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 ( 21 ).
(simplify your answer.)

(a) the average rate of change of ( f(x) ) over the interval 3 to 3.5 is ( 19.5 ).
(simplify your answer.)

(a) the average rate of change of ( f(x) ) over the interval 3 to 3.1 (square).
(simplify your answer.)

Explanation:

Part (a) - Interval 3 to 3.1

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, \( a = 3 \) and \( b = 3.1 \), and \( f(x)=3x^{2} \).

Step2: Calculate \( f(3) \) and \( f(3.1) \)

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

Next, find \( f(3.1) \):
\( f(3.1)=3\times(3.1)^{2}=3\times9.61 = 28.83 \)

Step3: Apply the average rate of change formula

Substitute \( f(3) = 27 \), \( f(3.1)=28.83 \), \( a = 3 \), and \( b = 3.1 \) into the formula:
\(\frac{f(3.1)-f(3)}{3.1 - 3}=\frac{28.83 - 27}{0.1}=\frac{1.83}{0.1}=18.3\)

Part (b) - Instantaneous rate of change at \( x = 3 \)

Step1: Recall the derivative formula for power functions

The derivative of \( f(x)=ax^{n} \) is \( f^{\prime}(x)=nax^{n - 1} \). For \( f(x)=3x^{2} \), \( a = 3 \) and \( n = 2 \).

Step2: Find the derivative of \( f(x) \)

Using the power rule, \( f^{\prime}(x)=2\times3x^{2 - 1}=6x \)

Step3: Evaluate the derivative at \( x = 3 \)

Substitute \( x = 3 \) into \( f^{\prime}(x) \):
\( f^{\prime}(3)=6\times3 = 18 \)

Final Answers

(a) The average rate of change over 3 to 3.1 is \(\boldsymbol{18.3}\)

(b) The instantaneous rate of change at \( x = 3 \) is \(\boldsymbol{18}\)

Answer:

Step1: Recall the derivative formula for power functions

The derivative of \( f(x)=ax^{n} \) is \( f^{\prime}(x)=nax^{n - 1} \). For \( f(x)=3x^{2} \), \( a = 3 \) and \( n = 2 \).

Step2: Find the derivative of \( f(x) \)

Using the power rule, \( f^{\prime}(x)=2\times3x^{2 - 1}=6x \)

Step3: Evaluate the derivative at \( x = 3 \)

Substitute \( x = 3 \) into \( f^{\prime}(x) \):
\( f^{\prime}(3)=6\times3 = 18 \)

Final Answers

(a) The average rate of change over 3 to 3.1 is \(\boldsymbol{18.3}\)

(b) The instantaneous rate of change at \( x = 3 \) is \(\boldsymbol{18}\)