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Average rate of change | $\frac{\Delta y}{\Delta x}=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}$ |
Can the average rate of change of a function be constant?
Yes, the average rate of change of all linear functions is constant.
If a function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is increasing on $\text{\hspace{0.17em}}(a,b)\text{\hspace{0.17em}}$ and decreasing on $\text{\hspace{0.17em}}(b,c),\text{\hspace{0.17em}}$ then what can be said about the local extremum of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}(a,c)?\text{\hspace{0.17em}}$
How are the absolute maximum and minimum similar to and different from the local extrema?
The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.
How does the graph of the absolute value function compare to the graph of the quadratic function, $\text{\hspace{0.17em}}y={x}^{2},\text{\hspace{0.17em}}$ in terms of increasing and decreasing intervals?
For the following exercises, find the average rate of change of each function on the interval specified for real numbers $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}h.$
$f\left(x\right)=4{x}^{2}-7\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}[1,\text{}b]$
$4\left(b+1\right)$
$g\left(x\right)=2{x}^{2}-9\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[4,\text{}b\right]$
$p\left(x\right)=3x+4\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}[2,\text{}2+h]$
3
$k\left(x\right)=4x-2\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}[3,\text{}3+h]$
$f\left(x\right)=2{x}^{2}+1\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}[x,x+h]$
$4x+2h$
$g\left(x\right)=3{x}^{2}-2\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}[x,x+h]$
$a\left(t\right)=\frac{1}{t+4}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}[9,9+h]$
$\frac{-1}{13\left(13+h\right)}$
$b\left(x\right)=\frac{1}{x+3}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}[1,1+h]$
$j\left(x\right)=3{x}^{3}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}[1,1+h]$
$3{h}^{2}+9h+9$
$r\left(t\right)=4{t}^{3}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}[2,2+h]$
$\frac{f\left(x+h\right)-f\left(x\right)}{h}\text{\hspace{0.17em}}$ given $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}-3x\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}[x,x+h]$
$4x+2h-3$
For the following exercises, consider the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ shown in [link] .
Estimate the average rate of change from $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x=4.$
Estimate the average rate of change from $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x=5.$
$\frac{4}{3}$
For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.
increasing on $\text{\hspace{0.17em}}\left(-\infty ,-2.5\right)\cup \left(1,\infty \right),\text{\hspace{0.17em}}$ decreasing on $\text{\hspace{0.17em}}(-2.5,\text{}1)$
increasing on $\text{\hspace{0.17em}}\left(-\infty ,1\right)\cup \left(3,4\right),\text{\hspace{0.17em}}$ decreasing on $\text{\hspace{0.17em}}\left(1,3\right)\cup \left(4,\infty \right)$
For the following exercises, consider the graph shown in [link] .
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