Video: Finding the Average Rate of Change of a Polynomial Function between Two Points

The distance traveled by a body in 𝑑 seconds is 𝑆 = 5𝑑² + 3𝑑 + 7. What is the average rate of change of 𝑆 when 𝑑 changes from 9 to 13 seconds?

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Video Transcript

The distance traveled by a body in 𝑑 seconds is 𝑆 is equal to five 𝑑 squared plus three 𝑑 plus seven. What is the average rate of change of 𝑆 when 𝑑 changes from nine to 13 seconds?

We’re given the distance traveled 𝑆, which is a function of time 𝑑, which is measured in seconds. And we’re asked for the average rate of change of 𝑆 when 𝑑 changes from nine to 13 seconds. Now, for a function 𝑓 of π‘₯ between π‘₯ is π‘Ž and π‘₯ is 𝑏, the average rate of change of 𝑓 is 𝑓 of 𝑏 minus 𝑓 of π‘Ž over 𝑏 minus π‘Ž. That is, the change in 𝑓 over the change in π‘₯. And this is effectively a version of the slope equation. Remember, the slope is 𝑦 two minus 𝑦 one over π‘₯ two minus π‘₯ one.

In our case, instead of 𝑓, which is a function of π‘₯, we have 𝑆, the distance, which is a function of time 𝑑. So that our average rate of change is the change in 𝑆, which is 𝑆 of 𝑏 minus 𝑆 of π‘Ž, over the change in 𝑑, which is 𝑏 minus π‘Ž, where π‘Ž is nine seconds and 𝑏 is 13 seconds. So that our average rate of change is 𝑆 evaluated at 𝑑 is equal to 13 minus 𝑆 evaluated at 𝑑 is equal to nine over 13 minus nine. We know that 13 minus nine is equal to four. And that’s our denominator.

So now let’s work out 𝑆 evaluated at 𝑑 is equal to 13 and 𝑆 evaluated at 𝑑 is equal to nine. And substituting 𝑑 is equal to 13 into our equation for 𝑆, we have 𝑆 at 𝑑 is equal to 13 is five times 13 squared plus three times 13 plus seven. That is, five times 169 plus 39 plus seven, which evaluates to 891. And next evaluating 𝑆 at 𝑑 is equal to nine, we have five times nine squared plus three times nine plus seven, which is five times 81 plus 27 plus seven. So that 𝑆 at 𝑑 is equal to nine is 439.

Our average rate of change is therefore 891, which is 𝑆 at 𝑑 is 13, minus 439, which is 𝑆 at 𝑑 is equal to nine, divided by four. Evaluating this, we have 452 divided by four, and that’s equal to 113.

If the distance traveled by a body in 𝑑 seconds is 𝑆 is equal to five 𝑑 squared plus three 𝑑 plus seven, then the average rate of change of 𝑆 when 𝑑 changes from nine to 13 seconds is 113. It’s worth noting here that, in general, we define speed as the rate of change of distance per unit time. So we might say here that the average speed the body traveled at between nine and 13 seconds is 113 distance units per second.

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