Given that the average rate of change for y = f(x) over the interval [0,3] is =-1, the...

Question:

Given that the average rate of change for {eq}y = f(x) {/eq} over the interval {eq}[0,3] {/eq} is =-1, the average rate of change over the interval {eq}[2,3] {/eq} is 5, and the average rate of change over the interval {eq}[2,6] {/eq} is 3, determine the average rate of change over the interval {eq}[0,6] {/eq}.

Average Rate of Change:

Let {eq}f(x) {/eq} be defined on the interval {eq}[a,b] {/eq}. The average rate of change on the interval is

{eq}\begin{align} f_{avg}=\frac{f(b)-f(a)}{b-a} \end{align} {/eq}.

If the function is continuous on {eq}[a,b] {/eq} and differentiable on {eq}(a,b) {/eq}, then the mean value theorem states that there is a {eq}c {/eq} in {eq}(a,b) {/eq} such that {eq}f'(c)=f_{avg}. {/eq}

Answer and Explanation: 1

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We are given the average rate of change on {eq}[0,3] {/eq} is -1. This can be stated as

{eq}-1=\frac{f(3)-f(0)}{3-0}\\ f(3)-f(0)=-3 {/eq}

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Average Rate of Change: Definition, Formula & Examples

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Chapter 20 / Lesson 5
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Finding the average rate of change is similar to finding the slope of a line. Study the definition of average rate of change, its formula, and examples of this concept.


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