Maywood Academy High School
Mathematics Department
Mrs. C. Van Room:204D
AP Calculus Course Outline
Prerequisites for Calculus
1. Coordinates and Graphs in the plane
Discussion Points: Difference between a pixel and a point
Student Expectations: Find a parabola’s vertex graphically, numerically and algebraically.
2. Slope and Equation for Lines
Discussion Points: Meaning of Slope
3. Relations, Functions, and Their Graphs
Discussion Points: Support symmetry graphically, numerically and algebraically
Distinguish properties of functions from properties of graphs
Establish “Calculus Library of Basic Functions”
Student Expectations: Sketch each function without the aid of graphing technology.
Determine the domain and range of a function from a sketch of the graph.
4. Geometric Transformations: Shifts, Reflections, Stretches, and Shrinks
Student Expectations: Given a graph of a function and a constant , sketch each of: .
5. Solving Equations and Inequalities Graphically
Discussion Points: Examples which can be solved graphically, but not algebraically
6. Relations, Functions, and Their Inverses
7. A Review of Trigonometric Functions
Student Expectations: Know sum and difference formulas for sine and cosine.
Be able to derive double- and half-angle formulas.
Limits and Continuity
1. Limits
Student Expectation: Find limits algebraically.
Use limit notation correctly.
2. Continuous Functions
Student Expectations: Know definition of continuous functions.
Know properties of continuous functions.
Explain continuity at a point in terms of limits.
Identify those functions from “Library” which are continuous and
those which are not.
3. The Squeeze Theorem and
Student Expectations: Explain the Squeeze Theorem graphically and numerically.
4. Limits involving Infinity
Student Expectations: Find horizontal and vertical asymptotes using limits.
Describe asymptotes in terms of graphs and limits.
5. Controlling Function Outputs: Target Values
Derivatives
1. Slopes, Tangent Lines, and Derivatives
Student Expectations: Be able to explain and derive formula for .
Explain relationship between differentiability and continuity.
2. Numerical Derivatives
Student Expectations: Estimate derivatives of at points from a graph.
Use numerical derivative capabilities of a calculator.
3. Differentiation Rules
Student Expectations: Be able to derive rule for derivative of a positive integer.
Estimate value of second derivative numerically.
4. Velocity, Speed, and Other Rates of Change
Student Expectations: Explain difference between average and instantaneous rate of change.
Approximate rates of change from graphs and tables.
Sketch graph of given the graph of .
5. Derivatives of Trigonometric Functions
Student Expectations: Be able to derive formula for derivative of .
6. The Chain Rule
7. Implicit Differentiation and Fractional Powers
(Note to teacher: Supplementing of textbook is necessary to present students wit the use of implicit differentiation to find the derivative of an inverse function.)
8. Linear Approximation and Differentials
Student Expectations: Use the tangent line as a local linear approximation.
Be able to illustrate graphically the difference between
Be able to use
Applications for Derivatives
1. Maxima, Minima, and the Mean Value Theorem
Student Expectations: Be able to explain how to find critical points and extreme values.
Be able to state and apply Mean Value Theorem.
2. Predicting the behavior of functions.
Student Expectation: Be able to sketch a graph of , given characteristics of
Be able to sketch a graph of from graph of .
3. Polynomial Functions, and Optimization.
4. Rational Functions and Economics Applications
5. Rational and Transcendental Functions
6. Related Rates of Change
7. Antiderivatives, Initial Value Problems, and Mathematical Modeling
FIRST SEMESTER EXAM
Integration
1. Calculus and Area
Student Expectations: Sketch diagrams for Left, Right, and Midpoint Rectangular Approximation Methods.
Calculate LH endpoints, RH endpoints, and MP with equal subdivisions.
2. Definite Integrals
Student Expectations: Be able to define integral as limit of Riemann sum.
3. Definite Integrals and Antiderivatives
4. The Fundamental Theorem of Calculus
5. Indefinite Integrals
6. Integration by substitution
7. Numerical Integration: Trapezoidal Rule
Applications of Definite Integrals
1. Areas between curves
Student Expectations: Write appropriate Riemann sum and set up definite integral.
2. Volumes of Solids of Revolution – Disks and Washers
Student Expectations: Be able to sketch 3-D figures illustrating disks.
3. Cylindrical Shells – An Alternative to Washers
Student Expectations: Be able to sketch 3-D figures illustrating cylindrical shells.
4. Write appropriate Riemann sum and set up definite integral.
5. The Basic Idea: Other Modeling Applications
Student Expectations: Be able to sketch 3-D figures illustrating cross-section.
Find volumes by cross sections.
Find total distance traveled by a particle along a line.
Calculus of Transcendental Functions
1. The Natural Logarithm Function
2. The Exponential Function
3. Other Exponential and Logarithmic Functions
4. The Law of Exponential Change Revisited
5. Indeterminate Forms
6. The Rates At Which Functions Grow
7. The Inverse Trigonometric Functions
8. Derivatives of Inverse Trigonometric Functions; Related Integrals
Techniques of Integration
1. Formulas for Elementary Integrals
Student Expectations: Be able to identify and discriminate between elementary forms.
2. Integral Involving Trigonometric Functions
3. Differential Equations
Student Expectation: Be able to solve separable differential equations.
Be able to model applications with separable differential equations.
Calculus Camp
AP readiness in UCLA. Details to follow.
Review for AP Exam
Throughout the year you will be exposed to the format of past AP Examination questions. In the days just prior to the administration of the spring exam, you will work as individuals and sometimes in groups to concentrate on entire past AP Calculus Exams. You will take pieces of exams and entire exams to give you a 3-hour test-taking experience. Quizzes, of both multiple-choice and essay types, will be given throughout the review weeks.
References and Materials
Calculus
Ross L. Finney, Franklin D. Demana
Bert K. Waits, Daniel Kennedy.
3rd Edition