Topic Name Description
Course Introduction 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.

Evaluating Functions Given Equations, Graphs, and Tables

Watch this video on evaluating functions given equations, graphs and tables.

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

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
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