#### How to use Ximera

This course is built in Ximera.

#### How is my work scored?

We explain how your work is scored.

#### 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.

#### How crazy could it be?

Two young mathematicians think about the plots of functions.

#### Polynomial functions

Polynomials are some of our favorite functions.

#### Rational functions

Rational functions are functions defined by fractions of polynomials.

#### Trigonometric functions

We review trigonometric functions.

#### Exponential and logarithmetic functions

Exponential and logarithmic functions illuminated.

#### Stars and functions

Two young mathematicians discuss stars and functions.

#### What is a limit?

We introduce limits.

#### 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’’ at certain points in their domain.

#### 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.

#### The derivative of the natural exponential function

We derive the derivative of the natural exponential function.

#### The derivative of sine

We derive the derivative of sine.

#### 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

#### Derivatives of trigonometric functions

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

#### 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 look at graphs of 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.

#### Standard form

Two young mathematicians discuss the standard form of a line.

#### Implicit differentiation

In this section we differentiate equations without expressing them in terms of a single variable.

#### Derivatives of inverse exponential functions

We derive the derivatives of inverse exponential functions using implicit differentiation.

#### Logarithmic differentiation

Two young mathematicians think about derivatives and logarithms.

#### Logarithmic differentiation

We use logarithms to help us differentiate.

#### A changing circle

Two young mathematicians discuss a circle that is changing.

#### More than one rate

Here we work abstract related rates problems.

#### Pizza and calculus, so cheesy

Two young mathematicians discuss tossing pizza dough.

#### Applied related rates

We work related rates problems in context.

#### We can figure it out

Two young mathematicians discuss the derivative of inverse functions.

#### Derivatives of inverse trigonometric functions

We derive the derivatives of inverse trigonometric functions using implicit differentiation.

#### The Inverse Function Theorem

We see the theoretical underpinning of finding the derivative of an inverse function at a point.

#### More coffee

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

#### Maximums and minimums

We use derivatives to help locate extrema.

#### What’s the graph look like?

Two young mathematicians discuss how to sketch the graphs of functions.

#### Concepts of graphing functions

We use the language of calculus to describe graphs of functions.

#### Wanted: graphing procedure

Two young mathematicians discuss how to sketch the graphs of functions.

#### Computations for graphing functions

We will give some general guidelines for sketching the plot of a function.

#### 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.

#### Replacing curves with lines

Two young mathematicians discuss linear approximation.

#### Linear approximation

We use a method called ‘‘linear approximation’’ to estimate the value of a (complicated) function at a given point.

#### Explanation of the product and chain rules

We give explanation for the product rule and chain rule.

#### 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.

#### A limitless dialogue

Two young mathematicians consider a way to compute limits using derivatives.

#### L’Hopital’s rule

We use derivatives to give us a ‘‘short-cut’’ for computing limits.

#### Modeling the spread of infectious diseases

Two young mathematicians discuss differential equations.

#### Differential equations

We study equations with that relate functions with their rates.

#### Jeopardy! Of calculus

Two young mathematicians discuss a ‘Jeopardy!’ version of calculus.

#### Basic antiderivatives

We introduce antiderivatives.

#### Falling objects

We study a special type of differential equation.

#### What is area?

Two young mathematicians discuss the idea of area.

#### Approximating area with rectangles

We introduce the basic idea of using rectangles to approximate the area under a curve.

#### Meaning of multiplication

A dialogue where students discuss multiplication.

#### Relating velocity and position, antiderivatives and areas

We see a connection between approximating antiderivatives and approximating areas.

#### Computing areas

Two young mathematicians discuss cutting up areas.

#### The definite integral

Definite integrals compute signed area.

#### What’s in a calculus problem?

Two young mathematicians discuss what calculus is all about.

#### The First Fundamental Theorem of Calculus

The rate that accumulated area under a curve grows is described identically by that curve.

#### A secret of the definite integral

Two young mathematicians discuss what calculus is all about.

#### The Second Fundamental Theorem of Calculus

The accumulation of a rate is given by the change in the amount.

#### A tale of three integrals

At this point we have three ‘‘different’’ integrals.

#### What could it represent?

Two young mathematicians discuss whether integrals are defined properly.

#### Applications of integrals

We give more contexts to understand integrals.

#### Geometry and substitution

Two students consider substitution geometrically.

#### The idea of substitution

We learn a new technique, called substitution, to help us solve problems involving integration.

#### Integrals are puzzles!

Two young mathematicians discuss how tricky integrals are puzzles.

#### Working with substitution

We explore more difficult problems involving substitution.

#### The Work-Energy Theorem

Substitution is given a physical meaning.