Mathematics 1210
Autumn 2006
Instructor: Mike Wills
Practice Midterm 3 Solutions
Problem #1 (5 points):
Let f(x) = 2 cos xsin x. Find the indefinite integral of f.
Solution
By the half angle formula,
(1) Z2 cos xsin xdx =Zsin(2x)dx =−cos(2x)
2+C
where Cis an arbitrary constant.
Problem #2 (5 points):
It costs Cut-Me-Own-Throat Dibbler 4 + 3x+x2Ankh-Morpork
dollars to procure 100xsausage-inna-buns. Find Dibbler’s average cost
and marginal cost. What procurement level will minimize the average
cost?
Let C(x) = 4 + 3x+x2. Then the average cost is given by A(x) =
C(x)
x=4
x+ 3 + x. The marginal cost will be given by c(x) = C0(x) =
3+2x. The minimum cost occurs when
c(x) = A(x)
3+2x=4
x+3+x
x=4
x
x2= 4.
(2)
Since x≥0 we conclude that x= 2. Thus, when Dibbler procures 200
sausage-inna-buns, his cost is minimized.
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