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

Instantaneous Rate of Change at x=a

The instantaneous rate of change at x = a measures the rate at which a function is changing at a specific point on its graph

The instantaneous rate of change at x = a measures the rate at which a function is changing at a specific point on its graph. It indicates the slope of the tangent line to the graph at that particular point.

To calculate the instantaneous rate of change at x = a, we can use the concept of the derivative. The derivative of a function represents the rate of change of the function with respect to its independent variable.

Let’s assume we have a function f(x) and we want to find the instantaneous rate of change at x = a. Here’s the step-by-step procedure:

1. Find the derivative of the function f(x) with respect to x. This can be done by using differentiation rules or techniques such as the power rule, product rule, chain rule, etc. The derivative of f(x) will represent the rate of change of the function.

2. Once you have the derivative, denote it as f'(x), where the ‘ symbol indicates differentiation.

3. Substitute x = a into the derivative f'(x) to find the value of the derivative at x = a. This gives us f'(a).

The value of f'(a) represents the instantaneous rate of change of the function at x = a. It tells us how much the function is changing at that specific point.

For example, let’s say we have the function f(x) = x^2. To find the instantaneous rate of change at x = 3, we can follow the steps above:

1. Take the derivative of f(x). Applying the power rule, we get f'(x) = 2x.

2. Substitute x = 3 into the derivative. f'(3) = 2(3) = 6.

Therefore, the instantaneous rate of change of the function f(x) = x^2 at x = 3 is 6.

By finding the instantaneous rate of change, we can better understand the behavior of a function at a specific point and make predictions about its behavior nearby.

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