Understanding Calculus: Problems, Solutions, and Tips

Watch Understanding Calculus: Problems, Solutions, and Tips

  • 2010
  • 1 Season

Understanding Calculus: Problems, Solutions, and Tips from The Great Courses Signature Collection is a comprehensive guide to learning calculus hosted by Bruce H. Edwards. The show is designed to teach students the fundamentals of calculus by providing a wide range of problems, solutions, and tips. The aim of the show is to simplify the complex nature of calculus by breaking it down into relatable concepts and applications.

Bruce H. Edwards is a distinguished professor of mathematics at the University of Florida, and has contributed significantly to calculus education. He is an experienced teacher, and his expertise shines through as he makes difficult calculus concepts accessible to anyone. Edwards is a charismatic host who is skilled in explaining calculus, making Understanding Calculus an engaging and informative show.

The show is split into thirty lessons that are each approximately thirty minutes in length. Each lesson is structured to teach one specific concept, which builds upon previous knowledge. Edwards introduces each lesson with a problem or challenge that students must solve. He then provides a step-by-step solution, carefully breaking down each stage and explaining why the solution works.

A great feature of the show is the use of visuals to complement Edwards' explanations. Animations aid in simplifying complex mathematical formulas and graphs, making them easier to comprehend. These visuals are especially helpful when it comes to visualizing curves, gradients, and tangent lines.

Understanding Calculus covers a wide range of calculus topics that will be extremely useful for anyone studying math, physics, or engineering. The first few episodes focus on curve sketching, differentiation, and applications of derivatives. This is followed by lessons on integrals, optimization, and multivariable calculus.

Throughout the show, Edwards provides students with tips that will help them become better problem-solvers. These tips emphasize the importance of understanding the fundamentals of calculus so that students can detect patterns and make connections between concepts. The tips also show students how to solve real-world problems using calculus.

Another excellent feature of this show is the diverse range of problems that Edwards presents. The problems range from basic to advanced applications, making it a great tool for students who are just starting their calculus journey as well as those who already have some calculus knowledge.

One of the best aspects of Understanding Calculus is the way it emphasizes the connections between different calculus concepts. Edwards shows how derivatives and integrals can be used together to solve a problem, and teaches students how to recognize patterns and structures in calculus problems. This not only makes calculus easier to learn, but also more enjoyable for students.

The show includes practice problems at the end of each lesson, allowing students to test their understanding of the content. The problems are also accompanied by detailed solutions and explanations, which are incredibly helpful for those who are struggling to understand a particular concept.

In conclusion, Understanding Calculus: Problems, Solutions, and Tips from The Great Courses Signature Collection is an excellent resource for anyone who wants to learn calculus. Bruce H. Edwards' expertise as a mathematics professor shines through, making otherwise inaccessible concepts understandable to all. The use of visuals, diverse range of problems, and tips for solving problems are just a few of the things that make this show a valuable tool for any calculus student.

Understanding Calculus: Problems, Solutions, and Tips is a series that is currently running and has 1 seasons (36 episodes). The series first aired on March 5, 2010.

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Applications of Differential Equations
36. Applications of Differential Equations
March 5, 2010
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.
Differential Equations and Slope Fields
35. Differential Equations and Slope Fields
March 5, 2010
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.
Other Techniques of Integration
34. Other Techniques of Integration
March 5, 2010
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.
Basic Integration Rules
33. Basic Integration Rules
January 1, 1970
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.
Applications - Arc Length and Surface Area
32. Applications - Arc Length and Surface Area
March 5, 2010
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.
Volume-The Shell Method
31. Volume-The Shell Method
January 1, 1970
Summary is not available.
Volume - The Disk Method
30. Volume - The Disk Method
March 5, 2010
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 episode, imagine adding up an infinite number of slices - in this case, of disks rather than rectangles - which yields a definite integral.
Area of a Region between 2 Curves
29. Area of a Region between 2 Curves
March 5, 2010
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.
Inverse Trigonometric Functions
28. Inverse Trigonometric Functions
March 5, 2010
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.
Bases other than
27. Bases other than "e"
March 5, 2010
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.
Exponential Function
26. Exponential Function
March 5, 2010
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.
Natural Logarithmic Function - Integration
25. Natural Logarithmic Function - Integration
March 5, 2010
Continue your investigation of logarithms by looking at some of the consequences of the integral formula developed in the previous episode. Next, change gears and review inverse functions at the precalculus level, preparing the way for a deeper exploration of the subject in coming episodes.
Natural Logarithmic Function-Differentiation
24. Natural Logarithmic Function-Differentiation
January 1, 1970
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.
Numerical Integration
23. Numerical Integration
March 5, 2010
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.
Integration by Substitution
22. Integration by Substitution
March 5, 2010
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.
The Fundamental Theorem of Calculus, Part 2
21. The Fundamental Theorem of Calculus, Part 2
March 5, 2010
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.
The Fundamental Theorem of Calculus, Part 1
20. The Fundamental Theorem of Calculus, Part 1
January 1, 1970
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.
The Area Problem and the Definite Integral
19. The Area Problem and the Definite Integral
March 5, 2010
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.
Antiderivatives and Basic Integration Rules
18. Antiderivatives and Basic Integration Rules
March 5, 2010
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.
Applications-Optimization Problems, Part 2
17. Applications-Optimization Problems, Part 2
January 1, 1970
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.
Applications - Optimization Problems, Part 1
16. Applications - Optimization Problems, Part 1
March 5, 2010
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.
Curve Sketching and Linear Approximations
15. Curve Sketching and Linear Approximations
March 5, 2010
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.
Concavity and Points of Inflection
14. Concavity and Points of Inflection
March 5, 2010
What does the second derivative reveal about a graph? It describes how the curve bends, whether it is concave upward or downward. Determine concavity much as you found the intervals where a graph was increasing or decreasing, except this time you'll use the second derivative.
Increasing and Decreasing Functions
13. Increasing and Decreasing Functions
March 5, 2010
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.
Extrema on an Interval
12. Extrema on an Interval
March 5, 2010
Having covered the rules for finding derivatives, embark on the first of five episodes 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.
Implicit Differentiation and Related Rates
11. Implicit Differentiation and Related Rates
March 5, 2010
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).
The Chain Rule
10. The Chain Rule
March 5, 2010
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.
Product and Quotient Rules
9. Product and Quotient Rules
March 5, 2010
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 episode. Also investigate higher-order derivatives, differential equations, and horizontal tangents.
Basic Differentiation Rules
8. Basic Differentiation Rules
March 5, 2010
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.
The Derivative and the Tangent Line Problem
7. The Derivative and the Tangent Line Problem
March 5, 2010
Building on what you've 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.
Infinite Limits and Limits at Infinity
6. Infinite Limits and Limits at Infinity
March 5, 2010
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.
An Introduction to Continuity
5. An Introduction to Continuity
March 5, 2010
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.
Finding Limits
4. Finding Limits
March 5, 2010
Jump into real calculus by going deeper into the concept of limits introduced in the first episode. 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.
Review - Functions and Trigonometry
3. Review - Functions and Trigonometry
March 5, 2010
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.
Review - Graphs, Models, and Functions
2. Review - Graphs, Models, and Functions
March 5, 2010
In the first of two review episodes 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.
A Preview of Calculus
1. A Preview of Calculus
March 5, 2010
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 series. Then get your feet wet by investigating the classic tangent line problem, which illustrates the concept of limits. #Science & Mathematics
Where to Watch Understanding Calculus: Problems, Solutions, and Tips
Understanding Calculus: Problems, Solutions, and Tips is available for streaming on the The Great Courses Signature Collection website, both individual episodes and full seasons. You can also watch Understanding Calculus: Problems, Solutions, and Tips on demand at Apple TV Channels and Amazon Prime and Amazon.
  • Premiere Date
    March 5, 2010