Our functions are usually measuring something, like distance, heat, luminacity, density, temperature, stress, weight - or the rate of change of these quantities. The numeric values of the functions tell us the amount of these quantities, but the sign of the measurement often adds additional information.

$\blacktriangleright$ Especially the sign of the rate of change.

An important aspect of a measurement is how it changes - increaing, decreasing, or remaining steady.

A major goal of function analysis is to measure how function values are changing - identify where functions are increasing and decreasing and where they switch between increasing and decreasing.

Introduced earlier, the derivative provides this rate-of-change information. We have also hinted that functions often switch between increasing and decreasing where the derivative equals $0$.

For this reason alone, locating zeros of functions is a huge motivation.

Learning Outcomes

In this section, students will

• identify zeros of functions.
• solve equations.
• solve inequalities.