Question

fall 2025
math 101 2m
mini quiz 3.3 - 2
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:

b. from - 1 to 1
answer:

(2) (5 points)
find the intervals on which the function is increasing, decreasing, or constant. write the answer in interval notation.
increasing:
decreasing:
constant:

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

Step2: Calculate for part a

Given $f(x)=\frac{5}{x^{2}}$, $a = 1$, $b = 2$. First, find $f(1)$ and $f(2)$.
$f(1)=\frac{5}{1^{2}}=5$, $f(2)=\frac{5}{2^{2}}=\frac{5}{4}$.
Then, $\frac{f(2)-f(1)}{2 - 1}=\frac{\frac{5}{4}-5}{1}=\frac{\frac{5 - 20}{4}}{1}=-\frac{15}{4}$.

Step3: Calculate for part b

Given $a=-1$, $b = 1$. $f(-1)=\frac{5}{(-1)^{2}} = 5$, $f(1)=\frac{5}{1^{2}}=5$.
Then, $\frac{f(1)-f(-1)}{1-(-1)}=\frac{5 - 5}{2}=0$.

Step4: Analyze increasing - decreasing - constant intervals from graph

For a function $y = f(x)$, if the slope of the tangent line is positive, the function is increasing; if negative, decreasing; if zero, constant.
From 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: $(-1,1)$