# Calculus

## Description of the Examination

The Calculus examination covers skills and concepts that are usually taught in a one-semester college course in calculus. The content of each examination 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, including algebra, geometry, trigonometry, and analytic geometry is assumed.

The examination contains 44 questions, in two sections, to be answered in approximately 90 minutes. Any time candidates spend on tutorials and providing personal information is in addition to the actual testing time.

- Section 1: 27 questions, approximately 50 minutes.

No calculator is allowed for this section. - Section 2: 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 is integrated into the exam software, and it is available to students during **Section 2** of the exam.

Only some of the questions actually require the graphing calculator. Students are expected to know how and when to make appropriate use of the calculator. The graphing calculator, together with brief video tutorials, is available to students as a free download for a 30-day trial period. **Students are expected to download the calculator and become familiar with its functionality prior to taking the exam.**

Get more information about the calculator and download a trial version.

In order to answer some of the questions in the calculator section 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, 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

## Knowledge and Skills Required

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

- Solving routine problems involving the techniques of calculus (approximately 50 percent of the exam)
- Solving nonroutine problems involving an understanding of the concepts and applications of calculus (approximately 50 percent 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.

### 10% Limits

- Statement of properties, e.g., limit of a constant, sum, product or quotient

### 40% Integral Calculus

Antiderivatives and Techniques of Integration

- Concept of antiderivatives
- Basic integration formulas
- Integration by substitution (use of identities, change of variable)

Applications of Antiderivatives

- Distance and velocity from acceleration with initial conditions
- Solutions of
*y*′**5**and applications to growth and decay*ky*

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:

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
- Limit calculations, including limits involving infinity, e.g.,

- Continuity

### 50% Differential Calculus

The Derivative

- Definitions of the derivative

- Derivatives of elementary functions
- Derivatives of sums, products and quotients (including
**tan**and*x***cot**)*x* - Derivative of a composite function (chain rule), e.g.,
**sin(***ax*1*b*),*ae*,ln(*kx*) - Implicit differentiation
- Derivative of the inverse of a function (including
**arcsin**and*x***arctan**)*x* - Higher order derivatives
- Corresponding characteristics of graphs of
**ƒ**,**ƒ**′ and**ƒ**″ - Statement of the Mean Value Theorem; applications and graphical illustrations
- Relation between differentiability and continuity
- Use of L'Hôpital'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

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