#### How to use Ximera

This course is built in Ximera.

#### How is my work scored?

We explain how your work is scored.

#### Sequences

We investigate sequences.

#### 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 integral test

Certain infinite series can be studied using improper integrals.

#### The divergence test

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

#### Dig-In: Estimating Series

We learn how to estimate the value of a series.

#### Dig-In: Remainders and the Integral Test

We investigate how the ideas of the Integral Test apply to remainders.

#### The ratio test

Some infinite series can be compared to geometric series.

#### The root test

Some infinite series can be compared to geometric series.

#### The comparison test

We compare infinite series to each other using inequalities.

#### The limit comparison test

We compare infinite series to each other using limits.

#### The alternating series test

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

#### Remainders for alternating series

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

#### Drawing a sphere

Learn how to draw a sphere.

#### Vectors

Vectors are lists of numbers that denote direction and magnitude.

#### The dot product

The dot product measures how aligned two vectors are with each other.

#### 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.

#### Drawing a torus

Learn how to draw a torus.

#### Functions of several variables

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

#### Continuity

We investigate what continuity means for real-valued functions of several variables.

#### Partial derivatives

We introduce partial derivatives and the gradient vector.

We use the gradient to approximate values for functions of several variables.

#### 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.

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

#### Taylor polynomials

We introduce Taylor polynomials for functions of several variables.

We will get to know some basic quadric surfaces.

#### Drawing paraboloids

Learn how to draw an elliptic and a hyperbolic paraboloid.

#### Maxima and minima

We see how to find extrema of functions of several variables.

#### Constrained optimization

We learn to optimize surfaces along and within given paths.

#### Lagrange multipliers

We give a new method of finding extrema.

#### Integrals over trivial regions

We study integrals over basic regions.

#### Integrals with trivial integrands

We study integrals over general regions by integrating $1$.

#### Polar coordinates

We integrate over regions in polar coordinates.

#### Cylindrical coordinates

We integrate over regions in cylindrical coordinates.

#### Spherical coordinates

We integrate over regions in spherical coordinates.

#### Surface area

We compute surface area with double integrals.

#### Mass, moments, and center of mass

We use integrals to model mass.

#### Computations and interpretations

We practice more computations and think about what integrals mean.

#### Vector fields

We introduce the idea of a vector at every point in space.

#### Line integrals

We accumulate vectors along a path.

#### Curl and Green’s Theorem

Green’s Theorem is a fundamental theorem of calculus.

#### Green’s Theorem as a planimeter

A planimeter computes the area of a region by tracing the boundary.

#### Divergence and Green’s Theorem

Divergence measures the rate field vectors are expanding at a point.

#### Surface integrals

We generalize the idea of line integrals to higher dimensions.

#### Drawing a Mobius strip

Learn how to draw a Möbius strip.

#### Divergence theorem

We introduce the divergence theorem.

#### Stokes’ theorem

We introduce Stokes’ theorem.