Question

determine the average rate of change over the given interval.
a. ( f(x) = 3x^2 - 3x + 1; -5, 5 )

-3

b. ( g(x) = 2x^2; -3, 3 )

0

c. (-3, 3)

x	f(x)
-3	0
-2	3
-1	-4
0	-3
1	0
2	5
3	12

type your answer...
d. interval (-2, 2)

(graph with points (-2,12), (6,12), (2,-1))

type your answer...

Explanation:

Part a:

Step1: Recall the average rate of change formula

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). Here, \( a=-5 \), \( b = 5 \) and \( f(x)=3x^{2}-3x + 1 \).

Step2: Calculate \( f(5) \)

Substitute \( x = 5 \) into \( f(x) \):
\( f(5)=3(5)^{2}-3(5)+1=3\times25-15 + 1=75-15 + 1=61 \)

Step3: Calculate \( f(-5) \)

Substitute \( x=-5 \) into \( f(x) \):
\( f(-5)=3(-5)^{2}-3(-5)+1=3\times25 + 15+1=75 + 15+1=91 \)

Step4: Apply the average rate of change formula

\(\frac{f(5)-f(-5)}{5-(-5)}=\frac{61 - 91}{5 + 5}=\frac{-30}{10}=- 3\)

Part b:

Step1: Recall the average rate of change formula

The average rate of change of a function \( g(x) \) over the interval \([a, b]\) is \(\frac{g(b)-g(a)}{b - a}\). Here, \( a=-3 \), \( b = 3 \) and \( g(x)=2x^{2} \).

Step2: Calculate \( g(3) \)

Substitute \( x = 3 \) into \( g(x) \):
\( g(3)=2(3)^{2}=2\times9 = 18 \)

Step3: Calculate \( g(-3) \)

Substitute \( x=-3 \) into \( g(x) \):
\( g(-3)=2(-3)^{2}=2\times9=18 \)

Step4: Apply the average rate of change formula

\(\frac{g(3)-g(-3)}{3-(-3)}=\frac{18 - 18}{3 + 3}=\frac{0}{6}=0\)

Part c:

Step1: Recall the average rate of change formula

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is \(\frac{f(b)-f(a)}{b - a}\). From the table, when \( x=-3 \) (i.e., \( a = - 3 \)), \( f(-3)=0 \) and when \( x = 3 \) (i.e., \( b=3 \)), \( f(3)=12 \).

Step2: Apply the average rate of change formula

\(\frac{f(3)-f(-3)}{3-(-3)}=\frac{12-0}{3 + 3}=\frac{12}{6}=2\)

Part d:

Answer:

s:
a. \(-3\)

b. \(0\)

c. \(2\)

d. \(-\frac{13}{4}\) (or \(-3.25\))