how to find the instantaneous rate of change
find first derivative, plug in given value to the first derivative
The instantaneous rate of change is the rate at which a function is changing at a specific point, or in other words, the slope of the tangent line of the function at that point. Here are the steps to find the instantaneous rate of change:
1. Identify the point at which you want to find the instantaneous rate of change.
2. Calculate the derivative of the function at that point using the appropriate differentiation rules.
3. Evaluate the derivative at the point to obtain the slope of the tangent line.
For example, suppose we want to find the instantaneous rate of change of the function f(x) = x^2 + 3x – 2 at the point x = 2. Here are the steps we would take:
1. The point we want to find the instantaneous rate of change is x = 2.
2. Calculate the derivative of the function: f'(x) = 2x + 3.
3. Evaluate the derivative at the point x = 2: f'(2) = 2(2) + 3 = 7.
Therefore, the instantaneous rate of change of the function f(x) = x^2 + 3x – 2 at the point x = 2 is 7. This means that the function is changing at a rate of 7 units per 1 unit of x at that point.
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