#### How to use Ximera

This course is built in Ximera.

#### How is my work scored?

We explain how your work is scored.

#### A review of differentiation

We review differentiation and integration.

#### A review of integration

We review differentiation and integration.

#### A review of integration techniques

We review common techniques to compute indefinite and definite integrals.

#### Area between curves

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

#### Accumulated cross-sections

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

#### What is a solid of revolution?

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

#### The washer method

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

#### The shell method

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

#### Comparing washer and shell method

We compare and contrast the washer and shell method.

#### Length of curves

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

#### Physical applications

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

#### Integration by parts

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.

#### Trigonometric integrals

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

#### Trigonometric substitution

We integrate by substitution with the appropriate trigonometric function.

#### Rational functions

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

#### Improper Integrals

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

#### Sequences

We investigate sequences.

#### Representing sequences visually

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

#### Limits of sequences

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

#### What is a series

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

#### Special Series

We discuss convergence results for geometric series and telescoping series.

#### The divergence test

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

#### The ratio test

Some infinite series can be compared to geometric series.

#### Higher Order Polynomial Approximations

We can approximate sufficiently differentiable functions by polynomials.

#### Power series

Infinite series can represent functions.

#### Introduction to Taylor series

We study Taylor and Maclaurin series.

#### Numbers and Taylor series

Taylor series are a computational tool.

#### Calculus and Taylor series

Power series interact nicely with other calculus concepts.

#### Parametric equations

We discuss the basics of parametric curves.

#### Calculus and parametric curves

We discuss derivatives of parametrically defined curves.

#### Introduction to polar coordinates

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

#### Gallery of polar curves

We see a collection of polar curves.

#### Derivatives of polar functions

We differentiate polar functions.

#### Integrals of polar functions

We integrate polar functions.

#### Working in two and three dimensions

We talk about basic geometry in higher dimensions.

#### Vectors

Vectors are lists of numbers that denote direction and magnitude.

#### The Dot Product

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

#### Projections and orthogonal decomposition

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

#### The cross product

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

#### Lines and curves in space

Vector-valued functions are parameterized curves.

#### Calculus and vector-valued functions

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

#### Motion and paths in space

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

#### Parameterizing by arc length

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

#### Unit tangent and unit normal vectors

We introduce two important unit vectors.

#### Planes in space

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

#### Parametric plots

Tangent and normal vectors can help us make interesting parametric plots.

#### Functions of several variables

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

#### Level sets

We introduce level sets.

#### Open and Closed Sets

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

#### Limits

We investigate limits of functions of several variables.

#### Continuity

We investigate what continuity means for functions of several variables.

#### Partial derivatives

We introduce partial derivatives and the gradient vector.

#### Tangent planes

We find tangent planes.

#### Differentiability

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

#### The directional derivative

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

#### The chain rule

We investigate the chain rule for functions of several variables.