How to use Ximera

This course is built in Ximera.

How is my work scored?

We explain how your work is scored.

Needed Score?

Two young mathematicians examine an equation.

Equations

We discuss solving equations.

Inequalities

We discuss inequalities.

Same or different?

Two young mathematicians examine one (or two!) functions.

For each input, exactly one output

We define the concept of a function.

Compositions of functions

We discuss compositions of functions.

Inverses of functions

Here we “undo” functions.

Stars and functions

Two young mathematicians discuss stars and functions.

What is a limit?

We introduce limits.

How crazy could it be?

Two young mathematicians think about the plots of functions.

Working with polynomials

Polynomials are some of our favorite functions.

End behavior

Polynomials are some of our favorite functions.

Graphs of polynomial functions

Polynomials are some of our favorite functions.

Working with rational functions

Rational functions are functions defined by fractions of polynomials.

Rational equations and inequalities

Equations and inequalities with Rational Functions

Graphs of rational functions

Rough graphs of rational functions

Equal or not?

Here we see a dialogue where students discuss combining limits with arithmetic.

Continuity

Continuity is defined by limits.

The limit laws

We give basic laws for working with limits.

The Squeeze Theorem

The Squeeze theorem allows us to exchange difficult functions for easy functions.

Could it be anything?

Two young mathematicians investigate the arithmetic of large and small numbers.

Limits of the form zero over zero

We want to evaluate limits where the Limit Laws do not directly apply.

Limits of the form nonzero over zero

We want to solve limits that have the form nonzero over zero.

Practice

Try these problems.

Zoom out

Two young mathematicians discuss what curves look like when one “zooms out.”

Vertical asymptotes

We explore functions that “shoot to infinity” near certain points.

Horizontal asymptotes

We explore functions that behave like horizontal lines as the input grows without bound.

Slant asymptotes

We explore functions that “shoot to infinity” at certain points in their domain.

Roxy and Yuri like food

Two young mathematicians discuss the eating habits of their cats.

Continuity of piecewise functions

Here we use limits to ensure piecewise functions are continuous.

The Intermediate Value Theorem

Here we see a consequence of a function being continuous.

Practice

Try these problems.

Limits and velocity

Two young mathematicians discuss limits and instantaneous velocity.

Instantaneous velocity

We use limits to compute instantaneous velocity.

Slope of a curve

Two young mathematicians discuss the novel idea of the “slope of a curve.”

The definition of the derivative

We compute the instantaneous growth rate by computing the limit of average growth rates.

Wait for the right moment

Two young mathematicians discuss derivatives as functions.

The derivative as a function

Here we study the derivative of a function, as a function, in its own right.

Differentiability implies continuity

We see that if a function is differentiable at a point, then it must be continuous at that point.

Patterns in derivatives

Two young mathematicians think about “short cuts” for differentiation.

Basic rules of differentiation

We derive the constant rule, power rule, and sum rule.

Derivatives of products are tricky

Two young mathematicians discuss derivatives of products and products of derivatives.

The Product rule and quotient rule

Here we compute derivatives of products and quotients of functions

An unnoticed composition

Two young mathematicians discuss the chain rule.

The chain rule

Here we compute derivatives of compositions of functions

Exponential and logarithmetic functions

Exponential and logarithmic functions illuminated.

Interesting changes

Two young mathematicians think about the plots of functions.

The derivative of the natural exponential function

We derive the derivative of the natural exponential function.

Derivatives of exponential and logarithmetic functions

Derivatives of exponential and logarithmic functions calculated.

Rates of rates

Two young mathematicians look at graph of a function, its first derivative, and its second derivative.

Higher order derivatives and graphs

Here we make a connection between a graph of a function and its derivative and higher order derivatives.

Concavity

Here we examine what the second derivative tells us about the geometry of functions.

Position, velocity, and acceleration

Here we discuss how position, velocity, and acceleration relate to higher derivatives.

Trigonometric Functions

Two young mathematicians think about periodic motion.

Trigonometric functions

We review trigonometric functions.

How fast was the pen going?

Two young mathematicians think about the rate of change of periodic motion.

The derivative of sine

We derive the derivative of sine.

Derivatives of trigonometric functions

We use the chain rule to unleash the derivatives of the trigonometric functions.

More coffee

Two young mathematicians witness the perils of drinking too much coffee.

Maximums and minimums

We use derivatives to help locate extrema.

Let’s run to class

Two young mathematicians race to math class.

The Extreme Value Theorem

We examine a fact about continuous functions.

The Mean Value Theorem

Here we see a key theorem of calculus.

A mysterious formula

Two young mathematicians discuss optimization from an abstract point of view.

Basic optimization

Now we put our optimization skills to work.

Volumes of aluminum cans

Two young mathematicians discuss optimizing aluminum cans.

Applied optimization

Now we put our optimization skills to work.