Instantaneous Rate of Change at x=a
The instantaneous rate of change of a function is the rate at which the function is changing at a specific point, in this case, when x = a
The instantaneous rate of change of a function is the rate at which the function is changing at a specific point, in this case, when x = a. To find the instantaneous rate of change at a specific point, we need to take the derivative of the function and evaluate it at that point.
Let’s say we have a function f(x), and we want to find the instantaneous rate of change at x = a. We can denote the instantaneous rate of change as f'(a) or dy/dx evaluated at x = a.
To find the instantaneous rate of change at x = a, follow these steps:
1. Differentiate the function f(x) with respect to x. This means finding the derivative of the function. The derivative, denoted f'(x) or dy/dx, represents the rate at which the function is changing at any given x.
2. Once you have the derivative function, substitute x = a into f'(x) or dy/dx. This will give you the instantaneous rate of change at x = a.
Let’s go through an example to illustrate this process:
Example:
Find the instantaneous rate of change of the function f(x) = 3x^2 + 2x – 1 at x = 2.
1. Differentiate the function f(x) with respect to x:
f'(x) = 6x + 2
2. Substitute x = 2 into f'(x) to find the instantaneous rate of change at x = 2:
f'(2) = 6(2) + 2 = 12 + 2 = 14
Therefore, the instantaneous rate of change of the function f(x) = 3x^2 + 2x – 1 at x = 2 is 14. This means that at x = 2, the function is changing at a rate of 14 units per unit change in x.
Remember that the instantaneous rate of change gives you the slope of the tangent line at a specific point on the graph of the function.
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