## Tutorial: Average rate of change

(This topic is also in Section 3.4 in*Applied Calculus*or Section 10.4 in

*Finite Mathematics and Applied Calculus*) #[I don't like this new tutorial. Take me back to the old tutorial!][No me gusta este nueve tutorial. ¡Regresame al tutorial más viejo!]#

#[Average rate of change numerically and graphically][Razón de cambio promedio numéricamente y gráficamente]#

In the %%slopetut we saw that the units of measurement of the slope of a line $y = mx+b$ are units of $y$ per unit of $x$, and in the %%applslopetut we saw how to interpret this fact in various applications: *The slope measures the rate at which the value of a linear function is changing.*

*non-linear*function, we can also calculate rates of change by mimicking the calculation of the slope:

**Change and average rate of change of a function**The

**change**in $f(x)$ over the interval $[a, b]$ is

Change in $f$ \t ${}= \Delta f$
\\ \t ${}={}$ Second value $-$ First value
\\ \t ${}=f(b) - f(a)$.

The **average rate of change**of $f(x)$ over the interval $[a, b]$ is

Average rate of change of $f$

\t $\displaystyle {}= \frac{\text{Change in }f}{\text{Change in }x}$
\\ \t $\displaystyle {}= \frac{\Delta f}{\Delta x} = \frac{f(b) - f(a)}{b - a}$
\\ \t $\displaystyle {}={}$ Slope of line through points $P$ and $Q$ (see the figure). **Units of measurement**As with slopes, we measure the average rate of change of $f(x)$ in units of $f$ per unit of $x$.

**%%Examples**

**1.**If $f(3) = -1$ zonars* and $f(5) = 0.5$ zonars, and if $x$ is measured in years, then the average rate of change of $f$ over the interval $[3, 5]$ is

Average rate of change of $f$ over $[3, 5]$

\t $\displaystyle {}= \frac{f(5) - f(3)}{5 - 3}$
\\ \t $\displaystyle {}= \frac{0.5 - (-1)}{2} = 0.75$ #[zonars per year][zonares por año]#
*The zonar, as we all should know, is the martian unit of currency.

**2.**The following table shows the distance $s$ of a freight train along a railway line at various times $t$:

Average rate of change over $[0,2]$

\t $\displaystyle {}= \frac{s(2) - s(0)}{2 - 0}$
\\ \t $\displaystyle {}= \frac{320 - 20}{2} = 150$ km/h
\\ Average rate of change over $[2,4]$

\t $\displaystyle {}= \frac{s(4) - s(2)}{4 - 2}$
\\ \t $\displaystyle {}= \frac{640 - 320}{2} = 160$ km/h
*velocity.*Had $s$ been a linear function of $t$, then the two velocities we calculated would have been equal. Here, however, they are different, and we call them

**average velocities;**the train has speeded up from an average velocity of 150 km/h in the first two hours to 160 km/h in the next two hours.

**One for you**

**Average rate of change graphically**We saw above that the average rate of change of $f$ over $[a, b]$ is the slope of the line through the points on the graph of $f$ where $x = a$ and $x = b$, and so we can calculate this rate from a graph.

**One for you**

Average rate of change algebraically

#[The formula][la fórmula]#
Average rate of change of $f$

\t $\displaystyle {}= \frac{f(b) - f(a)}{b - a}$
Average rates of change over smaller and smaller intervals

In preparation for the next section, we are going to look at the average rate of change of a function over smaller and smaller intervals and look for some kind of pattern or trend in the answers.
**%%Q**Think about how we should interpret this "limiting value."

Now try the exercises in Section 3.4 in

*Applied Calculus*or Section 10.4 in*Finite Mathematics and Applied Calculus*.*September 2022*

Copyright © 2019 Stefan Waner and Steven R. Costenoble