Understanding Calculus II: Problems, Solutions, and Tips

Watch Understanding Calculus II: Problems, Solutions, and Tips

  • 2013
  • 1 Season

Understanding Calculus II: Problems, Solutions, and Tips from The Great Courses Signature Collection is an outstanding educational program about calculus. The show is presented by Bruce H. Edwards, a well-known mathematician and a professor of mathematics. He delivers the lectures in an easy-to-understand, friendly manner making the entire experience of learning calculus an enjoyable one.

Calculus II is considered one of the most difficult courses in the field of mathematics. However, Bruce H. Edwards' presentation of the material is unique, concise, and clear, making it accessible even to those with limited math skills or rusty calculus skills. The course highlights sophisticated, challenging math topics and makes them digestible, fun, and interactive.

This program covers essential calculus II concepts, including integration techniques, sequences and series, power series, and functions in several variables. Bruce H. Edwards breaks down seemingly complex concepts into smaller, understandable parts, making it accessible for anyone, from novices to advanced students.

In each session, Bruce presents common problems and challenges that students typically face while studying Calculus II. He then provides easy-to-follow techniques and tips to help students solve these problems. Throughout the course, he emphasizes essential attributes of calculus, including abstract and critical reasoning, and mastery of critical concepts that are useful in life and the workplace.

The show provides examples of how to solve problems using both traditional paper-and-pencil methods and computer software, which is a valuable resource for students. Moreover, the program uses various interactive and graphing activities to help students visualize complex concepts, deftly bringing the material to life.

The presenters incorporate real-life applications of Calculus II, making the program even more relevant to students. For example, Calculus II is crucial for engineers, physicists, and other computational fields, and the program shows how it can be applied to solve practical problems, such as optimizing real-world processes.

Bruce H. Edwards' teaching style is one of the most valuable aspects of this series. He is engaging, approachable and clearly passionate about the subject, which is infectious. He is also an expert in Calculus II, which is evident in how he explains the material. The program touches on key areas of calcuclar knowledge that are both foundational as well as advanced but not beyond reach.

The show is delivered in a series of lectures, each around 30 minutes long, which makes it easy to follow along at home, in the park, on a plane or wherever is convenient. The program is produced professionally, with high-quality videography and clear sound that is easy to follow.

In conclusion, Understanding Calculus II: Problems, Solutions, and Tips from The Great Courses Signature Collection is a valuable resource for those who want to master Calculus II. Bruce H. Edwards' presentation style is both approachable and engaging, with examples and solutions that students can apply to their own learning. The interactive activities and software-based modeling are incredibly useful, and his real-world examples will help students see the relevancy of Calculus II in the practical world. Overall, the show is a must-watch for students who want to get better at calculus and for those who are interested in expanding their knowledge of the discipline.

Understanding Calculus II: Problems, Solutions, and Tips is a series that is currently running and has 1 seasons (36 episodes). The series first aired on May 31, 2013.

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Curvature and the Maximum Bend of a Curve
36. Curvature and the Maximum Bend of a Curve
May 31, 2013
See how the concept of curvature helps with analysis of the acceleration vector. Come full circle by using ideas from elementary calculus to determine the point of maximum curvature. Then close by looking ahead at the riches offered by the continued study of calculus.
Acceleration's Tangent and Normal Vectors
35. Acceleration's Tangent and Normal Vectors
May 31, 2013
Use the unit tangent vector and normal vector to analyze acceleration. The unit tangent vector points in the direction of motion. The unit normal vector points in the direction an object is turning. Learn how to decompose acceleration into these two components.
Velocity and Acceleration
34. Velocity and Acceleration
May 31, 2013
Combine parametric equations, curves, vectors, and vector-valued functions to form a model for motion in the plane. In the process, derive equations for the motion of a projectile subject to gravity. Solve several projectile problems, including whether a baseball hit at a certain velocity will be a home run.
Vector-Valued Functions
33. Vector-Valued Functions
May 31, 2013
Use your knowledge of vectors to explore vector-valued functions, which are functions whose values are vectors. The derivative of such a function is a vector tangent to the graph that points in the direction of motion. An important application is describing the motion of a particle.
The Dot Product of Two Vectors
32. The Dot Product of Two Vectors
May 31, 2013
Deepen your skill with vectors by exploring the dot product method for determining the angle between two nonzero vectors. Then turn to projections of one vector onto another. Close with some typical applications of dot product and projection that involve force and work.
Vectors in the Plane
31. Vectors in the Plane
May 31, 2013
Begin a series of episodes on vectors in the plane by defining vectors and their properties, and reviewing vector notation. Then learn how to express an arbitrary vector in terms of the standard unit vectors. Finally, apply what you've learned to an application involving force.
Area and Arc Length in Polar Coordinates
30. Area and Arc Length in Polar Coordinates
May 31, 2013
Continue your study of polar coordinates by focusing on applications involving integration. First, develop the polar equation for the area bounded by a polar curve. Then turn to arc lengths in polar coordinates, discovering that the formula is similar to that for parametric equations.
Polar Coordinates and the Cardioid
29. Polar Coordinates and the Cardioid
May 31, 2013
In the first of two episodes on polar coordinates, review the main properties and graphs of this specialized coordinate system. Consider the cardioids, which have a heart shape. Then look at the derivative of a function in polar coordinates, and study where the graph has horizontal and vertical tangents.
Parametric Equations and the Cycloid
28. Parametric Equations and the Cycloid
May 31, 2013
Parametric equations consider variables such as x and y in terms of one or more additional variables, known as parameters. This adds more levels of information, especially orientation, to the graph of a parametric curve. Examine the calculus concept of slope in parametric equations, and look closely at the equation of the cycloid.
Parabolas, Ellipses, and Hyperbolas
27. Parabolas, Ellipses, and Hyperbolas
May 31, 2013
Review parabolas, ellipses, and hyperbolas, focusing on how calculus deepens our understanding of these shapes. First, look at parabolas and arc length computation. Then turn to ellipses, their formulas, and the concept of eccentricity. Next, examine hyperbolas. End by looking ahead to parametric equations.
Taylor and Maclaurin Series
26. Taylor and Maclaurin Series
May 31, 2013
Finish your study of infinite series by exploring in greater depth the Taylor and Maclaurin series. Discover that you can calculate series representations in many ways. Close by using an infinite series to derive one of the most famous formulas in mathematics, which connects the numbers e, pi, and i.
Representation of Functions by Power Series
25. Representation of Functions by Power Series
May 31, 2013
Learn the steps for expressing a function as a power series. Experiment with differentiation and integration of known series. At the end of the episode, investigate some beautiful series formulas for pi, including one by the brilliant Indian mathematician Ramanujan.
Power Series and Intervals of Convergence
24. Power Series and Intervals of Convergence
May 31, 2013
Discover that a power series can be thought of as an infinite polynomial. The key question with a power series is to find its interval of convergence. In general, this will be a point, an interval, or perhaps the entire real line. Also examine differentiation and integration of power series.
Taylor Polynomials and Approximations
23. Taylor Polynomials and Approximations
May 31, 2013
Try out techniques for approximating a function with a polynomial. The first example shows how to construct the first-degree Maclaurin polynomial for the exponential function. These polynomials are a special case of Taylor polynomials, which you investigate along with Taylor's theorem.
The Ratio and Root Tests
22. The Ratio and Root Tests
May 31, 2013
Finish your exploration of convergence tests with the ratio and root tests. The ratio test is particularly useful for series having factorials, whereas the root test is useful for series involving roots to a given power. Close by asking if these tests work on the p-series, introduced in an earlier episode.
Alternating Series
21. Alternating Series
May 31, 2013
Having developed tests for positive-term series, turn to series having terms that alternate between positive and negative. See how to apply the alternating series test. Then use absolute value to look at the concepts of conditional and absolute convergence for series with positive and negative terms.
The Comparison Tests
20. The Comparison Tests
May 31, 2013
Develop more convergence tests, learning how the direct comparison test for positive-term series compares a given series with a known series. The limit comparison test is similar but more powerful, since it allows analysis of a series without having a term-by-term comparison with a known series.
Integral Test - Harmonic Series, p-Series
19. Integral Test - Harmonic Series, p-Series
May 31, 2013
Does the celebrated harmonic series diverge or converge? Discover a proof using the integral test. Then generalize to define an entire class of series called p-series, and prove a theorem showing when they converge. Close with the sum of the harmonic series, the fascinating Euler-Mascheroni constant, which is not known to be rational or irrational.
Series, Divergence, and the Cantor Set
18. Series, Divergence, and the Cantor Set
May 31, 2013
Explore an important test for divergence of an infinite series: If the terms of a series do not tend to zero, then the series diverges. Solve a bouncing ball problem. Then investigate a paradoxical property of the famous Cantor set.
Infinite Series - Geometric Series
17. Infinite Series - Geometric Series
May 31, 2013
Look at an example of a telescoping series. Then study geometric series, in which each term in the summation is a fixed multiple of the previous term. Next, prove an important convergence theorem. Finally, apply your knowledge of geometric series to repeating decimals.
Sequences and Limits
16. Sequences and Limits
May 31, 2013
Start the first of 11 episodes on one of the most important topics in Calculus II: infinite series. The concept of an infinite series is based on sequences, which can be thought of as an infinite list of real numbers. Explore the characteristics of different sequences, including the celebrated Fibonacci sequence.
Improper Integrals
15. Improper Integrals
May 31, 2013
So far, you have been evaluating definite integrals using the fundamental theorem of calculus. Study integrals that appear to be outside this procedure. Such "improper integrals" usually involve infinity as an end point and may appear to be unsolvable, until you split the integral into two parts.
Indeterminate Forms and L'Hôpital's Rule
14. Indeterminate Forms and L'Hôpital's Rule
May 31, 2013
Revisit the concept of limits from elementary calculus, focusing on expressions that are indeterminate because the limit of the function may not exist. Learn how to use L'Hopital's famous rule for evaluating indeterminate forms, applying this valuable theorem to a variety of examples.
Integration by Partial Fractions
13. Integration by Partial Fractions
May 31, 2013
Put your precalculus skills to use by splitting up complicated algebraic expressions to make them easier to integrate. Learn how to deal with linear factors, repeated linear factors, and irreducible quadratic factors. Finally, apply these techniques to the solution of the logistic differential equation.
Integration by Trigonometric Substitution
12. Integration by Trigonometric Substitution
May 31, 2013
Trigonometric substitution is a technique for converting integrands to trigonometric integrals. Evaluate several cases, discovering that you can conveniently represent these substitutions by right triangles. Also, what do you do if the solution you get by hand doesn't match the calculator's answer?
Trigonometric Integrals
11. Trigonometric Integrals
May 31, 2013
Explore integrals of trigonometric functions, finding that they are often easy to evaluate if either sine or cosine occurs to an odd power. If both are raised to an even power, you must resort to half-angle trigonometric formulas. Then look at products of tangents and secants, which also divide into easy and hard cases.
Integration by Parts
10. Integration by Parts
May 31, 2013
Begin a series of episodes on techniques of integration, also known as finding antiderivatives. After reviewing some basic formulas from Calculus I, learn to develop the method called integration by parts, which is based on the product rule for derivatives. Explore applications involving centers of mass and area.
Moments, Centers of Mass, and Centroids
9. Moments, Centers of Mass, and Centroids
May 31, 2013
Study moments and centers of mass, developing formulas for finding the balancing point of a planar area, or lamina. Progress from one-dimensional examples to arbitrary planar regions. Close with the famous theorem of Pappus, using it to calculate the volume of a torus.
Arc Length, Surface Area, and Work
8. Arc Length, Surface Area, and Work
May 31, 2013
Continue your exploration of the power of integral calculus. First, review arc length computations. Then, calculate the areas of surfaces of revolution. Close by surveying the concept of work, answering questions such as, how much work does it take to lift an object from Earth's surface to 800 miles in space?
Areas and Volumes
7. Areas and Volumes
May 31, 2013
Use integration to find areas and volumes. Begin by trying your hand at planar regions bounded by two curves. Then review the disk method for calculating volumes. Next, focus on ellipses as well as solids obtained by rotating ellipses about an axis. Finally, see how your knowledge of ellipsoids applies to the planet Saturn.
Linear Differential Equations
6. Linear Differential Equations
January 1, 1970
Investigate linear differential equations, which typically cannot be solved by separation of variables. The key to their solution is what Professor Edwards calls the "magic integrating factor." Try several examples and applications. Then return to an equation involving Euler's method, which was originally considered in an earlier lesson.
Applications of Differential Equations
5. Applications of Differential Equations
May 31, 2013
Continue your study of differential equations by examining orthogonal trajectories, curves that intersect a given family of curves at right angles. These occur in thermodynamics and other fields. Then develop the famous logistic differential equation, which is widely used in mathematical biology.
Differential Equations - Growth and Decay
4. Differential Equations - Growth and Decay
May 31, 2013
In the first of three episodes on differential equations, learn various techniques for solving these very useful equations, including separation of variables and Euler's method, which is the simplest numerical technique for finding approximate solutions. Then look at growth and decay models, with two intriguing applications.
Integration Warm-up
3. Integration Warm-up
May 31, 2013
Complete your review by going over the basic facts of integration. After a simple example of integration by substitution, turn to definite integrals and the area problem. Reacquaint yourself with the fundamental theorem of calculus and the second fundamental theorem of calculus. End the episode by solving a simple differential equation.
Differentiation Warm-up
2. Differentiation Warm-up
May 31, 2013
In your second warm-up episode, review the concept of derivatives, recalling the derivatives of trigonometric, logarithmic, and exponential functions. Apply your knowledge of derivatives to the analysis of graphs. Close by reversing the problem: Given the derivative of a function, what is the original function?
Basic Functions of Calculus and Limits
1. Basic Functions of Calculus and Limits
May 31, 2013
Learn what distinguishes Calculus II from Calculus I. Then embark on a three-episode review, beginning with the top 10 student pitfalls from precalculus. Next, Professor Edwards gives a refresher on basic functions and their graphs, which are essential tools for solving calculus problems. #Science & Mathematics
Where to Watch Understanding Calculus II: Problems, Solutions, and Tips
Understanding Calculus II: 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 II: Problems, Solutions, and Tips on demand at Amazon Prime and Amazon.
  • Premiere Date
    May 31, 2013