Find the instantaneous rate of change of the function [tex]f(x)=\sqrt{7 x}[/tex] when [tex]x=7[/tex].
Answer (1)
Find the derivative of f ( x ) = 7 x , which is f ′ ( x ) = 2 7 x 7 .
Evaluate the derivative at x = 7 : f ′ ( 7 ) = 2 7 × 7 7 .
Simplify the expression: f ′ ( 7 ) = 2 × 7 7 = 2 1 .
The instantaneous rate of change of the function at x = 7 is 2 1 .
Explanation
Problem Analysis We are asked to find the instantaneous rate of change of the function f ( x ) = 7 x when x = 7 . The instantaneous rate of change is given by the derivative of the function evaluated at the given point.
Finding the Derivative First, we need to find the derivative of the function f ( x ) = 7 x . We can rewrite this as f ( x ) = ( 7 x ) 2 1 . Using the power rule and the chain rule, we have:
f ′ ( x ) = 2 1 ( 7 x ) − 2 1 × 7 = 2 7 x 7
Evaluating the Derivative Now, we need to evaluate the derivative at x = 7 :
f ′ ( 7 ) = 2 7 × 7 7 = 2 49 7 = 2 × 7 7 = 2 1
Final Answer Therefore, the instantaneous rate of change of the function f ( x ) = 7 x at x = 7 is 2 1 .
Examples
In physics, if f ( x ) represents the position of an object at time x , then the instantaneous rate of change f ′ ( x ) represents the object's velocity at time x . For example, if f ( x ) = 7 x describes the position of a particle, finding f ′ ( 7 ) = 2 1 tells us the particle's velocity at time x = 7 . This concept is crucial in understanding motion and dynamics.
