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.
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.