Tutorial: Average rate of change

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(This topic is also in Section 3.4 in Applied Calculus or Section 10.4 in Finite Mathematics and Applied Calculus)

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#[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.

For a 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 $t$:

The average rate of change of $s$ over the intervals $[0.2]$ and $[2, 4]$ are given by

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

In the %%applslopetut, slopes like these were recognized as 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 $\displaystyle {}= \frac{f(b) - f(a)}{b - a}$

can be used to directly calculate average rates of change when we are given an algebraic formula for $f$:

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.