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 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.

Some infinite series can be compared to geometric series.

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.

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.

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 make interesting parametric plots.

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

We introduce level sets.

We generalize the notion of open and closed intervals to open and closed sets in $\R ^2$.

We investigate limits of functions of several variables.

We investigate what continuity means for functions of several variables.

We introduce partial derivatives and the gradient vector.

We find tangent planes.

We introduce differentiability for functions of several variables and find tangent planes.

We introduce a way of analyzing the rate of change in a given direction.

We investigate the chain rule for functions of several variables.

The gradient is the fundamental notion of a derivative for a function of several variables.