Understanding Calculus II: Problems, Solutions, and Tips

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.

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?

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.

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.

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.

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.

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.

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?

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.

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.

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.

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?

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.