Find the average rate of change of the function over the given interval.
-$$g ( t ) = 4 + \tan t , \left[ - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right]$$
A) $- \frac { 4 } { \pi }$
B) $\frac { 4 } { \pi }$
C) $- \frac { 3 } { 2 }$
D) 0

Question

Quizplus · Accepted Answer

The average rate of change of a function over an interval is given by the slope of the secant line connecting the endpoints of the interval. The slope of the secant line is given by the formula:$$m = \frac{f(b) - f(a)}{b - a}$$where $f(a)$ and $f(b)$ are the values of the function at the endpoints of the interval and $a$ and $b$ are the endpoints of the interval.In this case, we have $f(t) = 4 + \tan t$, $a = -\frac{\pi}{4}$, and $b = \frac{\pi}{4}$. So, the average rate of change of the function over the interval is:$$m = \frac{f\left(\frac{\pi}{4}\right) - f\left(-\frac{\pi}{4}\right)}{\frac{\pi}{4} - \left(-\frac{\pi}{4}\right)}$$$$= \frac{\left(4 + \tan \frac{\pi}{4}\right) - \left(4 + \tan \left(-\frac{\pi}{4}\right)\right)}{\frac{\pi}{4} + \frac{\pi}{4}}$$$$= \frac{\left(4 + 1\right) - \left(4 - 1\right)}{\frac{\pi}{2}}$$$$= \frac{4}{\frac{\pi}{2}}$$$$= \frac{8}{\pi}$$Therefore, the average rate of change of the function over the interval is $\frac{8}{\pi}$.

Find the Average Rate of Change of the Function Over g(t)=4+tan⁡t,[−π4,π4]g ( t ) = 4 + \tan t , \left[ - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right]g(t)=4+tant,[−4π​,4π​]

Find the Average Rate of Change of the Function Over $g ( t ) = 4 + \tan t , \left[ - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right]$