# How do you estimate the instantaneous rate of change?

The average rate of change of a function f(x) at point x=a is defined as

The instantaneous rate of change can then be estimated by computing the values of Δf/Δx, which we can then estimate by choosing values of x that are getting closer and closer to x=a (don't forget to chose them on both sides of x=a).

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To estimate the instantaneous rate of change, you can use the concept of calculus, specifically the derivative. The instantaneous rate of change of a function at a specific point is the slope of the tangent line to the curve at that point. Mathematically, it is represented by the derivative of the function with respect to the independent variable.

One common method to estimate the instantaneous rate of change is to calculate the average rate of change over smaller and smaller intervals around the point of interest. As the interval approaches zero, the average rate of change becomes closer to the instantaneous rate of change. This process is known as taking the limit, and it is the foundation of calculus.

Alternatively, if you have a function represented by a mathematical expression, you can use differentiation techniques to find the derivative analytically. Once you have the derivative, you can evaluate it at the specific point of interest to obtain the instantaneous rate of change.

In summary, to estimate the instantaneous rate of change, you can either approach it through the concept of limits by calculating the average rate of change over smaller intervals, or you can find the derivative of the function and evaluate it at the specific point.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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