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.

We review trigonometric functions.

Exponential and logarithmic functions illuminated.

Two young mathematicians discuss stars and functions.

We introduce limits.

Continuity is defined by limits.

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

We give basic laws for working with limits.

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

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.

Try these problems.

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

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

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

Two young mathematicians discuss the eating habits of their cats.

Here we use limits to ensure piecewise functions are continuous.

Here we see a consequence of a function being continuous.

Review questions for exam 1.

Two young mathematicians discuss limits and instantaneous velocity.

We use limits to compute instantaneous velocity.

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.

We derive the derivative of the natural exponential function.

We derive the derivative of sine.

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

Here we compute derivatives of products and quotients of functions

Two young mathematicians discuss the chain rule.

Here we compute derivatives of compositions of functions

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

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

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

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

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

Two young mathematicians discuss the standard form of a line.

In this section we differentiate equations that contain more than one variable on one side.

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

Two young mathematicians think about derivatives and logarithms.

We use logarithms to help us differentiate.

Two young mathematicians discuss the derivative of inverse functions.

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

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

Two young mathematicians discuss a circle that is changing.

Here we work abstract related rates problems.

Two young mathematicians discuss tossing pizza dough.

We solve related rates problems in context.

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

We use derivatives to help locate extrema.

Review questions for MIDTERM 2.

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 race to math class.

We examine a fact about continuous functions.

Here we see a key theorem of calculus.

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.

We give explanation for the product rule and chain rule.

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 consider a way to compute limits using derivatives.

We use derivatives to give us a “short-cut” for computing limits.

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

We introduce antiderivatives.

We study a special type of differential equation.

Two young mathematicians discuss the idea of area.

We introduce sigma notation.

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 net area.

A dialogue where students discuss multiplication.

We give an alternative interpretation of the definite integral and make a connection between areas and antiderivatives.

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.

Review questions for MIDTERM 3.

Two young mathematicians discuss whether integrals are defined properly.

We give more contexts to understand 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.

Substitution is given a physical meaning.