson Prentice Hall,
Boston, Massachusetts, 2007
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