Question

suppose that ( f(x) = 3x^2 ). (a) what is the average rate of change of ( f(x) ) over each of the following intervals: 3 to 4, 3 to 3.5, 3 and to 3.1? (b) what is the (instantaneous) rate of change of ( f(x) ) when ( x = 3 )? (a) the average rate of change of ( f(x) ) over the interval 3 to 4 is \\(\square\\). (simplify your answer.)

Explanation:

Step1: Recall the formula for average rate of change

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, \( a = 3 \) and \( b = 4 \), and \( f(x)=3x^{2}\).

Step2: Calculate \( f(4) \) and \( f(3) \)

First, find \( f(4) \):
\( f(4)=3\times(4)^{2}=3\times16 = 48 \)
Then, find \( f(3) \):
\( f(3)=3\times(3)^{2}=3\times9 = 27 \)

Step3: Apply the average rate of change formula

Using the formula \(\frac{f(b)-f(a)}{b - a}\) with \( a = 3 \), \( b = 4 \), \( f(4)=48 \) and \( f(3)=27 \):
\(\frac{f(4)-f(3)}{4 - 3}=\frac{48 - 27}{1}=\frac{21}{1}=21\)

Answer:

21