We review differentiation and integration.

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.

Some volumes of revolution are more easily computed with cylindrical shells.

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

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.

A sequence is an ordered list of numbers.

A sequence can be thought of as a function from the integers to the real numbers. There are two ways to establish whether a sequence has a limit.

A series is summation of a sequence.

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

Some infinite series can be compared to geometric series.

We can approximate smooth functions with 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.

We discuss the basics of parametric curves.

We discuss derivatives and integrals of parametric curves.

Polar coordinates are a special type of parametric curves.

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 measures how aligned two vectors are with each other.

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

Vector-valued functions are parameterized curves.

With one input, and vector outputs, we work component-wise.

We interpret vector-valued functions as paths of objects in space.

We find a new description of curves that trivializes arc length computations.

We introduce two important unit vectors.

We discuss how to find implicit and explicit formulas for planes.

Tangent and normal vectors can help us plot make interesting parametric functions.

We introduce functions that take vectors or points as inputs and output a number.

We investigate what continuity means for functions of several variables.

We introduce partial derivatives.

We introduce the gradient vector.

We use a method called “linear approximation” to estimate the value of a (complicated) function at a given point.

We investigate the chain rule for functions of several variables.