Average and Instantaneous Rate of Change

Since the rate of change is the slope at point P is calculatable by creating a point Q and finding the slope of \(\overline{PQ}\) as Q approaches P and slope can be modelled using rise over run, the rate of change can be modelled using \(\frac{\Delta y}{\Delta x} = \frac{f(x) - f(a)}{x - a}\)

Sandwich Theorem

As Q approaches P, the slope of \(\overline{PQ}\) approaches a certain value. Use the estimations to check what value the slope approaches and that value is an estimate for the value of the instantaneous rate of change.

Factor Theorem

For \(\frac{f(x) - f(a)}{x - a}\), if \(f(x) - f(a)\) (where a is the x-value of the tangent point) expanded is divisible by \((x - a)\) then factor out the \((x - a)\) and eliminate the terms.

Division by Zero

\(\frac{c}{0}\) where \(c\) is a constant value is undefined. \(\frac{0}{0}\) is interderminate which is to say there is still work that can be done. More extra steps (usually factoring and eliminating) are needed to get to the solution.

Key Words

If the word “during” or “between” is used in a question, it is asking for the average rate of change. If the word “at” is used in the question, it is asking for instantaneous rate of change.