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

The rate of change of a function with respect to a variable is given by its derivative. To find the derivative of $f(t) = \frac{7t - 5}{t + 2}$, we can use the quotient rule, which states that if $f(t) = \frac{g(t)}{h(t)}$, then $f'(t) = \frac{g'(t)h(t) - g(t)h'(t)}{[h(t)]^2}$. Applying this to our function:Let $g(t) = 7t - 5$ and $h(t) = t + 2$, so $g'(t) = 7$ and $h'(t) = 1$.Thus, $f'(t) = \frac{(7)(t + 2) - (7t - 5)(1)}{(t + 2)^2} = \frac{7t + 14 - 7t + 5}{(t + 2)^2} = \frac{19}{(t + 2)^2}$.To find the rate of change when $t = 2$, substitute $t = 2$ into $f'(t)$:$f'(2) = \frac{19}{(2 + 2)^2} = \frac{19}{16}$.Therefore, the rate of change of $f(t)$ with respect to $t$ 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 $f ( t ) = \frac { 7 t - 5 } { t + 2 }$