We review differentiation and integration.

We review differentiation and integration.

We review common techniques to compute indefinite and definite integrals.

We introduce the procedure of “Slice, Approximate, Integrate” and use it study the area of a region between two curves using the definite integral.

We can also use the procedure of “Slice, Approximate, Integrate” to set up integrals to compute volumes.

We define a solid of revolution and discuss how to find the volume of one in two different ways.

We use the procedure of “Slice, Approximate, Integrate” to develop the washer method to compute volumes of solids of revolution.

We use the procedure of “Slice, Approximate, Integrate” to develop the shell method to compute volumes of solids of revolution.

We compare and contrast the washer and shell method.

We can use the procedure of “Slice, Approximate, Integrate” to find the length of curves.

We compute surface area of a frustrum then use the method of “Slice, Approximate, Integrate” to find areas of surface areas of revolution.

We apply the procedure of “Slice, Approximate, Integrate” to model physical situations.

We learn a new technique, called integration by parts, to help find antiderivatives of certain types of products by reexamining the product rule for differentiation.

We can use substitution and trigonometric identities to find antiderivatives of certain types of trigonometric functions.

We integrate by substitution with the appropriate trigonometric function.

We discuss an approach that allows us to integrate rational functions.

We can use limits to integrate functions on unbounded domains or functions with unbounded range.

We investigate sequences.

We can graph the terms of a sequence and find functions of a real variable that coincide with sequences on their common domains.

There are two ways to establish whether a sequence has a limit.

A series is an infinite sum of the terms of sequence.

We discuss convergence results for geometric series and telescoping series.

If an infinite sum converges, then its terms must tend to zero.

Certain infinite series can be studied using improper integrals.

Alternating series are series whose terms alternate in sign between positive and negative. There is a powerful convergence test for alternating series.

We learn how to estimate the value of a series.

For a convergent geometric series or telescoping series, we can find the exact error made when approximating the infinite series using the sequence of partial sums.

There is a nice result for approximating the remainder of convergent alternating series.

There is a nice result for approximating the remainder for series that converge by the integral test.

Some infinite series can be compared to geometric series.

Some infinite series can be compared to geometric series.

We compare infinite series to each other using inequalities.

We compare infinite series to each other using limits.

The basic question we wish to answer about a series is whether or not the series converges. If a series has both positive and negative terms, we can refine this question and ask whether or not the series converges when all terms are replaced by their absolute values. This is the distinction between absolute and conditional convergence, which we explore in this section.

We can approximate sufficiently differentiable functions by polynomials.

Infinite series can represent functions.

We study Taylor and Maclaurin series.

Taylor series are a computational tool.

Power series interact nicely with other calculus concepts.

Differential equations show you relationships between rates of functions.

We describe numerical and graphical methods for understanding differential equations.

Separable differential equations are those in which the dependent and independent variables can be separated on opposite sides of the equation.

We discuss the basics of parametric curves.

We discuss derivatives of parametrically defined curves.

Polar coordinates are coordinates based on an angle and a radius.

We see a collection of polar curves.

We differentiate polar functions.

We integrate polar functions.

We talk about basic geometry in higher dimensions.

Vectors are lists of numbers that denote direction and magnitude.

The dot product is an important operation between vectors that captures geometric information.

Projections tell us how much of one vector lies in the direction of another and are important in physical applications.

The cross product is a special way to multiply two vectors in three-dimensional space.