3.3 Rates of Change and Behavior of Graphs

Using the data in the table, find the average rate of change of the price of gasoline between 2007 and 2009.

Using the data in the table, find the average rate of change between 2005 and 2010.

After picking up a friend who lives 10 miles away and leaving on a trip, Anna records her distance from home over time. The values are shown in the table. Find her average speed over the first 6 hours.

Compute the average rate of change of $f \left(x\right) = x^{2} - \frac{1}{x}$ on the interval $[\text{2}, \text{4}].$

Find the average rate of change of $f \left(x\right) = x - 2 \sqrt{x}$ on the interval $\left[1 , 9\right] .$

The electrostatic force $F ,$ measured in newtons, between two charged particles can be related to the distance between the particles $d ,$ in centimeters, by the formula $F \left(d\right) = \frac{2}{d^{2}} .$ Find the average rate of change of force if the distance between the particles is increased from 2 cm to 6 cm.

Find the average rate of change of $g \left(t\right) = t^{2} + 3 t + 1$ on the interval $\left[0 , a\right] .$ The answer will be an expression involving $a$ in simplest form.

Find the average rate of change of $f \left(x\right) = x^{2} + 2 x - 8$ on the interval $\left[5 , a\right]$ in simplest forms in terms of $a .$

Can the average rate of change of a function be constant?

How are the absolute maximum and minimum similar to and different from the local extrema?

Find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

$f \left(x\right) = 4 x^{2} - 7$ on $\left[1 , b\right]$

Find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

$p \left(x\right) = 3 x + 4$ on $\left[2 , 2 + h\right]$

Find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

$f \left(x\right) = 2 x^{2} + 1$ on $\left[x , x + h\right]$

Find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

$a \left(t\right) = \frac{1}{t + 4}$ on $\left[9 , 9 + h\right]$

Find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

$j \left(x\right) = 3 x^{3}$ on $\left[1 , 1 + h\right]$

Find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

Find $\frac{f \left(x + h\right) - f \left(x\right)}{h}$ given $f \left(x\right) = 2 x^{2} - 3 x$ on $\left[x , x + h\right]$

Estimate the average rate of change from $x = 2$ to $x = 5.$

Estimate the point(s) at which the graph of $f$ has a local maximum or a local minimum.

If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum.

the table gives the population of a town (in thousands) from 2000 to 2008. What was the average rate of change of population (a) between 2002 and 2004, and (b) between 2002 and 2006?

Find the average rate of change of each function on the interval specified.

$h \left(x\right) = 5 - 2 x^{2}$ on $\left[−2 , \text{4}\right]$

Find the average rate of change of each function on the interval specified.

$g \left(x\right) = 3 x^{3} - 1$ on $\left[−3 , \text{3}\right]$

Find the average rate of change of each function on the interval specified.

$p \left(t\right) = \frac{\left(t^{2} - 4\right) \left(t + 1\right)}{t^{2} + 3}$ on $\left[−3 , \text{1}\right]$

Use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

$f \left(x\right) = x^{4} - 4 x^{3} + 5$

Use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

$g \left(t\right) = t \sqrt{t + 3}$

Use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

$m \left(x\right) = x^{4} + 2 x^{3} - 12 x^{2} - 10 x + 4$

Let $f \left(x\right) = \frac{1}{x}$ . Find the number $b$ such that the average rate of change of $f$ on the interval $\left(2 , b\right)$ is $- \frac{1}{10} .$

A driver of a car stopped at a gas station to fill up their gas tank. They looked at their watch, and the time read exactly 3:40 p.m. At this time, they started pumping gas into the tank. At exactly 3:44, the tank was full and the driver noticed that they had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?