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Understanding Calculus: Problems, Solutions, and Tips

Season 1
These 36 episodes 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.
201036 episodesTV-PG
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Episodes

  1. S1 E1 - A Preview of Calculus
    March 4, 2010
    33min
    TV-PG
    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
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  2. S1 E2 - Review - Graphs, Models, and Functions
    March 4, 2010
    30min
    TV-PG
    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.
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  3. S1 E3 - Review - Functions and Trigonometry
    March 4, 2010
    30min
    TV-PG
    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.
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  4. S1 E4 - Finding Limits
    March 4, 2010
    31min
    TV-PG
    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.
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  5. S1 E5 - An Introduction to Continuity
    March 4, 2010
    31min
    TV-PG
    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.
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  6. S1 E6 - Infinite Limits and Limits at Infinity
    March 4, 2010
    31min
    TV-PG
    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.
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  7. S1 E7 - The Derivative and the Tangent Line Problem
    March 4, 2010
    31min
    TV-PG
    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.
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  8. S1 E8 - Basic Differentiation Rules
    March 4, 2010
    30min
    TV-PG
    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.
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  9. S1 E9 - Product and Quotient Rules
    March 4, 2010
    31min
    TV-PG
    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.
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  10. S1 E10 - The Chain Rule
    March 4, 2010
    31min
    TV-PG
    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.
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  11. S1 E11 - Implicit Differentiation and Related Rates
    March 4, 2010
    31min
    TV-PG
    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).
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  12. S1 E12 - Extrema on an Interval
    March 4, 2010
    30min
    TV-PG
    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.
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  13. S1 E13 - Increasing and Decreasing Functions
    March 4, 2010
    31min
    TV-PG
    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.
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  14. S1 E14 - Concavity and Points of Inflection
    March 4, 2010
    31min
    TV-PG
    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.
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  15. S1 E15 - Curve Sketching and Linear Approximations
    March 4, 2010
    32min
    TV-PG
    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.
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  16. S1 E16 - Applications - Optimization Problems, Part 1
    March 4, 2010
    31min
    TV-PG
    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.
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  17. S1 E17 - Applications - Optimization Problems, Part 2
    March 4, 2010
    31min
    TV-PG
    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.
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  18. S1 E18 - Antiderivatives and Basic Integration Rules
    March 4, 2010
    31min
    TV-PG
    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.
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  19. S1 E19 - The Area Problem and the Definite Integral
    March 4, 2010
    31min
    TV-PG
    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.
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  20. S1 E20 - The Fundamental Theorem of Calculus, Part 1
    March 4, 2010
    30min
    TV-PG
    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.
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  21. S1 E21 - The Fundamental Theorem of Calculus, Part 2
    March 4, 2010
    31min
    TV-PG
    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.
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  22. S1 E22 - Integration by Substitution
    March 4, 2010
    31min
    TV-PG
    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.
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  23. S1 E23 - Numerical Integration
    March 4, 2010
    31min
    TV-PG
    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.
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  24. S1 E24 - Natural Logarithmic Function - Differentiation
    March 4, 2010
    31min
    TV-PG
    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.
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  25. S1 E25 - Natural Logarithmic Function - Integration
    March 4, 2010
    31min
    TV-PG
    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.
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Extras

Bonus

Understanding Calculus: Problems, Solutions, and Tips
Understanding Calculus: Problems, Solutions, and Tips
1minTV-PG
These 36 episodes 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.
These 36 episodes 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.
These 36 episodes 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.

Details

More info

Subtitles
None available
Producers
The Great Courses
Cast
Bruce H. Edwards
Studio
The Great Courses
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