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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 4.8 out of 5 by 98.
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Rated 5 out of 5 by from I think a good bit did rub off Professor. It was also like climbing a mountain and I found about halfway thru that I started to anticipate or look forward to the class. (I kept thatto my self0. I had to repeat the first 10 classes and also began to do 10 or fifteen minutes in a sitting after that with more complete notetaking. My goal was to revive some math skill and calculus from decades ago. This more than did that, it was better than I remembered. I do not need AP prep but if an 18 yr old does, this will be a great help. Professor Edward’s is just a fantastic, humane and brilliant math teacher. My wish is that he would teach a shorter follow up class of everyday use or basic science applications so that we can practice and enjoy the benefits of math daily in non prof. life. Like the coffee or Chernobyl problem. Thank you
Date published: 2022-04-29
Rated 5 out of 5 by from Amazing Teacher! I have a degree in math and I took this course as a review. To be honest, I never really understood the concepts of Calculus, and just memorized theorems, and did few applications. Most math professors are not good teachers. Bruce Edwards is an exception. He not only knows his material but is enthusiastic in his approach. He gives examples, used graphs, teaches the theory, does problems and gives hints. I actually recommended the course to my son who has a PhD in math and is also a college professor to review Professor Edward’s technique.
Date published: 2021-10-08
Rated 5 out of 5 by from Made the concepts easily understandable As he warned, the concepts of calculus are understandable, particularly the way he presents them, but the math is messy. I have to admit, though, my math is not as messy as it was before I took Professor Edwards' Pre-Calculus and Calculus 1 courses, both very well done. On to Calculus 2!
Date published: 2021-08-25
Rated 5 out of 5 by from Excellent set of lectures! After retiring from my 40 year engineering career, I finally had time to go back and cover some of the material that had become foggy since my original study of calculus many years ago. I really appreciated the examples and shared Dr. Edwards' enthusiasm for the subject. I also remain amazed with the capabilities of the original pioneers like Newton and Leibniz who discovered calculus to help explain the world around them.
Date published: 2021-08-03
Rated 5 out of 5 by from Great Professor, Great Course Edwards is fun and engaging. Efficient in his teach and keeps things bearably simple. Everything is illustrated or presented on the screen. But I'd still recommend running through the coursebook to master what he teaches. Couldn't have asked for a better course.
Date published: 2021-06-17
Rated 4 out of 5 by from Slightly dry, but a good exposure to calculus Good review of calculus. This is very academic, and you have to want to learn more about calculus, and be willing to do work on your own, if you want this to be worthwhile.
Date published: 2021-03-07
Rated 5 out of 5 by from Kind of a quirky style, but effective nonetheless and entertaining. Keeps it simple, which is the key to absorbing material. Examples may seem trivial at first but in the end this is how you learn best.
Date published: 2021-03-05
Rated 4 out of 5 by from Back to College! I studied mathematics at university, so I have a fairly good grounding in the subject. However, while reading some recent publications about maths and physics, I began to realise how much I have forgotten of those distant student days. Professor Edwards course is a useful reminder of the ground rules of calculus. It is surprising how quickly some of the principles come back. I especially enjoyed renewing my acquaintance with methods of calculating area and volumes. I'll probably watch the lectures gain before moving on to Part 2 in the same series. Much enjoyed!
Date published: 2021-02-18
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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

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