Understanding Calculus: Problems, Solutions, and Tips
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Episodes
- S1 E1 - A Preview of Calculus
March 4, 201033minCalculus 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.Free trial of The Great Courses Signature Collection or buy - S1 E2 - Review - Graphs, Models, and Functions
March 4, 201030minIn 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.Free trial of The Great Courses Signature Collection or buy - S1 E3 - Review - Functions and Trigonometry
March 4, 201030minContinue 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.Free trial of The Great Courses Signature Collection or buy - S1 E4 - Finding Limits
March 4, 201031minJump 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.Free trial of The Great Courses Signature Collection or buy - S1 E5 - An Introduction to Continuity
March 4, 201031minBroadly 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.Free trial of The Great Courses Signature Collection or buy - S1 E6 - Infinite Limits and Limits at Infinity
March 4, 201031minInfinite 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.Free trial of The Great Courses Signature Collection or buy - S1 E7 - The Derivative and the Tangent Line Problem
March 4, 201031minBuilding 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.Free trial of The Great Courses Signature Collection or buy - S1 E8 - Basic Differentiation Rules
March 4, 201030minPractice 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.Free trial of The Great Courses Signature Collection or buy - S1 E9 - Product and Quotient Rules
March 4, 201031minLearn 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.Free trial of The Great Courses Signature Collection or buy - S1 E10 - The Chain Rule
March 4, 201031minDiscover 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.Free trial of The Great Courses Signature Collection or buy - S1 E11 - Implicit Differentiation and Related Rates
March 4, 201031minConquer 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).Free trial of The Great Courses Signature Collection or buy - S1 E12 - Extrema on an Interval
March 4, 201030minHaving 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.Free trial of The Great Courses Signature Collection or buy - S1 E13 - Increasing and Decreasing Functions
March 4, 201031minUse 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.Free trial of The Great Courses Signature Collection or buy - S1 E14 - Concavity and Points of Inflection
March 4, 201031minWhat 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.Free trial of The Great Courses Signature Collection or buy - S1 E15 - Curve Sketching and Linear Approximations
March 4, 201032minBy 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.Free trial of The Great Courses Signature Collection or buy - S1 E16 - Applications - Optimization Problems, Part 1
March 4, 201031minAttack 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.Free trial of The Great Courses Signature Collection or buy - S1 E17 - Applications - Optimization Problems, Part 2
March 4, 201031minConclude 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.Free trial of The Great Courses Signature Collection or buy - S1 E18 - Antiderivatives and Basic Integration Rules
March 4, 201031minUp 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.Free trial of The Great Courses Signature Collection or buy - S1 E19 - The Area Problem and the Definite Integral
March 4, 201031minOne 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.Free trial of The Great Courses Signature Collection or buy - S1 E20 - The Fundamental Theorem of Calculus, Part 1
March 4, 201030minThe 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.Free trial of The Great Courses Signature Collection or buy - S1 E21 - The Fundamental Theorem of Calculus, Part 2
March 4, 201031minTry 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.Free trial of The Great Courses Signature Collection or buy - S1 E22 - Integration by Substitution
March 4, 201032minInvestigate 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.Free trial of The Great Courses Signature Collection or buy - S1 E23 - Numerical Integration
March 4, 201031minWhen 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.Free trial of The Great Courses Signature Collection or buy - S1 E24 - Natural Logarithmic Function - Differentiation
March 4, 201031minReview 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.Free trial of The Great Courses Signature Collection or buy - S1 E25 - Natural Logarithmic Function - Integration
March 4, 201031minContinue 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.Free trial of The Great Courses Signature Collection or buy
Extras
Bonus
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.
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