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Understanding Calculus II: Problems, Solutions, and Tips

Understanding Calculus II: Problems, Solutions, and Tips

Professor Bruce H. Edwards Ph.D.
University of Florida
Course No.  1018
Course No.  1018
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Course Overview

About This Course

36 lectures  |  31 minutes per lecture

Calculus II is the payoff for mastering Calculus I. 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. Equipped with the skills of Calculus II, you can solve a wide array of problems in the physical, biological, and social sciences, engineering, economics, and other areas. Success at Calculus II also gives you a solid foundation for the further study of mathematics, and it meets the math requirement for many undergraduate majors.

But beyond these advantages, you will find that the methods you learn in Calculus II are practical, interesting, and elegant, involving ideas that are beautifully simple. Because it can model real-life situations, calculus has an amazing range of uses, and these applications come into full flower in Calculus II.

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Calculus II is the payoff for mastering Calculus I. 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. Equipped with the skills of Calculus II, you can solve a wide array of problems in the physical, biological, and social sciences, engineering, economics, and other areas. Success at Calculus II also gives you a solid foundation for the further study of mathematics, and it meets the math requirement for many undergraduate majors.

But beyond these advantages, you will find that the methods you learn in Calculus II are practical, interesting, and elegant, involving ideas that are beautifully simple. Because it can model real-life situations, calculus has an amazing range of uses, and these applications come into full flower in Calculus II.

Understanding Calculus II: Problems, Solutions, and Tips takes you on this exhilarating journey in 36 intensively illustrated half-hour lectures that cover all the major topics of the second full-year calculus course in high school at the College Board Advanced Placement BC level or a second-semester course in college. Drawing on decades of teaching experience, Professor Bruce H. Edwards of the University of Florida enriches his 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.

Few calculus teachers are as qualified, accessible, or entertaining as Professor Edwards, who has won multiple teaching awards and coauthored a best-selling series of calculus textbooks. Many calculus students give up trying to understand why a particular procedure works and resort to memorizing the steps to a solution. With Professor Edwards, the underlying concepts are always clear and constantly reinforced, which greatly eases the path to learning the material.

Get behind the Wheel of the “Limit Machine”

Professor Edwards begins with a three-lecture review of the fundamental ideas of calculus. He also includes brief reviews of major concepts throughout the course, which makes Understanding Calculus II a self-contained lecture series for anyone who is already familiar with the two main operations of calculus, differentiation and integration. Professor Edwards takes these ideas beyond the definitions, rules, and formulas that are the focus of first-semester calculus and applies them in intriguing ways. For example:

  • Differential equations: This far-reaching field puts derivatives to work—modeling population growth, nuclear decay, falling objects, and countless other processes involving change. Professor Edwards recalls that as a young mathematician, he spent summers working for NASA, solving differential equations for aircraft in flight.
  • Infinite series: Does adding an infinite sequence of numbers give an infinite result? Not necessarily. The series may converge on a specific value, or it may diverge to infinity. Calculus can provide the answer for different types of infinite series and represent familiar functions from algebra or trigonometry in surprising ways.
  • Vectors: Among the geometric applications of calculus is the analysis of vectors. These are quantities, such as velocity, that have both magnitude and direction. In Calculus II, you learn techniques for evaluating vectors in the plane, allowing you to solve problems involving moving and accelerating objects, whether they are on a straight or curved path.

Understanding Calculus II covers the above subjects in considerable depth, particularly infinite series, which you explore in 11 lectures. You also study other standard topics in second-semester calculus, including

  • integration formulas and techniques,
  • integrating areas and volumes,
  • Taylor and Maclaurin polynomials,
  • L’Hôpital’s rule for evaluating limits,
  • evaluating improper integrals,
  • calculus applied to parametric equations, and
  • calculus applied to polar coordinates.

These very different applications of calculus each involve the essential idea of the limit. Professor Edwards notes that calculus can be thought of as a “limit machine”—a set of procedures for approaching infinitely close to a value. One of the interesting features of calculus is its logical rigor combined with its creative use of the mysterious entity of the infinite. From this unusual marriage emerge astonishingly precise solutions to otherwise inaccessible problems.

Explore the Immense Riches of Calculus

Calculus is full of fascinating properties, baffling paradoxes, and entertaining problems. Among the many you investigate in Understanding Calculus II are these:

  • Gabriel’s Horn: Rotate a simple curve around its axis and you get a three-dimensional shape that looks like an infinitely long trumpet. Called Gabriel’s Horn, this geometric figure has an unusual property: It has infinite surface area but finite volume. See calculus prove that this must be so.
  • Baseball thriller: A baseball 3 feet above home plate is hit at 100 feet per second and at an angle of 45 degrees. Employ Newton’s second law of motion and the derivative of the position function to determine if the ball will be a home run, clearing a 10-foot-high fence 300 feet away.
  • Cantor Set: Remove the middle third of a line segment. Repeat with the two pieces that remain. Repeat again ad infinitum. The end points of all the pieces will form an infinite set. But what about the total length of all the line segments? Summing this infinite series reveals the surprising answer.

Master Calculus on Your Own Schedule

Understanding Calculus II is an immensely rewarding experience that you can study at your own pace. Professor Edwards often encourages you to pause the video and test yourself by solving a problem before he reveals the answer. Those who will benefit from this engaging and flexible presentation include

  • high school or college students currently, or about to be, enrolled in Calculus II who want personal coaching from an outstanding teacher;
  • high school students preparing for the College Board Advanced Placement test in Calculus at the BC level;
  • students in higher-level math courses or professionals who want a rigorous review of calculus; and
  • anyone interested in pursuing one of life’s greatest intellectual adventures, which has been solving difficult problems for over 300 years.
  • A three-time Teacher of the Year at the University of Florida, Professor Edwards knows how to help students surmount the stumbling blocks on their path to mastering calculus. In this course, he uses a steady stream of on-screen equations, graphs, and other visual aids to document the key steps in solving sample problems. The accompanying workbook is designed to reinforce each lecture with more practice problems and worked-out solutions, as well as lecture summaries, tips, and pitfalls; and formulas for derivatives, integration, and power series.

    Professor Edwards’s lectures also include a feature he calls “You Be the Teacher,” in which he reverses roles, challenging you to answer a typical question posed in the classroom, design a suitable problem to illustrate a principle, or otherwise put yourself in the instructor’s shoes—an invaluable exercise in learning to think for yourself in the language of calculus.

    Open Doors with Your New Fluency

    The place of calculus at the end of the high school math curriculum makes it seem like a final destination. But it is also only a beginning. Calculus is a world unto itself, an ever-expanding collection of tools that can solve the most intractable problems in ingenious and often surprising ways. The deeper you go into calculus, the richer it gets and the better you are prepared for even more advanced math courses that open doors of their own.

    In his last lecture, Professor Edwards looks ahead to where your math studies may take you after this course. It’s exciting terrain. Imagine arriving in a foreign country equipped with the ability to speak the nation’s language. Your opportunities for exploration, interaction, and further learning are almost limitless. That’s what Understanding Calculus II does for your fluency in one of the greatest achievements of the human mind.

    View Less
    36 Lectures
    • 1
      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. x
    • 2
      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? x
    • 3
      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. x
    • 4
      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. x
    • 5
      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. x
    • 6
      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. x
    • 7
      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. x
    • 8
      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? x
    • 9
      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. x
    • 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. x
    • 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. x
    • 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? x
    • 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. x
    • 14
      Indeterminate Forms and L'Hôpital'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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x
    • 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. x

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    Bruce H. Edwards
    Ph.D. Bruce H. Edwards
    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 Bogot·, 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.
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    Reviews

    Rated 4.7 out of 5 by 9 reviewers.
    Rated 5 out of 5 by Highly recommended Prof. Edwards is excellent, I bought the whole calculus series and I'll probably buy any other courses that he makes. The practice problems within the lectures and in the workbook are very good and he's a pro at explaining difficult concepts. I hope to see the rigorous college-level math courses continue, I'd vote for Real Analysis and Linear Algebra. July 20, 2014
    Rated 5 out of 5 by Excellent Calculus BC Course I have been using TGC math lessons to homeschool my children. The comprehensiveness of the material and clear presentations have helped my children advance several years beyond their respective grade levels. Professor Edwards' presentation style is our favorite and easily kept our attention throughout Precalc/Trig, Calculus I, and Calculus II. The workbook problems and solutions were excellent as I reviewed every problem/answer. Being an actuarial science/math major, I can't recommend this course enough, and I hope that TGC produces a Calculus III course with Prof. Edwards in the near future. It's amazing how, with the right teacher, my young children have learned these important yet challenging concepts. October 25, 2013
    Rated 5 out of 5 by Pofessor Edwards excellent as always I agree with the others reviewers here. Prof. Edwards is simply an excellent teacher. Some minor negative points which is probably not the good professors fault, but rather the producers: Some slightly annoying graphics in the video. Dark blue background animation when showing equation and the like is not necessary Graphs were visually better presented in the other courses by prof. Edward. TTC producers: please don't become like Discovery, History Channel and the like and annoy us with animations and flashy effects Be conservative (Like BBC) and keep it simple August 29, 2013
    Rated 5 out of 5 by Buy It If Your Need a Review I am not yet even done with this course, bit I have (purchased and) watched his Calculus I course, and both of them are outstanding. Over several years I have purchased about two dozen Great Courses on many(!) (including "humanities") subjects. Most of them (there have been two "clunkers") have been excellent. This is one of the excellent ones, together with his Calculus I course. He does not go into deep theory (and does not claim to), but with this one and the Calculus I course, if you need a practical refresher, buy it! August 7, 2013
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