# Calculus

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

## 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 scientific 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., $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}\left(x\right)}{x}=1$, $\underset{x\to 0}{\mathrm{lim}}\frac{1}{x}$ is nonexistent, and $\underset{x\to \infty}{\mathrm{lim}}\frac{\mathrm{sin}\left(x\right)}{x}=0$
- Continuity

### Differential Calculus (50%)

#### The Derivative

- Definitions of the derivative, e.g., $f\prime \left(a\right)=\underset{x\to a}{\mathrm{lim}}\frac{f\left(x\right)-f\left(a\right)}{x-a}$ and $f\prime \left(x\right)=\underset{h\to 0}{\mathrm{lim}}\frac{f(x+h)-f\left(x\right)}{h}$
- Derivatives of elementary functions
- Derivatives of sums, products and quotients (including $\mathit{t}\mathit{a}\mathit{n}\mathbf{}\mathit{x}$ and $\mathit{c}\mathit{o}\mathit{t}\mathbf{}\mathit{x}$
- Derivative of a composite function (chain rule), e.g., $\mathit{s}\mathit{i}\mathit{n}(ax+b)$, $\mathit{a}{\mathit{e}}^{\mathbf{k}\mathbf{x}}$, $\mathit{l}\mathit{n}\left(kx\right)$
- Implicit differentiation
- Derivative of the inverse of a function (including $\mathit{a}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{x}$ and $\mathit{a}\mathit{r}\mathit{c}\mathit{t}\mathit{a}\mathit{n}\mathbf{}\mathit{x}$)
- Higher order derivatives
- Corresponding characteristics of graphs of $\mathit{f}$, $\mathit{f}\mathbf{\text{'}}$ and $\mathit{f}\mathbf{\text{'}\text{'}}$
- 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 $\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathit{k}\mathit{y}$ 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: $\frac{d}{dx}{\int}_{a}^{x}f\left(t\right)dt=f\left(x\right)$ and ${\int}_{a}^{b}F\text{'}\left(x\right)dx=F\left(b\right)-F\left(a\right)$

#### 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

- 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.
- Unless otherwise specified, all angles are measured in radians, and all numbers used are real numbers.
- Unless otherwise specified, the domain of any function $f$ is assumed to be the set of all real numbers $x$ for which $f\left(x\right)$ is a real number. The range of $f$ is assumed to be the set of all real numbers $f\left(x\right)$ where $x$ is in the domain of $f$.
- In this test, $lnx$ denotes the natural logarithm of $x$ (that is, the logarithm to the base $e$).
- The inverse of a trigonometric function $f$ may be indicated using the inverse function notation ${f}^{-1}$ or with the prefix "arc" (e.g., ${\mathrm{sin}}^{-1}x=arc\mathrm{sin}x$)

## 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.