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Understanding Calculus II: Problems, Solutions, and Tips

Continue down the road to mastering calculus with this step-by-step guide to Calculus II, taught by an award-winning Professor of Mathematics.
Understanding Calculus II: Problems, Solutions, and Tips is rated 4.6 out of 5 by 37.
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Rated 5 out of 5 by from Beautifully explained survey of calc II topics Having used calculus throughout my career as a physical scientist, I noticed that the foundations were disappearing in the rear view mirror so to speak. This course is an excellent refresher, starting from the very beginning and clearly laying out the reasoning and the steps to solve a variety of problems. I wish that Prof. Edwards had taught me calculus back in the day. I appreciate the references to physics and engineering, which keep the course down to earth. Too much of my early calculus was directed toward future mathematicians rather than scientists or engineers. Edwards' approach is much superior to that. Thank you Prof. Edwards.
Date published: 2023-07-04
Rated 5 out of 5 by from Excellent Summary and Review It has been a long time since I took Calc and wanted to review my skills. I found this course to be an excellent summary of the key points I needed to refresh myself on. It was easy to match the course material to a calc textbook to try additional problems. The examples used by the professor really brought concepts home.
Date published: 2021-12-15
Rated 5 out of 5 by from wonderful I am in the process of re-study a subject that I seldom utilized in the past and found this a wonderful tool to re-introduce my mind to the subject. Thank you. Yvone
Date published: 2021-11-29
Rated 5 out of 5 by from 11/10 Bruce is an unbelievably clear and effective educator, and so deeply knowledgeable. I am extremely grateful to have stumbled across this lecture series.
Date published: 2021-02-11
Rated 5 out of 5 by from Clear explanations and good practice examples! I have learned so much Calculus in this course. It was very well explained and the Bruce Edwards always points out pitfalls too. My only complaint I guess was that the answers in the guidebook should have reiterated the questions so I didn't have to keep scrolling back and forth. I have purchased and am looking forward to Calculus III!
Date published: 2021-01-04
Rated 5 out of 5 by from Excellent Course! Professor Edwards rates at the top as a Great Courses professor. He was very understandable. He provided guidance on the importance of the homework and how to learn the material. He is an outstanding communicator on the subject of math. I felt I got a strong grasp of the material. I'm looking forward to taking his Multivariable Calculus.
Date published: 2020-09-02
Rated 5 out of 5 by from excellent job of presenting a difficult subject Dr. Edwards does an excellent job of presenting a difficult topic in a logical manner.
Date published: 2020-07-23
Rated 5 out of 5 by from Excellent teacher Excellent and logical presentation. Easy to follow. Would highly recommend to anyone taking Calc II.
Date published: 2020-06-20
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Solve a wide array of problems in the physical, biological, and social sciences, engineering, economics, and other areas with the skills you learn in Understanding Calculus II: Problems, Solutions, and Tips. This second course in the calculus sequence introduces you to exciting new techniques and applications of one of the most powerful mathematical tools ever invented. Professor Bruce H. Edwards of the University of Florida enriches these 36 lectures with crystal-clear explanations, frequent study tips, pitfalls to avoid, and-best of all-hundreds of examples and practice problems that are specifically designed to explain and reinforce key concepts.


Bruce H. Edwards

I love mathematics and tried to communicate this passion to others, regardless of their mathematical backgrounds.


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 Bogota, 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.

By This Professor

Prove It: The Art of Mathematical Argument
Understanding Multivariable Calculus: Problems, Solutions, and Tips
Understanding Calculus II: Problems, Solutions, and Tips
Understanding Calculus: Problems, Solutions and Tips
Mathematics Describing the Real World: Precalculus and Trigonometry
Understanding Calculus II: Problems, Solutions, and Tips


Basic Functions of Calculus and Limits

01: Basic Functions of Calculus and Limits

Learn what distinguishes Calculus II from Calculus I. Then embark on a three-lecture 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.

32 min
Differentiation Warm-up

02: Differentiation Warm-up

In your second warm-up lecture, 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?

30 min
Integration Warm-up

03: Integration Warm-up

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 lecture by solving a simple differential equation.

31 min
Differential Equations-Growth and Decay

04: Differential Equations-Growth and Decay

In the first of three lectures 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.

31 min
Applications of Differential Equations

05: Applications of Differential Equations

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.

31 min
Linear Differential Equations

06: Linear Differential Equations

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 Lecture 4.

31 min
Areas and Volumes

07: Areas and Volumes

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.

31 min
Arc Length, Surface Area, and Work

08: Arc Length, Surface Area, and Work

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?

30 min
Moments, Centers of Mass, and Centroids

09: Moments, Centers of Mass, and Centroids

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.

31 min
Integration by Parts

10: Integration by Parts

Begin a series of lectures on techniques of integration, also known as finding anti-derivatives. 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.

31 min
Trigonometric Integrals

11: Trigonometric Integrals

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.

31 min
Integration by Trigonometric Substitution

12: Integration by Trigonometric Substitution

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?

32 min
Integration by Partial Fractions

13: Integration by Partial Fractions

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.

32 min
Indeterminate Forms and L'Hopital's Rule

14: Indeterminate Forms and L'Hopital's Rule

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'Hôpital's famous rule for evaluating indeterminate forms, applying this valuable theorem to a variety of examples.

31 min
Improper Integrals

15: Improper Integrals

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.

31 min
Sequences and Limits

16: Sequences and Limits

Start the first of 11 lectures 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.

31 min
Infinite Series-Geometric Series

17: Infinite Series-Geometric Series

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.

32 min
Series, Divergence, and the Cantor Set

18: Series, Divergence, and the Cantor Set

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.

32 min
Integral Test-Harmonic Series, p-Series

19: Integral Test-Harmonic Series, p-Series

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.

31 min
The Comparison Tests

20: The Comparison Tests

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.

31 min
Alternating Series

21: Alternating Series

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.

31 min
The Ratio and Root Tests

22: The Ratio and Root Tests

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 Lecture 19.

32 min
Taylor Polynomials and Approximations

23: Taylor Polynomials and Approximations

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.

31 min
Power Series and Intervals of Convergence

24: Power Series and Intervals of Convergence

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.

31 min
Representation of Functions by Power Series

25: Representation of Functions by Power Series

Learn the steps for expressing a function as a power series. Experiment with differentiation and integration of known series. At the end of the lecture, investigate some beautiful series formulas for pi, including one by the brilliant Indian mathematician Ramanujan.

30 min
Taylor and Maclaurin Series

26: Taylor and Maclaurin Series

Finish your study of infinite series by exploring in greater depth the Taylor and Maclaurin series, introduced in Lecture 23. 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.

32 min
Parabolas, Ellipses, and Hyperbolas

27: Parabolas, Ellipses, and Hyperbolas

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.

30 min
Parametric Equations and the Cycloid

28: Parametric Equations and the Cycloid

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.

31 min
Polar Coordinates and the Cardioid

29: Polar Coordinates and the Cardioid

In the first of two lectures 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.

31 min
Area and Arc Length in Polar Coordinates

30: Area and Arc Length in Polar Coordinates

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.

32 min
Vectors in the Plane

31: Vectors in the Plane

Begin a series of lectures 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.

30 min
The Dot Product of Two Vectors

32: The Dot Product of Two Vectors

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.

31 min
Vector-Valued Functions

33: Vector-Valued Functions

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.

31 min
Velocity and Acceleration

34: Velocity and Acceleration

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.

31 min
Acceleration's Tangent and Normal Vectors

35: Acceleration's Tangent and Normal Vectors

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.

31 min
Curvature and the Maximum Bend of a Curve

36: Curvature and the Maximum Bend of a Curve

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.

31 min