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Consider a function y = f(t) for the items below: (a)Define the average rate of change of f...

Question:

Consider a function {eq}y = f(t) {/eq} for the items below:

(a) Define the average rate of change of {eq}f {/eq} between {eq}t = a {/eq} and {eq}t = b. {/eq}

(b) Define the instantaneous rate of change of {eq}f {/eq} at {eq}t = a. {/eq}

Answer and Explanation:

a) The average rate of change is the change in the function value relative to the change in t. It also can be thought of the slope between two points. Thus, if we want to find the average rate of change between two points a and b, we use the following ratio.

Average rate of change = {eq}\frac{f(b) - f(a)}{b - a} {/eq}

b) The instantaneous rate of change is the change in the function at a single moment. Thus, we aren't comparing one point to another. In order to compute this, we need to use the derivative. If we don't know how to take the derivative, we can use the following limit to define it.

Instantaneous rate of change = {eq}\begin{align*} f'(a) = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h} \end{align*} {/eq}


Learn more about this topic:

Average and Instantaneous Rates of Change

from PLACE Mathematics: Practice & Study Guide

Chapter 26 / Lesson 2
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