AP CALCULUS AB
Course Topics
UNIT 1: LIMITS AND THEIR PROPERTIES
1) Compute limits algebraically, numerically, graphically
2) Compute limits of piecewise functions
3) One-sided limits
4) Continuity and the Intermediate Value Theorem
5) Trigonometric limits
6) Limit Theorems
7) Limits involving infinity—algebraically, numerically, graphically
8) Removable and non-removable discontinuities—holes, vertical asymptotes
UNIT 2: DIFFERENTIATION
1) Secant line to tangent line
2) Definition of derivative with the limit of the change in x approaching 0, and with the limit of x approaching c
3) Derivative notation
4) Derivatives of polynomials
5) nDeriv with the use of the graphing calculator
6) Second and higher order derivatives
7) Derivative Rules—constant, power, constant multiple, sum/difference, product, quotient
8) Chain Rule—Leibniz and function notation
9) Rates of Change—average vs. instantaneous
10) Rates of Change algebraically, numerically, graphically
11) Position-Velocity-Acceleration
12) Derivatives of the trigonometric functions
13) Implicit Differentiation
UNIT 3: APPLICATIONS OF DIFFERENTIATION
1) Extrema on a closed interval
2) Critical numbers and increasing/decreasing intervals of a function
3) Rolle’s Theorem and the Mean Value Theorem
4) First Derivative Test to identify relative extrema
5) Inflection and concave up/concave down intervals of a function
6) Second Derivative Test to identify relative extrema
7) Curve sketching with the use of first and second derivative, including continuity vs. differentiability
8) Connection between the graphs of the original function, the original function’s first derivative, and the original function’s second derivative
9) Solve optimization problems
10) Related Rates
11) Approximate
a zero of a function with
12) Definition of differentials, local linearity, and Leibniz notation
UNIT 4: INTEGRATION
1) Antiderivative and indefinite integration
2) Basic integration rules to find antiderivatives
3) Pattern recognition to evaluate indefinite integrals
4) Change of variables to evaluate indefinite integrals
5) Antidifferentiation to find velocity and position functions
6) Solve differential equations by separation of variables
7) Particular solutions from initial conditions
8) fnint with the use of the graphing calculator
9) Riemann sums—left-hand, right-hand, midpoint
10) Use Riemann sums to approximate definite integrals of functions that are represented graphically and by tables of data
11) Algebra of definite integrals
12) Average value of a function
13) The integral as an accumulator function
14) Approximating the definite integral using the Trapezoidal and Simpson’s rule
UNIT 5: LOGARITHMIC, EXPONENTIAL, AND OTHER TRANSCENDENTAL FUNCTIONS
1) Derivative of functions involving common and natural logarithmic functions
2) Integration and the natural logarithmic function
3) Integration of trigonometric functions
4) Derivative of exponential functions, base e and other bases
5) Integration of exponential functions, base e and other bases
6) Solving differential equations by separation of variables, to include solutions involving the natural logarithmic function
7) Applications of differential equations in exponential growth and decay problems
8) Drawing slope fields and solution curves for differential equations
9) Use of slope fields to interpret a differential equation graphically
10) Using Euler’s Method to numerically approximate the particular solution to a differential equation
11) Derivatives of inverse trigonometric function
12) Integrate functions whose antiderivatives involve inverse trigonometric functions
UNIT 6: APPLICATIONS OF INTEGRATION
1) Area of a region between two curves
2) Find the volume of a solid of revolution—disk, washer, shell methods
3) Find the volume of a solid with known cross-section
4) Application involving total distance traveled vs. net change of position
5) Second Fundamental Theorem of Calculus—algebraic and graphical approaches
6) Connections of the Second Fundamental Theorem of Calculus with—Extreme Value Theorem, first and second derivative tests, average rate of change, position-velocity-acceleration of motion along a line
7) Find the arc length of a smooth curve
8) Find the area of a surface of revolution
UNIT 7: INTEGRATION TECHNIQUES, L’HOPITAL’S RULE, AND IMPROPER INTEGRALS
1) Find an Antiderivative using integration by parts, including the tabular method
2) Use L’Hopital’s Rule to determine limits
Teaching Strategies
Rule of Four
Students are presented with problems in a variety of ways:
analytical, graphical, numerical, and verbal.
- Students must be able to justify their work using proper symbolism and complete sentences where appropriate. Precise vocabulary and notation is stressed throughout the course. On tests, the students’ work is always evaluated in addition to the final answer.
- Students are expected to share both their results and their thought process with their classmates. This verbal exchange occurs on a daily basis and allows each student an opportunity to be a mentor to a peer. After the students have shared their thought process, they are then given time to formally write a justification.
- The concept of the limit is developed graphically and
numerically before the analytic techniques for evaluating a limit are
introduced.
- The definition of the derivative is introduced
graphically. From the limit definition of the derivative, the students
must first perform algebraic techniques to obtain the limit. Then the
students are presented with the analytic techniques (Rules) for
determining the derivative.
- Related rates, position-velocity-acceleration, optimization, average value of a function are all topics presented verbally.
- The indefinite integral is developed from a
graphical/analytical relationship (Slope fields and differential
equations) between the derivative of a function and the family of original
functions.
Rigor
Throughout the course, the students are required to understand the theory and logical underpinnings of the Calculus.
- Students must be able to find both
by the definition for appropriate functions. - Students must be able to work with the definite integral as the limit of a sequence of Riemann Sums.
- Students must be able to state completely important theorems such as Rolle’s Theorem, the Mean Value Theorem, and both parts of the Fundamental Theorem of Calculus.
Graphing Calculator
The use of the TI-83 graphing calculator enhances the development of visual understanding of calculus. Our students use the calculator on a daily basis.
Some of the graphing calculator capabilities used on a regular basis include:
- plotting graph of a function
- finding zeros, max/min points, intersections
- calculating the derivative of a function numerically, both on the home screen and the graph
- calculating the value of a definite integral numerically, both on the home screen and the graph
We also use programs on the graphing calculator which include:
- Riemann sums
- Rectangle – left, right, midpoint
- Trapezoid
- Slopefields
We feel that it is important and good preparation for the AP test, for students to work problems both by hand and using a graphing calculator. Class time is spent discussing the type of questions they must know how to work without a calculator and also with efficient use of a calculator. Tests are also written in both formats.
AP Review
Throughout the course, class examples, homework, and tests include problems from released AP exams. Prior to the AP exam, 3 ½ weeks is devoted to review. During this time, students work both Multiple Choice and Free Response items individually and cooperatively. Students are given an opportunity to evaluate their work to Free Response questions according to the 9-point scale used by the AP graders.
REFERENCES AND
MATERIALS
MAJOR TEXT
Larson, Ron, Robert P. Hostetler, Bruce H. Edwards. Calculus of a Single Variable. 7th ed. Houghton Mifflin Company, 2002
GRAPHING CALCULATOR