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

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

#### 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.

#### Vector-valued functions

#### The cross product

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

#### Parameterizing by arc length

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

#### Functions of several variables

#### Functions of several variables

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

#### Approximating with the gradient

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

#### Differentiability

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

#### Interpreting the gradient vector

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