Example 1: Composition Rates

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Problem. A student writing a term paper notices that she completed her  10th  page at  1 AM  and her  22nd  page at  4 AM. At what rate has she been writing?

Solution. The underlying function here is  P = P(t), giving the number of pages written  t  hours after, say, midnight. We know two values of the function, at two specific times:  P(1) = 10  pages and  P(4) = 22  pages.

We can find the change in value over the three hour period by calculating that  DP = 22 – 10 = 12  pages.

Finally, we can find the rate at which the student has been writing by calculating  DP/Dt = (22 – 10)/(4 – 1) = 12/3 = 4  pages per hour. We might expect that this rate would decrease as the morning wears on, due a decreased production  DP  over the next  Dt = three hours.