Ximera tutorial

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 integration

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.

Areas between curves

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

Accumulated cross-sections

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

Solids of revolution

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

Length of curves

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

Applications of integration

Physical applications

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

Integration by parts

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

Trigonometric integrals

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

Trigonometric substitution

Trigonometric substitution

We integrate by substitution with the appropriate trigonometric function.

Partial fractions

Rational functions

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

Improper integrals

Improper Integrals

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

Sequences

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.

Sequences as functions

Limits of sequences

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

Sums of sequences

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

The divergence test

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

Ratio test

The ratio test

Some infinite series can be compared to geometric series.

Approximating functions with polynomials

Approximating functions with polynomials

We can approximate smooth functions with polynomials.

Power series

Power series

Infinite series can represent functions.

Introduction to Taylor series

Introduction to Taylor series

We study Taylor and Maclaurin series.

Numbers and Taylor series

Numbers and Taylor series

Taylor series are a computational tool.

Calculus and Taylor series

Calculus and Taylor series

Power series interact nicely with other calculus concepts.

Parametric equations

Parametric equations

We discuss the basics of parametric curves.

Calculus and parametric curves

We discuss derivatives of parametrically defined curves.

Introduction to polar coordinates

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

Derivatives of polar functions

We differentiate polar functions.

Integrals of polar functions

Integrals of polar functions

We integrate polar functions.

Working in two and three dimensions

Working in two and three dimensions

We talk about basic geometry in higher dimensions.

Vectors

Vectors

Vectors are lists of numbers that denote direction and magnitude.

Dot products

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.

Cross products

The cross product

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

Lines and curves in space

Lines and curves in space

Vector-valued functions are parameterized curves.

Calculus and vector-valued functions

Calculus and vector-valued functions

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

Motion and paths in space

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.

Normal vectors

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

Functions of several variables

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

Level Sets

We introduce level sets.

Continuity of functions of several variables

Open and Closed Sets

We generalize the notion of open and closed intervals to open and closed sets in .

Limits

We investigate limits of functions of several variables.

Continuity

We investigate what continuity means for functions of several variables.

Partial derivatives and the gradient vector

Partial derivatives

We introduce partial derivatives and the gradient vector.

Tangent planes and differentiability

Tangent planes

We find tangent planes.

Differentiability

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

The directional derivative and the chain rule

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.

Interpreting the gradient

Interpreting the gradient vector

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

You can download a Certificate as a record of your successes.