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Understanding Calculus: Problems, Solutions, and Tips

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Understanding Calculus: Problems, Solutions, and Tips

Understanding Calculus: Problems, Solutions, and Tips

Professor Bruce H. Edwards Ph.D.
University of Florida
Course No.  1007
Course No.  1007
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Course Overview

About This Course

36 lectures  |  30 minutes per lecture

Calculus is the greatest mathematical breakthrough since the pioneering discoveries of the ancient Greeks. Without it, we wouldn't have spaceflight, skyscrapers, jet planes, economic modeling, accurate weather forecasting, modern medical technologies, or any of the countless other achievements we take for granted in today's world.

Indeed, calculus is so versatile and its techniques so diverse that it trains you to view problems, no matter how difficult, as solvable until proved otherwise. And the habit of turning a problem over in your mind, choosing an approach, and then working through a solution teaches you to think clearly—which is why the study of calculus is so crucial for improving your cognitive skills and why it is a prerequisite for admission to most top universities.

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Calculus is the greatest mathematical breakthrough since the pioneering discoveries of the ancient Greeks. Without it, we wouldn't have spaceflight, skyscrapers, jet planes, economic modeling, accurate weather forecasting, modern medical technologies, or any of the countless other achievements we take for granted in today's world.

Indeed, calculus is so versatile and its techniques so diverse that it trains you to view problems, no matter how difficult, as solvable until proved otherwise. And the habit of turning a problem over in your mind, choosing an approach, and then working through a solution teaches you to think clearly—which is why the study of calculus is so crucial for improving your cognitive skills and why it is a prerequisite for admission to most top universities.

Understanding Calculus: Problems, Solutions, and Tips

immerses you in the unrivaled learning adventure of this mathematical field in 36 half-hour lectures that 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. With crystal-clear explanations of the beautiful ideas of calculus, frequent study tips, pitfalls to avoid, and—best of all—hundreds of examples and practice problems that are specifically designed to explain and reinforce major concepts, this course will be your sure and steady guide to conquering calculus.

Your teacher for this intensively illustrated DVD set is Professor Bruce H. Edwards, an award-winning instructor at the University of Florida and the coauthor of a best-selling series of calculus textbooks.

Accomplish Mathematical Wonders

Calculus is one of the most powerful and astonishing tools ever invented, yet it is a skill that can be learned by anyone with an understanding of high school mathematics.

Among its many uses, calculus teaches you to

  • analyze a multitude of situations involving change, whether it's an accelerating rocket, the growth of a bacterial colony, or fluctuating stock prices;
  • calculate optimum values, such as the greatest volume for a box with a given surface area or the highest feasible profit from the sales of an item;
  • measure complex shapes—for example, the volume of a doughnut-shaped object called a torus or the area of a plot of land bounded by a river.
Learn about Precalculus and Limits . . .

Solving many types of calculus problems usually requires employing precalculus—algebra and trigonometry—to work out a solution. For this reason, Professor Edwards devotes the first few lectures to reviewing key topics in precalculus, then he covers some basic concepts such as limits and continuity before moving on to the two simple, yet brilliant ideas behind calculus—the derivative and the integral.

Despite the apparent differences between the derivative and integral, you discover that they are inextricably linked by the surprising fundamental theorem of calculus. Throughout the course, you will discover that simplicity is one of the hallmarks of the essential ideas of calculus.

. . . the Power of the Derivative . . .

The derivative is the foundation of differential calculus, which you study through Lecture 17, exploring its many applications in science, engineering, business, and other fields.

You start with a classic problem that illustrates one of the core ideas of calculus: Can you find the tangent line to a curve at a given point? This is the same as asking if the rate of change of the curve can be measured at that point—with a host of potential applications in situations where a quantity is changing, such as the speed of an accelerating vehicle. The answer is: Yes, and with amazing simplicity! After learning the steps involved, you have solved your first calculus problem.

You then

  • study a variety of ways to find derivatives, including the power rule, the constant multiple rule, the quotient rule, the chain rule, and implicit differentiation;
  • learn how to find extrema—the absolute maximum and minimum values of functions, using derivatives; and
  • apply derivatives to solve a variety of real-world problems.
. . . and the Importance of the Integral

Next, you are introduced to the integral, using a classic problem in which you are asked to find the area of a plot of land bounded by curves. To solve this problem, calculus provides us with the integral—a powerful tool that allows us to calculate areas, volumes, and other characteristics of complex shapes. The balance of the course is devoted to integral calculus and its applications. You study

  • arc length and surface area—two applications of calculus that are at the heart of engineering;
  • integration by substitution—a method that enables you to convert a difficult problem into one that's easier to solve; and
  • the formulas for continuous compound interest, radioactive decay, and a host of other real-world applications.

A Calculus Course for All

Understanding Calculus is well suited for anyone who wants to take the leap into one of history's greatest intellectual achievements, whether for the first time or for review. Those who will benefit include these learners:

  • Any student now studying calculus who would like personal coaching from a professor who has spent years honing his explanations for the areas that are most challenging to students. This course is specifically designed to 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.
  • Parents of students studying calculus, a subject with which they often give up trying to help their high-school-age children—at a critical turning point in their educational careers.
  • Those who have already taken calculus and who need a thorough review.
  • Anyone who didn't understand calculus on the first try and wants a lucid, in-depth presentation, with lots of interesting, well-explained practice problems.

The plentiful graphs, equations, and other visual aids in these lectures are clear and well-designed, allowing you to follow each step of Professor Edwards's presentation in detail. The accompanying workbook includes lecture summaries, sample problems and worked-out solutions, tips, and pitfalls; lists of formulas and theorems; a trigonometry review sheet; a glossary; and a removable study sheet to use for quick and easy reference during the lectures.

The Ideal Calculus Teacher

Professor Edwards is the ideal calculus teacher—friendly, animated, encouraging, and witty, but also focused on presenting the material in an organized and understandable way. For anyone who feels intimidated by calculus, there is a distinct joy in being able to calculate a derivative after just a few lessons. It's easier than one might have supposed, and it opens an amazing new world of insight.

As an educator who has been honored repeatedly, both for his teaching and for his textbooks, Professor Edwards is a fount of valuable advice. He offers frequent tips for success, including guidance for those preparing for the Advanced Placement Calculus AB exam, for which he has served as a grader and for which this course is excellent preparation. Among his suggestions are these:

  • Graphing calculators: While some calculus teachers prefer that their students not use graphing calculators, the Advanced Placement exam requires them. Professor Edwards points out the strengths of graphing calculators as well as the weaknesses—for example, that in certain situations they can fool you.
  • Memorization: Always memorize what your teacher assigns. However, no one can memorize all the formulas in calculus. A good approach is to commit to memory the idea behind a technique—for example, that the disk method of computing the volume of a solid involves slicing it into innumerable disks.

Ever since its inception in the 17th century, calculus has spawned a continuing flood of new ideas and techniques for solving problems. It's easy to be overwhelmed by the richness of this subject, which is why many beginning students find themselves struggling.

Through Professor Edwards's exceptional teaching in Understanding Calculus, you will come away with a deep appreciation for the extraordinary power of calculus, a grasp of which methods apply to different types of problems, and, with practice, a facility for unlocking the secrets of the ceaselessly changing world around us.

View Less
36 Lectures
  • 1
    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. x
  • 2
    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. x
  • 3
    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. x
  • 4
    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. x
  • 5
    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. x
  • 6
    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. x
  • 7
    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. x
  • 8
    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. x
  • 9
    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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x

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Bruce H. Edwards
Ph.D. Bruce H. Edwards
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 Bogot·, 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.
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Reviews

Rated 4.9 out of 5 by 43 reviewers.
Rated 5 out of 5 by Calculus can be made logical and clear Let me begin by stating I am a retired mathematics instructor who had a very rewarding career of 39 years including a highly successful Advanced Placement Calculus AB class for the last eighteen. I used the Calculus and PreCalculus textbooks, which I thought were the best available, written by Dr. Edwards and his coauthors. Dr. Edwards’ presentation in this excellent Teaching Company course is clear and logical. His carefully chosen examples are insightful and illuminating. I have always maintained mathematics is the worst taught subject in America but hopefully more instructors like Dr. Edwards will change that. This course should especially appeal to people who need a more in depth TTC class than the one offered by Dr. Michael Starbird. March 19, 2010
Rated 5 out of 5 by Excellent Review I am coming back to calculus after years as both a graduate student and as an engineering manager in the defense industry. Having now retired, I decided to go back and review some of the topics that I had studied, but had long since forgotten. I found the Calculus I course extremely well organized and well-presented. Dr. Edwards does an excellent job of presenting the material in a logical, well paced manner, which, when combined with the problems presented in the workbook lead the student naturally from one topic to the next. I have now purchased the Calculus II course and am working my way through that. I wish I had had this resource available when I was studying engineering as a grad student. September 6, 2014
Rated 5 out of 5 by Excellent Course that is well presented Dr. Edwards takes a difficult and complex math subject and makes is fun and easier to understand. This is a great refresher course even after 50 years since my last Calculus course. I would recommend it to AP high school students as a preparation for the AP exams. I am looking forward to Dr. Edwards next in line course of Calculus II. September 2, 2014
Rated 5 out of 5 by Great Calculus course Professor Edward and Professor Sellers are my favorite Math teachers ever. Am almost 70 and when I was young there was a difficult choice to make between Liberal Arts and Science. I opted for the Liberal because I felt that the profound social issues in my country required of me to be a journalist. That happened about 40 years ago. My love for Math stayed deep inside. Now I have the opportunity to get back on track. Professor Edwards is a brilliant mathematician but I haven't figured out whether his true talent is in his knowledge of Math or in his skill to explain the subject in such a clear, engaging and refreshing way. To make the story short, I started refreshing my Math using Algebra I and II by professor Sellers. Also, took on Precalculus with Prof. Edwards and now I am in the middle of Calculus I, with Calculus II in line. My old flame waked up and now I have a goal to accomplish: to be a Math teacher, for which I am enrolling at South New Hampshire Online University for a full bachelor's degree in Math. Honestly, it would have been impossible without Mr. Sellers and Mr. Edwards courses. Thank you both. My only complaint is the book. It contains several errors and omissions. In page 150 the following statement can be found: x^2 + y^2 = 25. It took me a good half hour to realize that it was a mistake. It should read X^2 + y^2 = 25^2. Granted, for the solution it didn't matter because the derivative would be 0 for the numeral value anyway. But... Also, sometimes the book omits important steps and the ride toward the solution is not always smooth. However that does not diminish the greatness of this course and I encourage the Teaching Company to give us more of him through new offerings, such as a Calculus III. This course easily tops the MIT course on Calculus I by a good margin. Everything is nice: Mr. Edward pristine presentation, the special effects and the use of his time which avoid the temptation to write on a blackboard. This is pure Math discourse. Almost perfect. . July 30, 2014
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