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.

Surface area

Surface areas of revolution

We compute surface area of a frustrum then use the method of “Slice, Approximate, Integrate” to find areas of surface areas of revolution.

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.

Series

We discuss convergence results for geometric series and telescoping series.

Integral and divergence tests

The integral test

Infinite sums can be studied using improper integrals.

The divergence test

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

Ratio and root tests

The ratio test

Some infinite series can be compared to geometric series.

The root test

Some infinite series can be compared to geometric series.

Comparison tests

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.

Alternating series

The alternating series test

Alternating series have nice properties.

Putting it all together

This section reinforces and synthesizes the ideas about sequences and 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.

Differential equations

Differential equations

Differential equations show you relationships between rates of functions.

Numerical methods

Slope fields and Euler’s method

We describe numerical and graphical methods for understanding differential equations.

Separable differential equations

Separable differential equations

Separable differential equations are those in which the dependent and independent variables can be separated on opposite sides of the equation.

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 measures how aligned two vectors are with each other.

Cross products

The cross product

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

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