Calculus

The Calculus exam assesses an intuitive understanding of calculus and features 60% limits and differential calculus and 40% integral calculus.

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Overview

The Calculus exam covers skills and concepts that are usually taught in a one-semester college course in calculus. The content of each exam is approximately 60% limits and differential calculus and 40% integral calculus. Algebraic, trigonometric, exponential, logarithmic, and general functions are included. The exam is primarily concerned with an intuitive understanding of calculus and experience with its methods and applications. Knowledge of preparatory mathematics is assumed, including algebra, geometry, trigonometry, and analytic geometry.

Knowledge and Skills Required

The exam contains 44 questions, in two sections, to be answered in approximately 90 minutes.

  • Section 1: approximately 27 questions, approximately 50 minutes.
    No calculator is allowed for this section.
  • Section 2: approximately 17 questions, approximately 40 minutes.
    The use of an online graphing calculator (non-CAS) is allowed for this section. Only some of the questions will require the use of the calculator.

Graphing Calculator

A graphing calculator, the TI-84 Plus CE, is integrated into the exam software and available to students during Section 2 of the exam. Only some of the questions actually require the graphing calculator.

To use the calculator during the exam, students need to select the Calculator icon. Information about how to use the calculator is available in the Help icon under the Calculator tab. Students are expected to know how and when to make appropriate use of the calculator.

Visit ETS to learn more and to practice using the graphing calculator.. In order to answer some of the questions in Section 2 of the exam, students may be required to use the online graphing calculator in the following ways:

  • Perform calculations (e.g., exponents, roots, trigonometric values, and logarithms)
  • Graph functions and analyze the graphs
  • Find zeros of functions
  • Find points of intersection of graphs of functions
  • Find minima/maxima of functions
  • Find numerical solutions to equations
  • Generate a table of values for a function

Questions on the exam require candidates to demonstrate the following abilities:

  • Solving routine problems involving the techniques of calculus (approximately 50% of the exam)
  • Solving nonroutine problems involving an understanding of the concepts and applications of calculus (approximately 50% of the exam)

The subject matter of the Calculus exam is drawn from the following topics. The percentages next to the main topics indicate the approximate percentage of exam questions on that topic.

Limits (10%)

  • Statement of properties, e.g., limit of a constant, sum, product or quotient
  • Limit calculations, including limits involving infinity, e.g., limit as x rightwards arrow 0 of fraction numerator sin open parentheses x close parentheses over denominator x end fraction equals 1, limit as x rightwards arrow 0 of 1 over x is nonexistent, and limit as x rightwards arrow infinity of fraction numerator sin open parentheses x close parentheses over denominator x end fraction equals 0
  • Continuity

Differential Calculus (50%)

The Derivative

  • Definitions of the derivative, e.g., f prime open parentheses a close parentheses equals limit as x rightwards arrow a of fraction numerator f left parenthesis x right parenthesis minus f left parenthesis a right parenthesis over denominator x minus a end fraction and f prime open parentheses x close parentheses equals limit as h rightwards arrow 0 of fraction numerator f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis over denominator h end fraction
  • Derivatives of elementary functions
  • Derivatives of sums, products and quotients (including bold italic t bold italic a bold italic n bold space bold italic x and bold italic c bold italic o bold italic t bold space bold italic x
  • Derivative of a composite function (chain rule), e.g., bold italic s bold italic i bold italic n begin bold style left parenthesis a x plus b right parenthesis end style, bold italic a bold italic e to the power of bold k bold x end exponent, bold italic l bold italic n begin bold style left parenthesis k x right parenthesis end style
  • Implicit differentiation
  • Derivative of the inverse of a function (including bold italic a bold italic r bold italic c bold italic s bold italic i bold italic n bold space bold italic x and bold italic a bold italic r bold italic c bold italic t bold italic a bold italic n bold space bold italic x)
  • Higher order derivatives
  • Corresponding characteristics of graphs of bold italic f, bold italic f bold apostrophe and bold italic f bold double apostrophe
  • Statement of the Mean Value Theorem; applications and graphical illustrations
  • Relation between differentiability and continuity
  • Use of L'Hospital's Rule (quotient and indeterminate forms)

Applications of the Derivative

  • Slope of a curve at a point
  • Tangent lines and linear approximation
  • Curve sketching: increasing and decreasing functions; relative and absolute maximum and minimum points; concavity; points of inflection
  • Extreme value problems
  • Velocity and acceleration of a particle moving along a line
  • Average and instantaneous rates of change
  • Related rates of change

Integral Calculus (40%)

Antiderivatives and Techniques of Integration

  • Concept of antiderivatives
  • Basic integration formulas
  • Integration by substitution (use of identities and change of variable)

Applications of Antiderivatives

  • Distance and velocity from acceleration with initial conditions
  • Solutions of bold italic y bold apostrophe bold equals bold italic k bold italic y and applications to growth and decay

The Definite Integral

  • Definition of the definite integral as the limit of a sequence of Riemann sums and approximations of the definite integral using areas of rectangles
  • Properties of the definite integral
  • The Fundamental Theorem: fraction numerator d over denominator d x end fraction integral subscript a superscript x f left parenthesis t right parenthesis d t equals f left parenthesis x right parenthesis and integral subscript a superscript b F apostrophe left parenthesis x right parenthesis d x equals F left parenthesis b right parenthesis minus F left parenthesis a right parenthesis

     

Applications of the Definite Integral

  • Average value of a function on an interval
  • Area, including area between curves
  • Other (e.g., accumulated change from a rate of change)

Notes and Reference Information

  1. Figures that accompany questions are intended to provide information useful in answering the questions. All figures lie in a plane unless otherwise indicated. The figures are drawn as accurately as possible except when it is stated in a specific question that the figure is not drawn to scale. Straight lines and smooth curves may appear slightly jagged.
  2. Unless otherwise specified, all angles are measured in radians, and all numbers used are real numbers.
  3. Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers x for which f left parenthesis x right parenthesis is a real number. The range of f is assumed to be the set of all real numbers f left parenthesis x right parenthesis where x is in the domain of f.
  4. In this test, l n space x denotes the natural logarithm of x (that is, the logarithm to the base e).
  5. The inverse of a trigonometric function f may be indicated using the inverse function notation f to the power of negative 1 end exponent or with the prefix "arc" (e.g., sin to the power of negative 1 end exponent space x equals a r c sin space x)

Score Information

ACE Recommendation for Calculus

Credit-granting Score 50
Semester Hours 4

Note: Each institution reserves the right to set its own credit-granting policy, which may differ from the American Council on Education (ACE). Contact your college to find out the score required for credit and the number of credit hours granted.

Add Study Guides

CLEP Calculus Examination Guide

The Calculus exam is approximately 60% limits and differential calculus and 40% integral calculus.

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2022 CLEP Official Study Guide

This study guide provides practice questions for all 34 CLEP exams. The ideal resource for taking more than one exam. Offered only by College Board.

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