Chapter 3: Functions and Graphs 3-7: Rates of Change

3.7: Rates of Change Average Rates of Change ◦ ◦ Really, it’s just like finding slope ◦ Example 1: If the equation for a falling rock is d(t) = 16t 2 where d(t) is measured in feet and t is measured in seconds  Find the average speed of the rock from  t = 1 to t = 4  t = 2 to t = 4.5

3.7: Rates of Change Again, this works just like slope: Example 2: A balloon’s volume in gallons is calculated by the function V(x) = x 3 /55 where x is the radius of the balloon in inches. Find the average change of the volume as the radius increases from 5 to 10 inches.

3.7: Rates of Change The difference quotient ◦ And you thought you were done with that… ◦ ◦ Example 5: Use the formula for a falling rock (d = 16t 2 ) to find the difference quotient. Use that formula to find the average speed from 3 to 3.1 seconds.

3.7: Rates of Change If the time change was 3 to 3.1 seconds, then x = 3, and h (the change) is 0.1 sec.

3.7: Rates of Change Why would I go through the trouble of finding the difference quotient? ◦ If I need to figure out multiple calculations, it’s easier (and faster) to plug in the x and h into the difference quotient than to plug x into the function every time. ◦ Calculus precursor: Finding smaller and smaller h values helps us calculate the instantaneous velocity

3.7: Rates of Change Assignment ◦ Page ◦ 1-25, odd problems ◦ Show work