For $f ( x ) = - \frac { 3 } { \sqrt { x } }$ , find the average rate of change of f (x) with respect to x as x changes from 144 to 145. Then use calculus to find the instantaneous rate of change at x = 144. Round your answer to six decimal places, if necessary.
A) Average rate of change: 0.000864; Instantaneous rate of change: 0.000868
B)Average rate of change: -0.000864; Instantaneous rate of change: 0.125
C) Average rate of change: 0.000864; Instantaneous rate of change: -0.125
D) Average rate of change: -0.000864; Instantaneous rate of change: 0.000868

Question

Quizplus · Accepted Answer

The average rate of change is the slope of the secant line connecting the points (144, f(144)) and (145, f(145)). The slope of the secant line is:$$\frac{f(145) - f(144)}{145 - 144} = \frac{-\frac{3}{\sqrt{145}} - \left(-\frac{3}{\sqrt{144}}\right)}{1} = 0.000864$$The instantaneous rate of change is the slope of the tangent line to the graph of f(x) at x = 144. The slope of the tangent line is the derivative of f(x) at x = 144:$$f'(x) = \frac{d}{dx}\left(-\frac{3}{\sqrt{x}}\right) = \frac{3}{2x^{3/2}}$$$$f'(144) = \frac{3}{2(144)^{3/2}} = 0.000868$$

For f(x)=−3xf ( x ) = - \frac { 3 } { \sqrt { x } }f(x)=−x​3​

For $f ( x ) = - \frac { 3 } { \sqrt { x } }$