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Understanding Calculus: Problems, Solutions and Tips

Succeed at calculus-the most feared of all math subjects-with this thorough and easy-to-follow guide taught by an award-winning math educator.
Understanding Calculus: Problems, Solutions and Tips is rated 4.9 out of 5 by 107.
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Rated 5 out of 5 by from Enthusiastic Professor TEACHES Calculus! This Calculus I course is very well constructed! The problem sets are selected well such that the concepts are solidified through effective problem solving. I've been out of school for awhile... Just finished a 32 year career as an electrical engineer. My sophomore year I was deciding between physics and engineering. My physics prof said "You could be an average physicist, but a good engineer. So EE it was!! Physics at UC Berkeley is HARD! Electrical Engineering there was HARD! So being retired I can go back to physics as a hobby. From what I gathered of this course, I ordered all the calculus, linear algebra, discrete math, differential equations courses offered in order to begin my physics quest!
Date published: 2024-03-10
Rated 5 out of 5 by from Excellent! If you want to learn something, anything, get an instructor who is enthusiastic about what they teach. Prof. Edwards is very very inspiring. A master teacher! A beast! You will not want to see his lectures end. He takes you in an amazing journey. Recommended to anyone interested in mathematics. I can only imagine how productive his classrooms are with in-person lectures. Thanks Prof Edwards, and hopefully we will se more of your courses here.
Date published: 2023-11-09
Rated 5 out of 5 by from Excellent Oh, my! If I only had this course fifty-eight years ago when I struggled with my first calculus course as a senior in secondary school. Prof. Edwards covers all of the important basics of differentiation, integration and differential equations with excellent graphics and a broad range of examples. And the course workbook is as good as any calculus textbook on the subject. HWF, Mesa AZ.
Date published: 2023-04-09
Rated 5 out of 5 by from Excellent review of calculus! The professor is very enthusiastic and this makes what could be a dry and difficult subject much more enjoyable.
Date published: 2022-12-18
Rated 5 out of 5 by from Excellent instructor Understanding Calculus is so much better than the other calculus course that was offered by Great Courses. I took this course as a review of my first calculus course that I took 65 years ago.
Date published: 2022-11-26
Rated 5 out of 5 by from Though provoking explanations Crystal clear, thought provoking, carefully selected material and examples
Date published: 2022-10-02
Rated 5 out of 5 by from Pure Mathematics Pleasure! I am really enjoying this course because Dr. Edwards teaches Calculus in a fluid, well polished, PURE Mathematics perspective. I received my B.A. in Mathematics years ago and this is a delightful refresher.
Date published: 2022-08-22
Rated 5 out of 5 by from This course emphasizes Understanding Calculus Professor Edwards uses graphics perfectly to emphasize the geometric meanings of the various concepts of calculus. As a result, the student truly understands the concepts without a need to first decipher the mathematical jargon. When I first studied calculus decades ago, I viewed calculus as a series of rules and formulae. Now I truly understand the simple geometric concepts behind each of these rules and formulae. The course was very well presented indeed. I would like to suggest one minor improvement: I would very much like to Professor Edwards to publish a small vignette dedicated to the history and meaning of the epsilon-delta definition of limits. I would particularly like to hear his reasoning behind the sentence "for any epsilon, no matter how small,....." I have searched in vain for a cogent explanation of this definition and have so far found no PhD who can really explain this part of the definition. If possible, I would like to speak to Professor Edwards about this.
Date published: 2022-08-15
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Immerse yourself in the unrivaled experience of learning&;amp;-and grasping&;amp;-calculus with Understanding Calculus: Problems, Solutions, and Tips. These 36 lectures cover all the major topics of a full-year calculus course in high school at the College Board Advanced Placement AB level or a first-semester course in college. Award-winning Professor Bruce H. Edwards guides you through hundreds of examples and problems, each of which is designed to explain and reinforce the major concepts of this vital mathematical field.


Bruce H. Edwards

I love mathematics and tried to communicate this passion to others, regardless of their mathematical backgrounds.


University of Florida

Dr. Bruce H. Edwards is Professor of Mathematics at the University of Florida. Professor Edwards received his B.S. in Mathematics from Stanford University and his Ph.D. in Mathematics from Dartmouth College. After his years at Stanford, he taught mathematics at a university near Bogota, Colombia, as a Peace Corps volunteer. Professor Edwards has won many teaching awards at the University of Florida, including Teacher of the Year in the College of Liberal Arts and Sciences, Liberal Arts and Sciences Student Council Teacher of the Year, and the University of Florida Honors Program Teacher of the Year. He was selected by the Office of Alumni Affairs to be the Distinguished Alumni Professor for 1991-1993. Professor Edwards has taught a variety of mathematics courses at the University of Florida, from first-year calculus to graduate-level classes in algebra and numerical analysis. He has been a frequent speaker at research conferences and meetings of the National Council of Teachers of Mathematics. He has also coauthored a wide range of mathematics textbooks with Professor Ron Larson. Their textbooks have been honored with various awards from the Text and Academic Authors Association.

By This Professor

Prove It: The Art of Mathematical Argument
Understanding Multivariable Calculus: Problems, Solutions, and Tips
Understanding Calculus II: Problems, Solutions, and Tips
Understanding Calculus: Problems, Solutions and Tips
Mathematics Describing the Real World: Precalculus and Trigonometry
Understanding Calculus: Problems, Solutions and Tips


A Preview of Calculus

01: A Preview of Calculus

Calculus is the mathematics of change, a field with many important applications in science, engineering, medicine, business, and other disciplines. Begin by surveying the goals of the course. Then get your feet wet by investigating the classic tangent line problem, which illustrates the concept of limits.

33 min
Review—Graphs, Models, and Functions

02: Review—Graphs, Models, and Functions

In the first of two review lectures on precalculus, examine graphs of equations and properties such as symmetry and intercepts. Also explore the use of equations to model real life and begin your study of functions, which Professor Edwards calls the most important concept in mathematics.

30 min
Review—Functions and Trigonometry

03: Review—Functions and Trigonometry

Continue your review of precalculus by looking at different types of functions and how they can be identified by their distinctive shapes when graphed. Then review trigonometric functions, using both the right triangle definition as well as the unit circle definition, which measures angles in radians rather than degrees.

30 min
Finding Limits

04: Finding Limits

Jump into real calculus by going deeper into the concept of limits introduced in Lecture 1. Learn the informal, working definition of limits and how to determine a limit in three different ways: numerically, graphically, and analytically. Also discover how to recognize when a given function does not have a limit.

31 min
An Introduction to Continuity

05: An Introduction to Continuity

Broadly speaking, a function is continuous if there is no interruption in the curve when its graph is drawn. Explore the three conditions that must be met for continuity—along with applications of associated ideas, such as the greatest integer function and the intermediate value theorem.

31 min
Infinite Limits and Limits at Infinity

06: Infinite Limits and Limits at Infinity

Infinite limits describe the behavior of functions that increase or decrease without bound, in which the asymptote is the specific value that the function approaches without ever reaching it. Learn how to analyze these functions, and try some examples from relativity theory and biology.

31 min
The Derivative and the Tangent Line Problem

07: The Derivative and the Tangent Line Problem

Building on what you have learned about limits and continuity, investigate derivatives, which are the foundation of differential calculus. Develop the formula for defining a derivative, and survey the history of the concept and its different forms of notation.

31 min
Basic Differentiation Rules

08: Basic Differentiation Rules

Practice several techniques that make finding derivatives relatively easy: the power rule, the constant multiple rule, sum and difference rules, plus a shortcut to use when sine and cosine functions are involved. Then see how derivatives are the key to determining the rate of change in problems involving objects in motion.

30 min
Product and Quotient Rules

09: Product and Quotient Rules

Learn the formulas for finding derivatives of products and quotients of functions. Then use the quotient rule to derive formulas for the trigonometric functions not covered in the previous lecture. Also investigate higher-order derivatives, differential equations, and horizontal tangents.

31 min
The Chain Rule

10: The Chain Rule

Discover one of the most useful of the differentiation rules, the chain rule, which allows you to find the derivative of a composite of two functions. Explore different examples of this technique, including a problem from physics that involves the motion of a pendulum.

31 min
Implicit Differentiation and Related Rates

11: Implicit Differentiation and Related Rates

Conquer the final strategy for finding derivatives: implicit differentiation, used when it's difficult to solve a function for "y". Apply this rule to problems in related rates—for example, the rate at which a camera must move to track the space shuttle at a specified time after launch.

31 min
Extrema on an Interval

12: Extrema on an Interval

Having covered the rules for finding derivatives, embark on the first of five lectures dealing with applications of these techniques. Derivatives can be used to find the absolute maximum and minimum values of functions, known as extrema, a vital tool for analyzing many real-life situations.

30 min
Increasing and Decreasing Functions

13: Increasing and Decreasing Functions

Use the first derivative to determine where graphs are increasing or decreasing. Next, investigate Rolle's theorem and the mean value theorem, one of whose consequences is that during a car trip, your actual speed must correspond to your average speed during at least one point of your journey.

31 min
Concavity and Points of Inflection

14: Concavity and Points of Inflection

What does the second derivative reveal about a graph? It describes how the curve bends—whether it is concave upward or downward. You determine concavity much as you found the intervals where a graph was increasing or decreasing, except this time you use the second derivative.

31 min
Curve Sketching and Linear Approximations

15: Curve Sketching and Linear Approximations

By using calculus, you can be certain that you have discovered all the properties of the graph of a function. After learning how this is done, focus on the tangent line to a graph, which is a convenient approximation for values of the function that lie close to the point of tangency.

32 min
Applications—Optimization Problems, Part 1

16: Applications—Optimization Problems, Part 1

Attack real-life problems in optimization, which requires finding the relative extrema of different functions by differentiation. Calculate the optimum size for a box, and the largest area that can be enclosed by a circle and a square made from a given length of wire.

31 min
Applications—Optimization Problems, Part 2

17: Applications—Optimization Problems, Part 2

Conclude your investigation of differential calculus with additional problems in optimization. For success with such word problems, Professor Edwards stresses the importance of first framing the problem with precalculus, reducing the equation to one independent variable, and then using calculus to find and verify the answer.

31 min
Antiderivatives and Basic Integration Rules

18: Antiderivatives and Basic Integration Rules

Up until now, you've calculated a derivative based on a given function. Discover how to reverse the procedure and determine the function based on the derivative. This approach is known as obtaining the antiderivative, or integration. Also learn the notation for integration.

31 min
The Area Problem and the Definite Integral

19: The Area Problem and the Definite Integral

One of the classic problems of integral calculus is finding areas bounded by curves. This was solved for simple curves by the ancient Greeks. See how a more powerful method was later developed that produces a number called the definite integral, and learn the relevant notation.

31 min
The Fundamental Theorem of Calculus, Part 1

20: The Fundamental Theorem of Calculus, Part 1

The two essential ideas of this course—derivatives and integrals—are connected by the fundamental theorem of calculus, one of the most important theorems in mathematics. Get an intuitive grasp of this deep relationship by working several problems and surveying a proof.

30 min
The Fundamental Theorem of Calculus, Part 2

21: The Fundamental Theorem of Calculus, Part 2

Try examples using the second fundamental theorem of calculus, which allows you to let the upper limit of integration be a variable. In the process, explore more relationships between differentiation and integration, and discover how they are almost inverses of each other.

31 min
Integration by Substitution

22: Integration by Substitution

Investigate a straightforward technique for finding antiderivatives, called integration by substitution. Based on the chain rule, it enables you to convert a difficult problem into one that's easier to solve by using the variable "u" to represent a more complicated expression.

31 min
Numerical Integration

23: Numerical Integration

When calculating a definite integral, the first step of finding the antiderivative can be difficult or even impossible. Learn the trapezoid rule, one of several techniques that yield a close approximation to the definite integral. Then do a problem involving a plot of land bounded by a river.

31 min
Natural Logarithmic Function—Differentiation

24: Natural Logarithmic Function—Differentiation

Review the properties of logarithms in base 10. Then see how the so-called natural base for logarithms, "e," has important uses in calculus and is one of the most significant numbers in mathematics. Learn how such natural logarithms help to simplify derivative calculations.

31 min
Natural Logarithmic Function—Integration

25: Natural Logarithmic Function—Integration

Continue your investigation of logarithms by looking at some of the consequences of the integral formula developed in the previous lecture. Next, change gears and review inverse functions at the precalculus level, preparing the way for a deeper exploration of the subject in coming lectures.

31 min
Exponential Function

26: Exponential Function

The inverse of the natural logarithmic function is the exponential function, perhaps the most important function in all of calculus. Discover that this function has an amazing property: It is its own derivative! Also see the connection between the exponential function and the bell-shaped curve in probability.

31 min
Bases other than e

27: Bases other than e

Extend the use of the logarithmic and exponential functions to bases other than "e," exploiting this approach to solve a problem in radioactive decay. Also learn to find the derivatives of such functions, and see how "e" emerges in other mathematical contexts, including the formula for continuous compound interest.

31 min
Inverse Trigonometric Functions

28: Inverse Trigonometric Functions

Turn to the last set of functions you will need in your study of calculus, inverse trigonometric functions. Practice using some of the formulas for differentiating these functions. Then do an entertaining problem involving how fast the rotating light on a police car sweeps across a wall and whether you can evade it.

31 min
Area of a Region between 2 Curves

29: Area of a Region between 2 Curves

Revisit the area problem and discover how to find the area of a region bounded by two curves. First imagine that the region is divided into representative rectangles. Then add up an infinite number of these rectangles, which corresponds to a definite integral.

30 min
Volume—The Disk Method

30: Volume—The Disk Method

Learn how to calculate the volume of a solid of revolution—an object that is symmetrical around its axis of rotation. As in the area problem in the previous lecture, you imagine adding up an infinite number of slices—in this case, of disks rather than rectangles—which yields a definite integral.

30 min
Volume—The Shell Method

31: Volume—The Shell Method

Apply the shell method for measuring volumes, comparing it with the disk method on the same shape. Then find the volume of a doughnut-shaped object called a torus, along with the volume for a figure called Gabriel's Horn, which is infinitely long but has finite volume.

31 min
Applications—Arc Length and Surface Area

32: Applications—Arc Length and Surface Area

Investigate two applications of calculus that are at the heart of engineering: measuring arc length and surface area. One of your problems is to determine the length of a cable hung between two towers, a shape known as a catenary. Then examine a peculiar paradox of Gabriel's Horn.

32 min
Basic Integration Rules

33: Basic Integration Rules

Review integration formulas studied so far, and see how to apply them in various examples. Then explore cases in which a calculator gives different answers from the ones obtained by hand calculation, learning why this occurs. Finally, Professor Edwards gives advice on how to succeed in introductory calculus.

30 min
Other Techniques of Integration

34: Other Techniques of Integration

Closing your study of integration techniques, explore a powerful method for finding antiderivatives: integration by parts, which is based on the product rule for derivatives. Use this technique to calculate area and volume. Then focus on integrals involving products of trigonometric functions.

31 min
Differential Equations and Slope Fields

35: Differential Equations and Slope Fields

Explore slope fields as a method for getting a picture of possible solutions to a differential equation without having to solve it, examining several problems of the type that appear on the Advanced Placement exam. Also look at a solution technique for differential equations called separation of variables.

30 min
Applications of Differential Equations

36: Applications of Differential Equations

Use your calculus skills in three applications of differential equations: first, calculate the radioactive decay of a quantity of plutonium; second, determine the initial population of a colony of fruit flies; and third, solve one of Professor Edwards's favorite problems by using Newton's law of cooling to predict the cooling time for a cup of coffee.

31 min