# Question about a Rate of Change Problem

I am currently studying for a calculus test, and came across a tricky problem in the study guide. I double and triple checked my math, but my answer is not the same as in the study guide answer key. My father recommended I post here for help. The problem is as follows:

A men's suit manufacturer finds that the cost, in dollars, of producing x suits is given by C(x) = 841 + 18√(x) Find the rate at which the average cost is changing when 500 suits have been produced. Round the answer to four decimal places.

My logic that I have repeatedly checked is as follows:

C(x) = 841 + 18√(x) = 18x^(1/2) + 841
C(x) = 9x^(-1/2)
C(x) = (-9/2)x^(-3/2) = -9/(2x√(x))
C(500) = -0.000402492  ≈  -$0.0004/suit The answer that was in the back of the study guide was -$0.0042/suit

I also tried finding C(500) but am fairly certain that it is not the answer as it gave me a positive answer, and also due to the wording of the problem "rate at which the average cost is changing"

Was this possibly a rounding error on the end of the author of the problem, or am I missing something entirely? Thanks

Note: I just posted this over on math overflow, but was informed in no uncertain terms that this is the forum I need to post on. I have since removed the original post on overflow.

• The average cost is the total cost divided by the number of items $\bar{C}(x) = \frac{C(x)}{x}$, you need the derivative of that – Triatticus Feb 2 '18 at 6:59

The total cost of $x$ suits is as given by the formula, but you need to work out the average cost per suit. Let's call that $A(x)$.
$A(x) = \frac{C(x)}{x} = 841x^{-1} + 18x^{-\frac 12}$.
$A'(x) = -841x^{-2} - \frac 12(18)x^{-\frac 32} = -841x^{-2} - 9x^{-\frac 32}$
$A'(500)= -(841)500^{-2} - 9(500)^{-\frac 32} \approx -0.0042$, which is the required solution.