Question

The mathematics editor at a major publishing house, estimates that if x thousand complimentary copies are distributed to professors, the first-year sales of a certain new text will be f(x)=20- 15e^ -0.2x thousand copies. Currently, the editor is planning to distribute 10,000 complimentary copies.

a. Use marginal analysis to estimate the increase in first-year sales that Jane should expect if 1,000 additional complimentary copies are distributed. (Hint: find f'(x)
b. Calculate the actual increase first-year sales that will result from the distribution of the additional 1,000 complimentary copies. Is the estimate in part (a) a good one?

Answer 1

The estimate in part (a) using marginal analysis is approximately 1.64 thousand copies, and the actual increase in first-year sales is also approximately 1.64 thousand copies.

Step-by-step explanation:

a. To estimate the increase in first-year sales if 1,000 additional complimentary copies are distributed, we need to find the derivative of the sales function. The derivative of f(x) = 20 - 15e^(-0.2x) is f'(x) = 3e^(-0.2x).
By plugging in x=10 (since 10,000 complimentary copies have already been distributed) into the derivative, we can find the rate of increase: f'(10) ≈ 3e^(-0.2(10)) ≈ 1.64 thousand copies.
So, Jane could expect an increase in first-year sales of approximately 1.64 thousand copies if 1,000 additional complimentary copies are distributed.

b. To calculate the actual increase in first-year sales, we need to find the value of f(11) - f(10) (since 11,000 complimentary copies will be distributed after the addition).
Using the given function f(x) = 20 - 15e^(-0.2x), we have:
f(11) = 20 - 15e^(-0.2(11)) ≈ 19.69 thousand copies
f(10) = 20 - 15e^(-0.2(10)) ≈ 18.05 thousand copies
The actual increase in first-year sales is approximately 19.69 - 18.05 = 1.64 thousand copies, which matches the estimate from part (a). Therefore, the estimate in part (a) is a good one.