#### Content for the First Exam

#### Working with rational functions

Rational functions are functions defined by fractions of polynomials.

#### 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.

#### Horizontal asymptotes

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

#### Content for the Second Exam

#### The definition of the derivative

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

#### 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.

#### 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

#### Content for the Third Exam

#### 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.

#### Position, velocity, and acceleration

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

#### How fast was the pen going?

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

#### Derivatives of trigonometric functions

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

#### Additional content for Autumn Final Exam

#### Content for the First Exam

#### Linear approximation

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

#### Computations for graphing functions

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

#### Implicit differentiation

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

#### Finding dx dy

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

#### Content for the Second Exam

#### Derivatives of inverse trigonometric functions BreakGround

Two young mathematicians think about the plots of 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.

#### Indeterminate mutterings

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

#### Content for the Third Exam

#### Approximating area with rectangles

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

#### The First Fundamental Theorem of Calculus

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

#### The Second Fundamental Theorem of Calculus

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