Calculating Instantaneous Rate of Change: Finding the Slope of a Function at a Specific Point

Instantaneous Rate of Change at x=a

The instantaneous rate of change of a function at a particular point, denoted as x=a, represents the slope of the function at that point

The instantaneous rate of change of a function at a particular point, denoted as x=a, represents the slope of the function at that point. It tells us how much the function is changing per unit change in x when x=a.

To calculate the instantaneous rate of change at x=a, we need to find the derivative of the function and then evaluate it at x=a.

Let’s say we have a function f(x). The derivative of f(x), denoted as f'(x) or dy/dx, represents the rate of change of the function at any given point.

To find the instantaneous rate of change at x=a, follow these steps:

1. Find the derivative of the function f'(x).
2. Substitute x=a into the derivative obtained in step 1.
3. Evaluate the derivative at x=a to find the instantaneous rate of change.

For example, let’s say we have the function f(x) = 3x^2 + 2x – 1 and we want to find the instantaneous rate of change at x=2.

1. Find the derivative of f(x):
f'(x) = d/dx(3x^2 + 2x – 1)
= 6x + 2

2. Substitute x=2 into the derivative obtained in step 1:
f'(2) = 6(2) + 2
= 12 + 2
= 14

3. The instantaneous rate of change at x=2 is 14. This means that when x=2, the function is changing at a rate of 14 units per unit change in x.

So, the instantaneous rate of change at x=2 for the function f(x) = 3x^2 + 2x – 1 is 14.

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