Find the limit as x approaches c of the average rate of change of the function from c to x.
-$$c = 3 ; \quad f ( x ) = 3 x ^ { 2 } + 39 x$$
A) 108
B) 57
C) 30
D) 90

Question

Quizplus · Accepted Answer

The average rate of change of a function $f(x)$ from $c$ to $x$ is given by the formula $\frac{f(x) - f(c)}{x - c}$. For the given function $f(x) = 3x^2 + 39x$ and $c = 3$, we calculate the average rate of change as follows:$$f(c) = 3(3)^2 + 39(3) = 27 + 117 = 144$$Thus, the average rate of change from $c$ to $x$ is:$$\frac{f(x) - f(c)}{x - c} = \frac{3x^2 + 39x - 144}{x - 3}$$Factoring the numerator:$$= \frac{3(x^2 + 13x - 48)}{x - 3}$$$$= \frac{3(x + 16)(x - 3)}{x - 3}$$After canceling $x - 3$, we get:$$= 3(x + 16)$$Substituting $x = 3$, we find the limit as $x$ approaches $c$:$$= 3(3 + 16) = 3(19) = 57$$Therefore, the correct answer is 57.

Find the Limit as X Approaches C of the Average c=3;f(x)=3x2+39xc = 3 ; \quad f ( x ) = 3 x ^ { 2 } + 39 xc=3;f(x)=3x2+39x

Find the Limit as X Approaches C of the Average $c = 3 ; \quad f ( x ) = 3 x ^ { 2 } + 39 x$