We investigate sequences.

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.

Certain infinite series can be studied using improper integrals.

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

We learn how to estimate the value of a series.

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

Some infinite series can be compared to geometric series.

Some infinite series can be compared to geometric series.

We compare infinite series to each other using inequalities.

We compare infinite series to each other using limits.

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

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

Learn how to draw a sphere.

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 make interesting parametric plots.

Learn how to draw a torus.

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

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

We introduce partial derivatives and the gradient vector.

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

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.

We introduce Taylor polynomials for functions of several variables.

We will get to know some basic quadric surfaces.

Learn how to draw an elliptic and a hyperbolic paraboloid.

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

We learn to optimize surfaces along and within given paths.

We give a new method of finding extrema.

We study integrals over basic regions.

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

We integrate over regions in polar coordinates.

We integrate over regions in cylindrical coordinates.

We integrate over regions in spherical coordinates.

We compute surface area with double integrals.

We use integrals to model mass.

We practice more computations and think about what integrals mean.

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

We accumulate vectors along a path.

Green’s Theorem is a fundamental theorem of calculus.

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

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

We generalize the idea of line integrals to higher dimensions.

Learn how to draw a Möbius strip.

We introduce the divergence theorem.

We introduce Stokes’ theorem.