Question

Consider the graphs of f(x) and g(x) below:For each interval given below, calculate the average rate of change of f(x) or g(x) and compare the results.i. [0, 4]ii. [0, 8]iii. [0, 2.2]iv. [5.2, 6.1]v. [5.2, 6.9]

Answer 1

Answer:

$\begin{gathered} 1)\text{ Rf(x)=0 Rg(x)=0.5} \\ 2.\text{ Rf(x)=0 Rg(X)=0.5} \\ 3)\text{ Rf(x)=1.32 Rg(x)=0.5} \\ 4)\text{ Rf(x)=2.22 Rg(x)=0.5} \\ 5)\text{ Rf(x)=0 Rg(x)=0.5} \end{gathered}$

Step-by-step explanation:

To determine the average rate of change of an interval [a,b], we must use the following equation:

$\frac{f(b)-f(a)}{b-a}$

Therefore, for the interval [0,4]:

$ratef(x)=\frac{0-0}{4-0}=0$

For g(x) we need to find the equation of the line by the slope-point form:

$\begin{gathered} y-y_0=m(x-x_0) \\ y-2.9=(\frac{4.9-2.9}{6.1-2.2})(x-2.2) \\ y-2.9=\frac{20}{39}(x-2.2) \\ y=\frac{20}{39}x-\frac{44}{39}+2.9 \\ y=\frac{20}{39}x+\frac{691}{390} \end{gathered}$

To find the value to x=4, substitute it into the equation:

$\begin{gathered} y=\frac{20}{39}(4)+\frac{691}{390} \\ y=3.82 \\ \text{ Rate of change:} \\ \text{ Rate g(x)=}\frac{3.8-1.77}{4-0}=0.5 \end{gathered}$

Now, for [0,8]:

$\begin{gathered} Rf(x)=\frac{0-0}{8-0}=0 \\ Rg(x)=\frac{5.87-1.77}{8-0}=0.5 \end{gathered}$

[0,2.2]:

Since g(x) is a linear function, it will have a constant rate of change 0.5.

f(x):

$Rf(x)=\frac{2.9-0}{2.2-0}=1.32$

[5.2, 6.1]:

$\begin{gathered} Rf(x)=\frac{4.9-2.9}{6.1-5.2}=2.22 \\ Rg(x)=0.5 \end{gathered}$

[5.2,6.9]:

$\begin{gathered} Rf(x)=\frac{2.9-2.9}{6.9-5.2}=0 \\ Rg(x)=0.5 \end{gathered}$