Find the instantaneous rate of change for the function at the given value.
[tex]f(x)=x^2+2 x \text { at } x=2[/tex]
The instantaneous rate of change at [tex]x=2[/tex] is $\square$
Answer (1)
Find the derivative of the function f ( x ) = x 2 + 2 x , which is f ′ ( x ) = 2 x + 2 .
Evaluate the derivative at x = 2 : f ′ ( 2 ) = 2 ( 2 ) + 2 .
Simplify the expression: f ′ ( 2 ) = 4 + 2 = 6 .
The instantaneous rate of change at x = 2 is 6 .
Explanation
Problem Analysis We are given the function f ( x ) = x 2 + 2 x and we want to find the instantaneous rate of change at x = 2 . 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 ) = x 2 + 2 x . Using the power rule, we have:
d x d ( x 2 ) = 2 x
d x d ( 2 x ) = 2
So, the derivative of f ( x ) is:
f ′ ( x ) = 2 x + 2
Evaluating the Derivative Now, we need to evaluate the derivative at x = 2 :
f ′ ( 2 ) = 2 ( 2 ) + 2 = 4 + 2 = 6
Final Answer Therefore, the instantaneous rate of change of the function f ( x ) = x 2 + 2 x at x = 2 is 6.
Examples
Imagine you are driving a car, and your distance from the starting point is given by the function f ( x ) = x 2 + 2 x , where x is the time in seconds. The instantaneous rate of change at x = 2 seconds tells you how fast your car is moving at that exact moment. In this case, the car is moving at 6 units of distance per second at x = 2 seconds. This concept is useful in physics, engineering, and economics to analyze how quantities change over time or with respect to other variables.
