Estimate the instantaneous rate of change of [tex]$P(x)=3 x^2+6$[/tex] at the point [tex]$x=-1$[/tex]
Answer (2)
Find the derivative of P ( x ) = 3 x 2 + 6 , which is P ′ ( x ) = 6 x .
Evaluate the derivative at x = − 1 : P ′ ( − 1 ) = 6 ( − 1 ) .
Calculate the result: P ′ ( − 1 ) = − 6 .
The instantaneous rate of change of P ( x ) at x = − 1 is − 6 .
Explanation
Problem Analysis We are given the function P ( x ) = 3 x 2 + 6 and asked to estimate the instantaneous rate of change at the point x = − 1 . The instantaneous rate of change is given by the derivative of the function evaluated at that point.
Finding the Derivative First, we need to find the derivative of P ( x ) with respect to x . Using the power rule, we have:
d x d ( 3 x 2 + 6 ) = 3 ⋅ 2 x + 0 = 6 x
Evaluating the Derivative Now, we evaluate the derivative at x = − 1 :
P ′ ( − 1 ) = 6 ( − 1 ) = − 6
Final Answer Therefore, the instantaneous rate of change of P ( x ) at x = − 1 is − 6 .
Examples
In physics, if P ( x ) represents the position of an object at time x , then the instantaneous rate of change at x = − 1 would represent the object's velocity at that time. For example, if P ( x ) is in meters and x is in seconds, then the instantaneous velocity at x = − 1 is -6 meters per second, indicating the object is moving in the negative direction.
The instantaneous rate of change of P ( x ) = 3 x 2 + 6 at x = − 1 is − 6 . This is determined by finding the derivative and evaluating it at the given point. The negative sign indicates that the function is decreasing at that point.
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