CALCULUS
SELFPACED COURSE
1. About this Course
It reviews the fundamentals taught in a onesemester college course in calculus. Our lessons are aligned to the content of the CLEP exam, which covers approximately 60% limits and differential calculus and 40% integral calculus. Algebraic, trigonometric, exponential, logarithmic, and general functions are included.
"Calculus is synonymous with serious mathematics, and if you watch my course videos, read the supplemental readings, and work hard on the practice problems, you’ll be on your way to being a serious student of mathematics.
The Calculus course –completely selfpaced and entirely free– is organized into six topical chapters or modules that contain short video lessons, exercises, readings and other interactive resources.
The goal of the creator of this course –Modern States Education Alliance, a nonprofit organization– is to prepare you to pass the College Board's CLEP examination and obtain college credit for free.
2. About the Examination
It requires
The examination 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 2: approximately 17 questions, approximately 40 minutes. An online graphing calculator (nonCAS) is integrated into the exam software, although only some of the following questions actually require its use:
• 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
To become familiar with the functionality of the calculator prior to taking the exam, download it for free for a 30day trial period.
3. Required Knowledge and Skills
Questions on the Calculus examination will require you 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)
4. Percentage of Exam Questions
Limits (10%)

Continuity
Differential Calculus (50%)
The Derivative

Definitions of the derivative
 Derivatives of elementary functions
 Derivatives of sums,
products and quotients (including tan x and cot x)  Derivative of a
composite function (chain rule), e.g., sin(ax + b),ae ^{kx} , ln(kx )  Implicit differentiation
 Derivative of the inverse of a function (including arcsin x and 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'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, change of variable)
Applications of Antiderivatives
 Distance and velocity from acceleration with initial conditions
 Solutions of y′ =
ky 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:
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 ( x ) is a real number. The range of f is assumed to be the set of all real numbers f ( x ) where x is in the domain
of f .  In this test, ln x 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., sin^{1} x = arcsin x)
5. Course Modules
Following are the main topics and percentages of the exam’s questions, mostly based on the College Board's description of the course:
Module 1: Limits
1.0 Introduction
1.1 Definition of a Limit
1.2 Computing Basic Limits
1.3 Continuity
1.4 Squeeze Theorem
Module 2: Theory of the Derivative
2.0 Introduction
2.1 Tangent Lines
2.2 Definition of Derivative
2.3 Rates of Change
2.4 Derivative Rules
2.4.1 Fundamental Derivative Rules
2.4.2 Chain Rule
2.4.3 Derivatives of Exponential and Logarithmic Functions
2.4.4 Trigonometric Derivatives
2.4.5 Derivatives of Inverse Trigonometric Functions
2.5 Higher Order Derivatives
2.6 Implicit Differentiation
2.7 L’Hôpital’s Rule
2.8 Some Classic Theoretical Results
2.9 Derivatives of Inverse Functions
Module 3: Applications of the Derivative
3.0 Introduction
3.1 Plotting with Derivatives
3.1.1 Increasing and Decreasing Functions
3.1.2 Extrema
3.1.3 Concavity
3.2 Rate of Change
3.3 Some Physics Problems
Module 4: Theory of the Integral
4.0 Introduction
4.1 Antidifferentiation
4.2 Definite Integral
4.3 Riemann Sums
4.3.1
4.3.2
4.4 The Fundamental Theorem of Calculus
4.5 Basic Integral Rules
4.5.1 Basic Integral Rules I
4.5.2 Basic Integral Rules II
4.6 USubstitutions
Module 5: Applications of the Integral
5.0 Introduction
5.1 Area Under Curves
5.1.1 Area Under Curves Part I
5.1.2 Area Under Curves Part II
5.2 Average Value
5.3 Growth and Decay Models
5.4 Return to Physics
6. About James Murphy
Dr. James Murphy is an applied mathematician. His research interests include harmonic analysis, machine learning, and the analysis of remote sensing data. Recent work has focused on fast algorithms for unsupervised learning and development of distance metrics for stochastic data models, with applications to hyperspectral imagery.
Dr. Murphy has taught at some of the world’s finest academic institutions, including the University of Chicago, the University of Maryland College Park, Duke University, and most recently at the Johns Hopkins University. His courses range from introductory statistics and precalculus to research topics courses for Ph.D. students. He was honored with the Aziz/Osborn Gold Medal
7. How CLEP Works
Developed by the College Board, CLEP (CollegeLevel Examination Program®) is the most widely accepted creditbyexamination program.
CLEP’s credits are accepted by 2,900 colleges and universities, according to the College Board. These tests assess collegelevel knowledge in 33 subject areas.
Modern States Education Alliance is the nonprofit organization behind these edXstyle courses. Its project is called “Freshman Year for Free” and its mission is to make college more accessible and affordable through free, highquality online education.
• 'Passing the CLEP and Learning with Modern States' orientation course