Content for the First Exam
6.2Working with rational functions
Rational functions are functions defined by fractions of polynomials.
7.4The Squeeze Theorem
The Squeeze theorem allows us to exchange difficult functions for easy functions.
8.1Could it be anything?
Two young mathematicians investigate the arithmetic of large and small
numbers.
8.2Limits of the form zero over zero
We want to evaluate limits where the Limit Laws do not directly apply.
8.3Limits of the form nonzero over zero
We want to solve limits that have the form nonzero over zero.
9.2Vertical asymptotes
We explore functions that “shoot to infinity” at certain points in their domain.
9.3Horizontal asymptotes
We explore functions that behave like horizontal lines as the input grows without
bound.
10.2Continuity of piecewise functions
Here we use limits to ensure piecewise functions are continuous.
Content for the Second Exam
12.2The definition of the derivative
We compute the instantaneous growth rate by computing the limit of average growth
rates.
13.2The derivative as a function
Here we study the derivative of a function, as a function, in its own right.
13.3Differentiability implies continuity
We see that if a function is differentiable at a point, then it must be continuous at
that point.
15.1Derivatives of products are tricky
Two young mathematicians discuss derivatives of products and products of
derivatives.
15.2The Product rule and quotient rule
Here we compute derivatives of products and quotients of functions
Content for the Third Exam
18.2The derivative of the natural exponential function
We derive the derivative of the natural exponential function.
18.3Derivatives of exponential and logarithmetic functions
Derivatives of exponential and logarithmic functions calculated.
19.1Rates of rates
Two young mathematicians look at graph of a function, its first derivative, and its
second derivative.
19.2Higher order derivatives and graphs
Here we make a connection between a graph of a function and its derivative and
higher order derivatives.
19.4Position, velocity, and acceleration
Here we discuss how position, velocity, and acceleration relate to higher
derivatives.
21.1How fast was the pen going?
Two young mathematicians think about the rate of change of periodic motion.
21.3Derivatives of trigonometric functions
We use the chain rule to unleash the derivatives of the trigonometric functions.