What is the rate of change of $f ( t ) = \frac { 7 t - 5 } { t + 2 }$ with respect to t when t = 2?
A) $\frac { 9 } { 4 }$
B)4
C) $\frac { 19 } { 16 }$
D) $\frac { 17 } { 4 }$

Question

Quizplus · Accepted Answer

To find the rate of change of $f(t) = \frac{7t - 5}{t + 2}$ with respect to $t$ when $t = 2$, we need to compute the derivative of $f(t)$ with respect to $t$ and then substitute $t = 2$ into the derivative. The derivative of $f(t)$, $f'(t)$, can be found using the quotient rule: $$f'(t) = \frac{(t + 2)(7) - (7t - 5)(1)}{(t + 2)^2} = \frac{14 + 14 - 7t + 5}{(t + 2)^2} = \frac{19 - 7t}{(t + 2)^2}.$$ Substituting $t = 2$ into $f'(t)$ gives $$f'(2) = \frac{19 - 7(2)}{(2 + 2)^2} = \frac{19 - 14}{16} = \frac{5}{16}.$$ It appears there was a mistake in my calculation. Let's correct that:The correct derivative calculation should be: $$f'(t) = \frac{(7)(t + 2) - (7t - 5)}{(t + 2)^2} = \frac{7t + 14 - 7t + 5}{(t + 2)^2} = \frac{19}{(t + 2)^2}.$$ Substituting $t = 2$ into the corrected $f'(t)$ gives $$f'(2) = \frac{19}{(2 + 2)^2} = \frac{19}{16}.$$ Therefore, the correct rate of change when $t = 2$ is $\frac{19}{16}$.

What Is the Rate of Change Of f(t)=7t−5t+2f ( t ) = \frac { 7 t - 5 } { t + 2 }f(t)=t+27t−5​

What Is the Rate of Change Of $f ( t ) = \frac { 7 t - 5 } { t + 2 }$