Understanding the Instantaneous Rate of Change in Calculus | Definition, Calculation, and Examples

Instantaneous Rate of Change at x=a

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

The instantaneous rate of change at x = a represents the rate at which a function is changing at a specific point, a, on its graph. It is also known as the derivative of the function at that point.

To calculate the instantaneous rate of change at x = a, we need to find the slope of the tangent line to the graph of the function at that point. This can be done using calculus.

The derivative of a function f(x) with respect to x is denoted by f'(x) or dy/dx. It represents the rate of change of the function with respect to x. So, to find the instantaneous rate of change at x = a, we evaluate the derivative at that point, f'(a).

In mathematical notation, the instantaneous rate of change at x = a is given by f'(a) or dy/dx|_(x=a).

For example, let’s say we have a function f(x) = x^2. To find the instantaneous rate of change at x = 3, we take the derivative of f(x) with respect to x:

f'(x) = 2x

Now, substitute x = 3 into the derivative:

f'(3) = 2(3) = 6

The instantaneous rate of change at x = 3 for the function f(x) = x^2 is 6. This means that at x = 3, the function is changing at a rate of 6 units for every unit change in x.

In summary, the instantaneous rate of change at x = a represents the rate at which a function is changing at that specific point. It is calculated by finding the derivative of the function and evaluating it at x = a.

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