- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Differentiation rules practice question 16
A print shop’s daily cost of printing posters is dollars, given below. The average cost per poster is . Find .
is over a single power of . Is there a way to rewrite it first so that no quotient rule is needed?
Divide first: . Each term is now a constant times a power of .
The derivative of is . Substitute at the end.
A number.
Theory UBC Math 100 §4.1CLP-1 §2.6Key results
Worked solution
Strategy
Write the average cost as a sum of powers of q, differentiate term by term, then substitute. The quotient rule on C(q)/q gives the same result.
Working
Divide each term of C by q.
Power rule; the constant 20 drops out.
Negative: near 100 posters, each extra poster lowers the average cost by about 45 cents.
Answer
The answer is negative 0.45: the average cost falls by about 45 cents per poster for each extra poster.
Check
The marginal cost C′(100) = 30 is below the average cost A(100) = 75, so the average must be falling, and (30 − 75)/100 = −0.45.
Where this goes wrong
Answering the marginal cost C′(100) = 30. That is roughly the cost of one more poster; the question asks how the cost per poster changes, which is the derivative of C(q)/q.
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