# Course Syllabus for AP Calculus

• AP CALCULUS COURSE SYLLABUS

INDIVIDUAL INSTRUCTOR SECTION

BASIC COURSE INFORMATION

Course Title:  AP Calculus

Credit Hours:  8

Instructor:  Mrs. Michelle Ann Cinicola

INSTRUCTIONAL MATERIALS

Text:  Calculus of a Single Variable, Larson, Hostetler and Edwards, Eighth Edition, Houghton Mifflin Company.

PERFORMANCE OBJECTIVES – a statement of what the learner must do to master a competency

Limits, their properties, and functions

Given a function, the student will be able to calculate the domain and range.

Given a function, the student will be able to recall the properties of limits and various techniques for evaluating the limits, to calculate the limit of the function when x approaches a certain value.

Given a function, the student will be able to determine its continuity and one-sided limit.

Differentiation

Given a function, the student will be able to calculate the derivative of the function.

Given a function that has an inverse function, the student will be able to determine the inverse of the function and its derivative.

Given a problem involving velocity, the student will be able to solve the problem using derivatives.

Given a composite function, the student will be able to determine its derivative by applying the chain rule.

Given a product of functions, the student will be able to determine the derivative by applying the product rule.

Given a function, the student will be able to determine the first derivative, the second derivative, the third derivative, etc.

Given a quotient of two functions, the student will be able to determine its derivative by applying the quotient rule.

Given trigonometric functions, the student will be able to determine its derivative.

Given a polynomial, the student will be able to apply the rules to calculate the derivative of a polynomial.

Given an equation, the student will be able to perform the implicit differentiation to calculate the derivative.

Applications of Differentiation

Given necessary information, the student will be able to set up equations and use differentiation to solve related rate problems.

Given a non-continuous function, the student will be able to determine the points of discontinuity.

Given a function, the student will be able to determine whether the function is increasing or decreasing on a given interval by using the sign of the first derivative.

Given a function, the student will be able to determine the point of inflection using the second derivative.

Given a function, the student will be able to sketch its graph by using concepts like domain and range, x-intercepts and y-intercepts, symmetry, points of discontinuity, vertical asymptotes, horizontal asymptotes, relative extrema, concavity, and points of inflection.

Given a function, the student will be able to determine whether the graph is concave upward or downward by using the sign of the second derivative.

Given a problem, the student will be able to apply Rolle’s Theorem and the Mean Value Theorem to solve it.

Given a function, the student will be able to calculate its differential from the derivative and use differentials to obtain reasonable approximation to a problem.

Given necessary information, the student will be able to use the maxima and minima theory to determine the maximum and minimum values.

Integration

Given the derivative of a function, the student will be able to use the notation for antiderivatives and will be able to determine the function.

Given the derivative of a function, the student will be able to determine the function by using basic integration rules.

The student will be able to use sigma notation to calculate the area of a plane region.

The student will be able to calculate the area of a plane region by the integration method.

The student will be able to apply the Fundamental Theorem of Calculus and the Mean Value Theorem for integrals to solve a given problem.

The student will be able to apply the u- substitution, change of variables, and the general power rules for integration to solve a given problem.

Applications of  the Definite Integral

The student will be able to calculate the area of a region between two curves by the integration method.

Given a three-dimensional solid, the student will be able to calculate the volume of a solid of revolution using the disc method or the washer method.

Given a three-dimensional solid, the student will be able to calculate the volume of a solid of revolution using the shell method.

Given the equation of a plane curve, the student will be able to calculate the arc length and the area of a surface of revolution.

The student will be able solve problems involving work with constant and variable force.

The student will be able to apply integration techniques to solve problems with rectilinear motion.

Logarithm and Exponential Functions

Given a logarithmic function, the student will be able to determine the derivative.

Given a logarithmic function, the student will be able to determine the antiderivative.

Given an algebraic function, the student will be able to determine the existence of an inverse function, find the inverse function and determine the derivative of the inverse function.

Given an exponential function, the student will be able to determine the derivative and the antiderivative.

Given a hyperbolic function, the student will be able to determine the derivative and antiderivative.

Inverse Trigonometric and Hyperbolic Functions

Given an inverse trigonometric function, the student will be able to determine the derivative and antiderivative.

Given an inverse hyperbolic function, the student will be able to determine the derivative and antiderivative.

Techniques of Integration

The student will be able to apply the basic rules of integration to solve a given problem.

The student will be able to apply the rules of integration by parts to solve a given problem.

Given a trigonometric function involving powers, the student will be able to determine the antiderivative.

The student will be able to use trigonometric substitution to determine an antiderivative.

The student will be able to use partial fractions to determine an antiderivative.

The student will be able to apply L’Hopital’s Rule to determine the limit of an indeterminate form.

Infinite Series

Given a sequence, the student will be able to determine the limit of the sequence, recognize the sequence pattern and determine if the sequence is monotonic and/or bounded.

Given an infinite series, the student will be able to determine whether it is convergent or divergent.

Given a function, the student will be able to calculate the polynomial approximation of it by Taylor and Maclaurin series method.

Given a power series, the student will be able to differentiate and integrate the power series.

The student will be able to represent a function using a power series.

The student will be able to find a Taylor or Maclaurin series for a given function.

A             94 – 100                D   70 – 75

B             86 – 93                  F    0 – 69

C             76 – 85

COURSE DESCRIPTION

The major topics in this course are:  limits, continuity and differentiability, derivative applications in curve sketching, related rates, maxima and minima problems, indefinite and definite integration, transcendental functions, techniques of integration, applications of definite integrals, indeterminate forms, improper integrals, simple differential equations and infinite series.

COURSE RATIONALE

This course is to ready any student for majors at college which require a strong math background.

COURSE COMPETENCIES

The student will be able to:

Determine the equation of a straight line when given the necessary information.

Determine the domain and range of a function.

Apply various techniques for evaluating limits.

Determine those values of x in the domain of f at which the function possesses a derivative.

Recall the various ways that the derivative of y = f(x) with respect to x can be written.

Solve velocity and rate problems using the derivative.

Recall and apply the rules to find the derivative of a polynomial.

Perform implicit differentiation.

Review the trigonometric functions, identities, law of cosines, addition formulas, and graphing of trigonometric functions.

Calculate the derivatives of trigonometric functions.

Use the sign of the first derivative, point of inflection to sketch the graph of a function.

Use maxima and minima theory to solve problems.

Use Rolle’s Theorem and Mean Value Theorem to solve problems.

Solve problems involving the indefinite integral including applications.

Use the sigma notation to calculate the area of a plane region.

Determine the area of a plane region by the limit definition and the definite integral method.

Apply the Fundamental Theorem of Calculus and the Mean Value Theorem for integrals.

Do integration by the substitution method.

Apply the log rule for integration and the integrals for trigonometric functions.

Apply the rule for differentiation and integration of exponential functions.

Apply the rules for differentiation of inverse trigonometric functions.

Calculate the area of a region between two curves by the integration method.

Calculate the volume by solid of revolution using the disc or shell method.

Calculate arc length and surfaces of revolution.

Apply various integration techniques.

Determine the limit of a sequence.

Apply Taylor and Maclaurin series to represent functions.