#### 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.

#### Follow the bouncing pen.

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.

You can download a  as a record of your successes.