Q. 16 - Calculus

Question

For the graph of g(x) shown next at the right, list the following quantities in order from least to greatest:
(a) the average rate of change of g on $[0,1]$
(b) the instantaneous rate of change of g at $x=1$ is $0.4$
(c) $\frac{g(-1+0.1)-g(-1)}{0.1}$

(d) $\frac{g\left(1\right)-g(-1)}{1-(-1)}=$

Step-by-Step Solution

Answer

(a) The average rate of change of the function on $[0,1]$ is $2$

(b) The instantaneous rate of change of the function of the function at $x=1$ is 

(c) The value of $\frac{g(-1+0.1)-g(-1)}{0.1}$ is $-1$

(d) $\frac{g\left(1\right)-g(-1)}{1-(-1)}=0$

From the values, the quantities in  order from the least to the greatest is $\left(c\right), \left(d\right), \left(b\right), \left(a\right)$

Part (a) Step 1. Given Information.

The graph of the function.

Part (a) Step 2. Calculate average rate of change.

Consider the interval $[0,1]$

$$\begin{gathered} \frac{g\left(1\right)-g\left(0\right)}{1-0}=\frac{1-(-1)}{1} \\ =2 \end{gathered}$$

Part (b) Determine the instantaneous rate of change.

Consider an interval $[0,0.5]$

$$\begin{gathered} \frac{g(0.5)-g\left(0\right)}{0.5-0}=\frac{-0.8-(-1)}{0.5} \\ =\frac{0.2}{0.5} \\ =0.4 \end{gathered}$$

Consider an interval $[0,0.25]$

$$\begin{gathered} \frac{f(0.25)-f\left(0\right)}{0.25-0}=\frac{-0.9-(-1)}{0.25} \\ =\frac{0.1}{0.25} \\ =0.4 \end{gathered}$$

So the instantaneous rate of change is $0.4$

Part (c) Step 1. Find the given value.

From the graph, 

$$\begin{gathered} \frac{g(-1+0.1)-g(-1)}{0.1}=\frac{g(0.9)-g(-1)}{0.1} \\ =\frac{-0.1-0}{0.1} \\ =-1 \end{gathered}$$

Part (d) Step 1. Find the given function value.

From the graph,

$$\begin{gathered} \frac{g\left(1\right)-g(-1)}{1-(-1)}=\frac{0-0}{1+1} \\ =0 \end{gathered}$$

Part (d) Step 2. Arrange in order.

From the found values, the function of the list from the least to the greatest is:

$\left(c\right), \left(d\right), \left(b\right), \left(a\right)$