Question

assignment 3.3: rates of change and behavior of graphs
score: 11.5/13 answered: 12/13
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question 13
0/1 pt 5 19 details
let ( f(x)=\frac{1}{x} ). find the number ( b ) such that the average rate of change of ( f ) on the interval (2,b) is (-\frac{1}{6}).
your answer:
b =

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of $y = f(x)$ on $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 2$, $f(x)=\frac{1}{x}$, so $\frac{f(b)-f(2)}{b - 2}=-\frac{1}{6}$.

Step2: Substitute $f(x)$ values

Since $f(b)=\frac{1}{b}$ and $f(2)=\frac{1}{2}$, we have $\frac{\frac{1}{b}-\frac{1}{2}}{b - 2}=-\frac{1}{6}$.

Step3: Simplify the left - hand side

$\frac{\frac{2 - b}{2b}}{b - 2}=-\frac{1}{6}$, which simplifies to $\frac{2 - b}{2b(b - 2)}=-\frac{1}{6}$. Notice $2 - b=-(b - 2)$, so $-\frac{1}{2b}=-\frac{1}{6}$.

Step4: Solve for $b$

Cross - multiply: $2b = 6$, so $b = 3$.

Answer:

$3$