This course is built in Ximera.

We explain how your work is scored.

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

We define the concept of a function.

We discuss compositions of functions.

Here we ‘‘undo’’ functions.

Two young mathematicians think about the plots of functions.

Polynomials are some of our favorite functions.

Rational functions are functions defined by fractions of polynomials.

Exponential and logarithmic functions illuminated.

Two young mathematicians discuss stars and functions.

We introduce limits.

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

Continuity is defined by limits.

We give basic laws for working with limits.

Here we use limits to ensure piecewise functions are continuous.

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

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

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

Here we see a dialogue where students discuss solving inequalities as they learned in precalculus.

Here we see a consequence of a function being continuous.

Two young mathematicians discuss the novel idea of the ‘‘slope of a curve.’’

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

Two young mathematicians discuss derivatives as functions.

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

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

Two young mathematicians think about ‘‘short cuts’’ for differentiation.

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

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

Here we compute derivatives of products and quotients of functions

Here we compute derivatives of compositions of functions

We derive the derivative of the natural exponential function.

We derive the derivatives of general exponential functions using the chain rule.

Two young mathematicians discuss the derivative of inverse functions.

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

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

Two young mathematicians discuss the standard form of a line.

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

Two young mathematicians think about derivatives and logarithms.

We use logarithms to help us differentiate.

Here we look at higher order derivatives.

Two young mathematicians discuss linear approximation.

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

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

We use derivatives to help locate extrema.

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

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

Here we look at the second derivative test.

We examine a fact about continuous functions.

We show a procedure to find global extrema on closed intervals.

Two young mathematicians discuss what curves look like when one ‘‘zooms out.’’

We explore functions that ‘‘shoot to infinity’’ at certain points in their domain.

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

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

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

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

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

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

Now we put our optimization skills to work.

Two young mathematicians discuss optimizing aluminum cans.

Now we put our optimization skills to work.

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

We introduce antiderivatives.

Two young mathematicians discuss the idea of area.

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

Two young mathematicians discuss cutting up areas.

Definite integrals compute signed area.

Two young mathematicians discuss what calculus is all about.

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

Two young mathematicians discuss what calculus is all about.

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

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

Two students consider substitution geometrically.

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

Two young mathematicians discuss how tricky integrals are puzzles.

We explore more difficult problems involving substitution.

We compute the area of a region between two curves using the definite integral.

Two young mathematicians discuss differential equations.

We study equations with that relate functions with their rates.

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

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

We introduce partial derivatives.