MA005: Resources | Saylor Academy

Topic	Name	Description
	Course Syllabus

1.1: Preview of Calculus	Preview of Calculus	Read this section for an introduction to calculus.
	Derivatives and Tangent Lines	Watch this video (until 8:12) on how to find the slopes of tangent lines from a graph.
	Find the Area of an Irregular Shape	Watch this video on how to find the area of an irregular shape.
	Practice Problems	Work through the odd-numbered problems 1-7. Once you have completed the problem set, check your answers.

1.2: Lines in the Plane	Lines in the Plane	Read this section and work through practice problems 1-9.
	Find a Slope Given Two Points	Watch this video on how to find the slope of a line given two points on a graph.
	Find the Equation of a Line Given Two Points	Watch this video on how to write the equation of a line through two points using the slope-point formula, and converting the equation to slope-intersect form.
	Equations of Circles: Radius and Center	Watch this video on how to write an equation of a circle with the given center and radius.
	How to Find the Equation of a Line from a Slope and Point	Watch this video on how to write an equation of a line through a point given the slope.
	Determine the Equation of a Line Parallel to a Line in General Form	Watch this video on how to find the equation of a line going through a point and parallel to another line.
	Determine the Equation of a Line Perpendicular to a Line in Slope-Intercept Form	Watch this video on how to find the equation of a line going through a point and perpendicular to another line.
	Practice Problems	Work through the odd-numbered problems 1-29. Once you have completed the problem set, check your answers.

1.3: Functions and Their Graphs	Functions and Their Graphs	Read this section for an introduction to functions and their graphs. Work through practice problems 1-5.
	Evaluate Function at a Point	Watch this video on evaluating function at a point.
	Practice Problems	Work through the odd-numbered problems 1-23. Once you have completed the problem set, check your answers.

1.4: Combinations of Functions	Combinations of Functions	Read this section for an introduction to combinations of functions, then work through practice problems 1-9.
	More on the Composition of Functions	Watch the first 16 minutes of this video on evaluating composition of functions.
	Combine Functions using Algebraic Operations	Watch this video on combining functions using algebraic operations.
	Practice Problems	Work through the odd-numbered problems 1-31. Once you have completed the problem set, check your answers.

1.5: Mathematical Language	Mathematical Language	Read this section for an introduction to mathematical language, then work through practice problems 1-4.
	Negating Quantified Statements	Watch this video on how to negate a quantified statement.
	Converse and Contrapositive	Watch this video on converse and contrapositive of a conditional statement and how to write them.
	Practice Problems	Work through the odd-numbered problems 1-25. Once you have completed the problem set, check your answers for the odd-numbered questions.

2.1: Tangent Lines, Velocities, and Growth	Tangent Lines, Velocities, and Growth	Read this section for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1-4.
	Slope of a Function at a Point	Watch this video on how to find the slope of the line tangent to the graph of a function at a point.
	Practice Problems	Work through the odd-numbered problems 1-9. Once you have completed the problem set, check your answers for the odd-numbered questions.

2.2: The Limit of a Function	The Limit of a Function	Read this section for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1-4.
	Evaluating Limits Using a Graph	Watch this video on evaluating limits using a graph.
	More on the Limit of a Function	Watch this video on how to evaluate limit of a function.
	The Limit of Laws	Read this section.
	Practice Problems	Work through the odd-numbered problems 1-19. Once you have completed the problem set, check your answers.

2.3: Properties of Limits	Properties of Limits	Read this section to learn about the properties of limits. Work through practice problems 1-6.
	Solving Limits (Rationalization)	Watch this video on finding limits algebraically. Be warned that removing $x-4$ from the numerator and denominator in Step 4 of this video is only legal inside this limit. The function $\frac{x - 4}{x - 4}$ is not defined at $x = 4$ ; however, when $x$ is not 4, it simplifies to 1. Because the limit as $x$ approaches 4 depends only on values of $x$ different from 4, inside that limit $\frac{x - 4}{x - 4}$ and 1 are interchangeable. Outside that limit, they are not! However, this kind of cancellation is a key technique for finding limits of algebraically complicated functions.
	Calculating Slope of Tangent Line Using the Definition of a Derivative	Watch this video on limits as the slopes of tangent lines. Before you watch, know that, for this problem, the limit that gives the slope of the tangent line to a curve is $y = f(x)$ at a point $x = a$ , which is the derivative of $f(x)$ at $a$ . We will talk about this more in Unit 3.
	Practice Problems	Work through the odd-numbered problems 1-21. Once you have completed the problem set, check your answers.

2.4: Continuous Functions	Continuous Functions	Read this section for an introduction to what we mean when we say a function is continuous. Work through practice problems 1 and 2.
	More on Continuous Functions	Watch this video on continuous functions.
	Intermediate Value Theorem	Watch this video on the Intermediate Value Theorem.
	Practice Problems	Work through the odd-numbered problems 1-23. Once you have completed the problem set, check your answers.

2.5: Definition of a Limit	Definition of a Limit	Read this section to learn how a limit is defined. Work through practice problems 1-6.
	Epsilon-Delta Limits	Watch this video to learn the epsilon-delta definition of a limit.
	Practice Problems	Work through the odd-numbered problems 1-23. Once you have completed the problem set, check your answers.

3.1: Introduction to Derivatives	Introduction to Derivatives	Read this section to lay the groundwork for introducing the concept of a derivative. Work through practice problems 1-5.
	The Tangent Line	Watch this video on the slope of secant lines converging to tangent line slopes.
	Practice Problems	Work through the odd-numbered problems 1-17. Once you have completed the problem set, check your answers.

3.2: The Definition of a Derivative	The Definition of a Derivative	Read this section to understand the definition of a derivative. Work through practice problems 1-8.
	Derivatives from the Definition	Watch this video on how to find a derivative from the definition.
	Practice Problems	Work through the odd-numbered problems 1-37. Once you have completed the problem set, check your answers.

3.3: Derivatives, Properties, and Formulas	Derivatives, Properties, and Formulas	Read this section to understand the properties of derivatives. Work through practice problems 1-11.
	Applying the Product Rule for Derivatives	Watch this video on the product rule for differentiation.
	Practice Problems	Work through the odd-numbered problems 1-55. Once you have completed the problem set, check your answers.

3.4: Derivative Patterns	Derivative Patterns	Read this section to learn about patterns of derivatives. Work through practice problems 1-8.
	The Power Rule for Polynomials	Watch this video on how to find derivatives of polynomial functions using the power rule.
	Derivatives of Exponential Functions	Watch this video on derivatives of exponential functions.
	Derivatives of Trigonometric Functions	Watch this video on how to find the derivatives of trig functions.
	Higher-Ordered Derivatives	Watch this video on how to calculate and intepret higher ordered derivatives.
	Practice Problems	Work through the odd-numbered problems 1-47. Once you have completed the problem set, check your answers.

3.5: The Chain Rule	The Chain Rule	Read this section to learn about the Chain Rule. Work through practice problems 1-8.
	Chain Rule Examples	Watch these videos for an introduction to the chain rule for differentiation and examples of the chain rule for differentiation.
	Practice Problems	Work through the odd-numbered problems 1-83. Once you have completed the problem set, check your answers.

3.6: Some Applications of the Chain Rule	Some Applications of the Chain Rule	Read this section to learn how to apply the Chain Rule. Work through practice problems 1-8.
	Solving a Related Rates Problem	Watch this video on using the chain rule to solve rates problems.
	Practice Problems	Work through the odd-numbered problems 1-49. Once you have completed the problem set, check your answers.

3.7: Related Rates	Related Rates	Read this section to learn to connect derivatives to the concept of the rate at which things change. Work through practice problems 1-3.
	Derivatives as Rates of Change	Watch this video on application of derivatives
	Differentiating Parametric Functions	Watch this video on differentiating parametric functions.
	Practice Problems	Work through the odd-numbered problems 1-21. Once you have completed the problem set, check your answers.

3.8: Newton's Method for Finding Roots	Newton's Method for Finding Roots	Read this section. Work through practice problems 1-6.
	Using Newton's Method	Watch this video on Newton's method.
	Determining Differentiability Graphically	Watch this video on how to determine differentiability graphically.
	Continuity on Intervals	Watch this video on continuity of intervals and continuous functions.
	Practice Problems	Work through the odd-numbered problems 1-21. Once you have completed the problem set, check your answers.

3.9: Linear Approximation and Differentials	Linear Approximation and Differentials	Read this section to learn how linear approximation and differentials are connected. Work through practice problems 1-10.
	More on Linear Approximation	Watch this video on linear approximation and differentials.
	The Properties of a Differential of a Function	Watch this video on how to calculate the differential of a function using derivatives.
	Practice Problems	Work through the odd-numbered problems 1-19. Once you have completed the problem set, check your answers.

3.10: Implicit and Logarithmic Differentiation	Implicit and Logarithmic Differentiation	Read this section to learn about implicit and logarithmic differentiation. Work through practice problems 1-6.
	Implicit Differentiation and the Derivative of x^(x^x)	Watch these videos on implicit and logarithmic differentiation.
	Practice Problems	Work through the odd-numbered problems 1-55. Once you have completed the problem set, check your answers.

4.1: Finding Maximums and Minimums	Finding Maximums and Minimums	Read this section to learn about maximums, minimums, and extreme values for functions. Work through practice problems 1-5.
	Maxima and Minima	Watch this video on how to identify minimum/maximum points of a function.
	Practice Problems	Work through the odd-numbered problems 1-43. Once you have completed the problem set, check your answers.

4.2: The Mean Value Theorem and Its Consequences	The Mean Value Theorem and Its Consequences	Read this section to learn about the Mean Value Theorem and its consequences. Work through practice problems 1-3.
	Critical Points and the Mean Value Theorem	Watch this video on local maximums/minimums and proves Rolle's Theorem and the Mean Value Theorem.
	Practice Problems	Work through the odd-numbered problems 1-35. Once you have completed the problem set, check your answers.

4.3: The First Derivative and the Shape of a Function f(x)	The First Derivative and the Shape of a Function f(x)	Read this section to learn how the first derivative is used to determine the shape of functions. Work through practice problems 1-9.
	Examples of the First Derivative Test	Watch both parts of this video on the first derivative test.
	Practice Problems	Work through the odd-numbered problems 1-29. Once you have completed the problem set, check your answers.

4.4: The Second Derivative and the Shape of a Function f(x)	The Second Derivative and the Shape of a Function f(x)	Read this section to learn how the second derivative is used to determine the shape of functions. Work through practice problems 1-9.
	Concavity Across Intervals	Watch this video on the second derivative test. This video describes a way to identify critical points as minima or maxima other than the first derivative test, using the second derivative.
	Concavity and the Second Derivative	Watch this video, which works through an example of the second derivative test.
	Practice Problems	Work through the odd-numbered problems 1-17. Once you have completed the problem set, check your answers.

4.5: Applied Maximum and Minimum Problems	Applied Maximum and Minimum Problems	Read this section to learn how to apply previously learned principles to maximum and minimum problems. Work through practice problems 1-3. There is no review for this section; instead, make sure to study the problems carefully to become familiar with applied maximum and minimum problems.
	Minimizing Sum of Squares	Watch this video on optimization.
	Minimum Triangle Area	Watch this video on optimization.
	Practice Problems	Work through the odd-numbered problems 1-33. Once you have completed the problem set, check your answers.

4.6: Infinite Limits and Asymptotes	Infinite Limits and Asymptotes	Read this section to learn how to use and apply infinite limits to asymptotes. Work through practice problems 1-8.
	Limits at Infinity and Asymptotes	Watch this video on finding limits at infinite, and graphing using first and second derivative tests.
	Practice Problems	Work through the odd-numbered problems 1-59. Once you have completed the problem set, check your answer

4.7: L'Hopital's Rule	L'Hopital's Rule	Read this section to learn how to use and apply L'Hopital's Rule. Work through practice problems 1-3.
	Introduction to L'Hopital's Rule	Watch this video for an introduction to L'Hopital's Rule.
	Practice Problems	Work through the odd-numbered problems 1-29. Once you have completed the problem set, check your answers.

5.1: Introduction to Integration	Introduction to Integration	Read this section to learn about area. Work through practice problems 1-9.
	Find the Exact and Approximate Area of a Region	Watch this video on finding the exact and approximate area of a region.
	Practice Problems	Work through the odd-numbered problems 1-15. Once you have completed the problem set, check your answers.

5.2: Sigma Notation and Riemann Sums	Sigma Notation and Riemann Sums	Read this section to learn about area. Work through practice problems 1-9.
	Riemann Sums in Summation Notation	Watch this video on riemann sums in summation notation.
	Practice Problems	Work through the odd-numbered problems 1-61. Once you have completed the problem set, check your answers.

5.3: The Definite Integral	The Definite Integral	Read this section to learn about the definite integral and its applications. Work through practice problems 1-6.
	Find the Area between Two Curves Using Definite Integrals	Watch this video on how to find the area between two curves using definite integrals.
	Practice Problems	Work through the odd-numbered problems 1-29. Once you have completed the problem set, check your answers.

5.4: Properties of the Definite Integral	Properties of the Definite Integral	Read this section to learn about properties of definite integrals and how functions can be defined using definite integrals. Work through practice problems 1-5.
	Integrals and the Graph of a Function	Watch this video on definite integral and a function's graph.
	Practice Problems	Work through the odd-numbered problems 1-51. Once you have completed the problem set, check your answers for the odd-numbered questions against here.

5.5: Areas, Integrals, and Antiderivatives	Areas, Integrals, and Antiderivatives	Read this section to learn about the relationship among areas, integrals, and antiderivatives. Work through practice problems 1-5.
	Integration and Antiderivatives	Watch this video on Power Rule of Integration, Antiderivative of Polynomial Functions, Integrating Square Root Functions, and Antiderivatives of Rational Functions.
	Practice Problems	Work through the odd-numbered problems 1-25. Once you have completed the problem set, check your answers.

5.6: The Fundamental Theorem of Calculus	The Fundamental Theorem of Calculus	Read this section to see the connection between derivatives and integrals. Work through practice problems 1-5.
	Integrals and the Fundamental Theorem of Calculus	Watch this video on Integrals and Fundamental theorem of calculus.
	Differentiating an Integral Function	Watch this video on differentiating integral functions.
	Practice Problems	Work through the odd-numbered problems 1-67. Once you have completed the problem set, check your answers.

5.7: Finding Antiderivatives	Finding Antiderivatives	Read this section to see how you can (sometimes) find an antiderivative. In particular, we will discuss the change of variable technique. Change of variable, also called substitution or u-substitution (for the most commonly-used variable), is a powerful technique that you will use time and again in integration. It allows you to simplify a complicated function to show how basic rules of integration apply to the function. Work through practice problems 1-4.
	Integrating with U-Substitution	Watch these videos on change of variable, also called substitution or U-substitution.
	Practice Problems	Work through the odd-numbered problems 1-69. Once you have completed the problem set, check your answers.

5.8: First Application of Definite Integral	First Application of Definite Integral	Read this section to see how some applied problems can be reformulated as integration problems. Work through practice problems 1-4.
	Practice Problems	Work through the odd-numbered problems 1-41. Once you have completed the problem set, check your answers.

5.9: Using Tables to Find Antiderivatives	Using Tables to Find Antiderivatives	Read this section to learn how to use tables to find antiderivatives. See the Calculus Reference Facts for the table of integrals. Work through practice problems 1-5.
	Practice Problems	Work through the odd-numbered problems 1-55. Once you have completed the problem set, check your answers.

Study Guide	MA005 Study Guide

Course Feedback Survey	Course Feedback Survey

COURSE INTRODUCTION

Course Syllabus

Unit 1: Preview and Review

1.1: Preview of Calculus

Preview of Calculus

Derivatives and Tangent Lines

Find the Area of an Irregular Shape

Practice Problems

1.2: Lines in the Plane

Lines in the Plane

Find a Slope Given Two Points

Find the Equation of a Line Given Two Points

Equations of Circles: Radius and Center

How to Find the Equation of a Line from a Slope and Point

Determine the Equation of a Line Parallel to a Line in General Form

Determine the Equation of a Line Perpendicular to a Line in Slope-Intercept Form

Practice Problems

1.3: Functions and Their Graphs

Functions and Their Graphs

Evaluate Function at a Point

Practice Problems

1.4: Combinations of Functions

Combinations of Functions

More on the Composition of Functions

Combine Functions using Algebraic Operations

Practice Problems

1.5: Mathematical Language

Mathematical Language

Negating Quantified Statements

Converse and Contrapositive

Practice Problems

Unit 2: Functions, Graphs, Limits, and Continuity

2.1: Tangent Lines, Velocities, and Growth

Tangent Lines, Velocities, and Growth

Slope of a Function at a Point

Practice Problems

2.2: The Limit of a Function

The Limit of a Function

Evaluating Limits Using a Graph

More on the Limit of a Function

The Limit of Laws

Practice Problems

2.3: Properties of Limits

Properties of Limits

Solving Limits (Rationalization)

Calculating Slope of Tangent Line Using the Definition of a Derivative

Practice Problems

2.4: Continuous Functions

Continuous Functions

More on Continuous Functions

Intermediate Value Theorem

Practice Problems

2.5: Definition of a Limit

Definition of a Limit

Epsilon-Delta Limits

Practice Problems

Unit 2 Assessment

Unit 2 Assessment

Unit 3: Derivatives

3.1: Introduction to Derivatives

Introduction to Derivatives

The Tangent Line

Practice Problems

3.2: The Definition of a Derivative

The Definition of a Derivative

Derivatives from the Definition

Practice Problems

3.3: Derivatives, Properties, and Formulas

Derivatives, Properties, and Formulas

Applying the Product Rule for Derivatives

Practice Problems

3.4: Derivative Patterns

Derivative Patterns

The Power Rule for Polynomials

Derivatives of Exponential Functions

Derivatives of Trigonometric Functions

Higher-Ordered Derivatives

Practice Problems

3.5: The Chain Rule

The Chain Rule