determine the average rate of change for the function $f(x)=\frac{1}{2}x^{2}+3x$ from $x = 1$ to $x =…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$.

First question:

For $f(x)=\frac{1}{2}x^{2}+3x$, $a = 1$ and $b = 4$.

Step2: Calculate $f(4)$

$f(4)=\frac{1}{2}(4)^{2}+3\times4=\frac{1}{2}\times16 + 12=8 + 12=20$.

Step3: Calculate $f(1)$

$f(1)=\frac{1}{2}(1)^{2}+3\times1=\frac{1}{2}+3=\frac{1 + 6}{2}=\frac{7}{2}$.

Step4: Calculate average rate of change

$\frac{f(4)-f(1)}{4 - 1}=\frac{20-\frac{7}{2}}{3}=\frac{\frac{40 - 7}{2}}{3}=\frac{\frac{33}{2}}{3}=\frac{33}{2}\times\frac{1}{3}=\frac{11}{2}=5.5$ (There seems to be an error in the options for this part. The correct way to calculate is as above).

Second question:

For $f(x)=\frac{1}{x}$, $a = 1$ and $b = 3$.

Step5: Calculate $f(3)$

$f(3)=\frac{1}{3}$.

Step6: Calculate $f(1)$

$f(1)=1$.

Step7: Calculate average rate of change

$\frac{f(3)-f(1)}{3 - 1}=\frac{\frac{1}{3}-1}{2}=\frac{\frac{1 - 3}{3}}{2}=\frac{-\frac{2}{3}}{2}=-\frac{1}{3}$.

Answer:

First question: (No correct option among given ones based on calculation) Second question: b. $-\frac{1}{3}$