Math 116: Calculus 2

Course Content

Broadly speaking, we will be studying sequences and series, advanced integration methods, and applications of integration to calculating volumes, along with the various applications of those topics to understanding real-world phenomena in the sciences and social sciences. In slightly more detail, the course is divided into the following six units. 

Unit 1: Review of Precalculus, Limits, and Differentiation

As a review unit, the goal of the unit is to recall the most important results, procedures, and concepts from precalculus and the limit and differentiation portions of calculus. We'll be drawing on that material heavily throughout the course so we'll spend this and the next lesson revisiting that content.

Unit 2: Sequences and Series

The driving question of this unit is: Does an infinite sum have a sum? That is, if we sum an infinite number of numbers (e.g., 1+1+1+...), does that yield a number? In the example just given your gut should tell you the answer: "no." But, surprisingly, certain infinite sums do add up to a number. Knowing when that's true and how to determine what the sum is constitutes the bulk of this unit.

Unit 3: Power Series and Taylor Series

The driving question of this unit is: Can a function be approximated by a polynomial? That is, can we say that f(x) is approximately p(x), where f is a function and p a polynomial? We know from Calculus 1—and the theory of linear approximation, specifically—that if f is differentiable then f(x) is approximately L(x), where L is the equation of the tangent line at x = a, with this approximation being most accurate for x near a. But is this the best we can do? Can we get higher-accuracy approximations? We’ll soon find out that in many cases the answer is: yes! Furthermore, we’ll discover that in some cases we’ll be able to say that f(x) = sum[T_n(x)], where the T_n are special polynomials. Working all that out, and exploring the implications and applications of what we’ll develop, constitutes the bulk of this unit.

Unit 4: Integration -- Numerical Integration

This unit and the remaining ones in the course are all directed at the third driving question of the course: How can we calculate the volume enclosed by a surface? We'll get to that question in earnest in Unit 6. But first we need to review integration, discuss various techniques for integrating functions, and learn how to approximate definite integrals for the cases where we cannot find an antiderivative using those techniques. The first and last topics on that list are where we'll focus in this unit.

Unit 5: Integration -- Techniques of Integration

The lessons in this unit focus on building a repository of integration techniques that help us evaluate definite integrals. We'll start off simple by reviewing the basics of integration from Calculus 1 and also u-substitution. After that, we'll learn new integration techniques (not covered in a typical Calculus 1 course). At the end of this unit we'll be ready to tackle the last driving question in the course (How can we calculate the volume enclosed by a surface?), which we'll get to in the next unit.

Unit 6: Applications of Integration

This final unit in the course investigates the last driving question in the course, How can we calculate the volume enclosed by a surface?, and explores other applications of integration that result from the techniques and results that we'll develop in trying to answer that question. We'll begin the unit by learning how to calculate the area between two curves, which will set us up later for calculating the volumes of certain types of solids. We'll also learn how to calculate the arc length of a curve and the surface area of a surface. Finally, we'll explore some real-world applications of integration. The unit is in many ways a capstone one, since it draws on the techniques of integration we learned in Unit 5 and in some cases the numerical integration content learned in Unit 4. Finally, we'll also be learning how to think and sketch in 3D. This is perhaps the first time in the mathematics curriculum you'll have done this, so it will likely take some time and practice to get good at it.