Instantenous Rate of Change
The instantaneous rate of change, also known as the derivative, measures how a function’s output (“dependent variable”) changes with respect to the input (“independent variable”) at a specific point
The instantaneous rate of change, also known as the derivative, measures how a function’s output (“dependent variable”) changes with respect to the input (“independent variable”) at a specific point. It calculates the slope of the function at that particular point.
To find the instantaneous rate of change of a function at a specific point, you need to compute the derivative of the function.
Here’s a step-by-step explanation of how to find the instantaneous rate of change:
1. Start with a function f(x) that represents the relationship between two variables, typically denoted as x and y.
2. Use the limit definition of the derivative to find the rate of change at a specific point. The formula for the derivative is:
f'(x) = lim(h->0) [f(x + h) – f(x)] / h
where h is a small change in the x-value.
3. Simplify the formula by substituting the specific value of x into the equation.
For example, if you want to find the instantaneous rate of change at x = 2, substitute 2 for x:
f'(2) = lim(h->0) [f(2 + h) – f(2)] / h
4. Evaluate the limit by taking smaller and smaller values of h. This will help approach the instantaneous rate of change at the desired point.
5. Solve the limit expression to find the derivative, which represents the instantaneous rate of change.
Once you obtain the derivative as a function, you can also evaluate it at specific points to find the instantaneous rate of change at those points.
It’s important to note that the instantaneous rate of change can vary at different points on the function. It provides information about how the function behaves locally at a specific point. Additionally, the derivative can be interpreted as the slope of the tangent line to the function at that point.
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