Question

check your understanding

1. calculate the average rate of change for the function (g(x)=4x^{2}-5x + 1) over each interval.

a) (2leq xleq4) d) (2leq xleq2.25)
b) (2leq xleq3) e) (2leq xleq2.1)
c) (2leq xleq2.5) f) (2leq xleq2.01)

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = g(x)$ over the interval $[a,b]$ is $\frac{g(b)-g(a)}{b - a}$.

Step2: Calculate $g(a)$ and $g(b)$ for part a) ($a = 2$, $b = 4$)

First, find $g(2)$:
$g(2)=4\times2^{2}-5\times2 + 1=4\times4-10 + 1=16-10 + 1=7$.
Then, find $g(4)$:
$g(4)=4\times4^{2}-5\times4 + 1=4\times16-20 + 1=64-20 + 1=45$.
Now, calculate the average rate of change:
$\frac{g(4)-g(2)}{4 - 2}=\frac{45 - 7}{2}=\frac{38}{2}=19$.

Step3: Calculate $g(a)$ and $g(b)$ for part b) ($a = 2$, $b = 3$)

$g(2)=7$ (calculated above).
$g(3)=4\times3^{2}-5\times3 + 1=4\times9-15 + 1=36-15 + 1=22$.
The average rate of change is $\frac{g(3)-g(2)}{3 - 2}=\frac{22 - 7}{1}=15$.

Step4: Calculate $g(a)$ and $g(b)$ for part c) ($a = 2$, $b = 2.5$)

$g(2)=7$.
$g(2.5)=4\times(2.5)^{2}-5\times2.5 + 1=4\times6.25-12.5 + 1=25-12.5 + 1=13.5$.
The average rate of change is $\frac{g(2.5)-g(2)}{2.5 - 2}=\frac{13.5 - 7}{0.5}=\frac{6.5}{0.5}=13$.

Step5: Calculate $g(a)$ and $g(b)$ for part d) ($a = 2$, $b = 2.25$)

$g(2)=7$.
$g(2.25)=4\times(2.25)^{2}-5\times2.25 + 1=4\times5.0625-11.25 + 1=20.25-11.25 + 1=10$.
The average rate of change is $\frac{g(2.25)-g(2)}{2.25 - 2}=\frac{10 - 7}{0.25}=\frac{3}{0.25}=12$.

Step6: Calculate $g(a)$ and $g(b)$ for part e) ($a = 2$, $b = 2.1$)

$g(2)=7$.
$g(2.1)=4\times(2.1)^{2}-5\times2.1 + 1=4\times4.41-10.5 + 1=17.64-10.5 + 1=8.14$.
The average rate of change is $\frac{g(2.1)-g(2)}{2.1 - 2}=\frac{8.14 - 7}{0.1}=\frac{1.14}{0.1}=11.4$.

Step7: Calculate $g(a)$ and $g(b)$ for part f) ($a = 2$, $b = 2.01$)

$g(2)=7$.
$g(2.01)=4\times(2.01)^{2}-5\times2.01 + 1=4\times4.0401-10.05+1=16.1604 - 10.05 + 1=7.1104$.
The average rate of change is $\frac{g(2.01)-g(2)}{2.01 - 2}=\frac{7.1104 - 7}{0.01}=\frac{0.1104}{0.01}=11.04$.

Answer:

a) 19
b) 15
c) 13
d) 12
e) 11.4
f) 11.04