Question

fall 2025
math 101 2m
mini quiz 3.3 - 2
name: sidney martin
wednesday, october 8, 2025
show your work to get full points.
if your work is correct, you will get 5 points
if your work is partially correct, you will get 2 points
if your work is incorrect, you will get a participation 1 point
(1) (10 points)
find the average rate of change of $f(x)=\frac{5}{x^{2}}$
a. from 1 to 2
answer: $-\frac{15}{4}$
b. from -1 to 1
answer: 0
(2) (5 points)
find the intervals on which the function is increasing, decreasing, or constant. write the answer in interval notation.
increasing: $(-3, -1)$
decreasing: $(1, 3)$
constant: $(-1, 1)$

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 given by $\frac{f(b)-f(a)}{b - a}$.

Step2: Solve part (a)

Given $f(x)=\frac{5}{x^{2}}$, $a = 1$, and $b=2$. First, find $f(1)$ and $f(2)$.
$f(1)=\frac{5}{1^{2}}=5$ and $f(2)=\frac{5}{2^{2}}=\frac{5}{4}$.
Then, calculate the average rate of change: $\frac{f(2)-f(1)}{2 - 1}=\frac{\frac{5}{4}-5}{1}=\frac{\frac{5 - 20}{4}}{1}=-\frac{15}{4}$.

Step3: Solve part (b)

Given $a=-1$ and $b = 1$, find $f(-1)$ and $f(1)$.
$f(-1)=\frac{5}{(-1)^{2}}=5$ and $f(1)=\frac{5}{1^{2}}=5$.
Then, calculate the average rate of change: $\frac{f(1)-f(-1)}{1-(-1)}=\frac{5 - 5}{2}=0$.

Step4: Analyze increasing, decreasing, and constant intervals for part (2)

For an increasing interval, if $x_1f(x_2)$. For a constant interval, if $x_1Looking at the graph, the function is increasing on the interval $(-3,-1)$.
The function is decreasing on the interval $(1,3)$.
The function is constant on the interval $(-1,1)$.

Answer:

(1) a. $-\frac{15}{4}$
b. $0$
(2) Increasing: $(-3,-1)$
Decreasing: $(1,3)$
Constant: $(-1,1)$