Download ICNS 103 Final Exam Practice 1. If f(x) = x3 ln(2x

ICNS 103 Final Exam Practice
1. If f (x) = x3 ln(2x − 1), find f ′ (1) + f ′′ (1).
2. If f (x, y) =
4xy
, find fx (1, −2).
x2 y + 1
3. A company’s production function is given by P = 15ℓk − 3ℓ2 + 5k 2 + 500, where P is the
total output generated by ℓ units of labor and k units of capital. Determine the marginal
production functions.
4. Sketch the graph of the function y = x(x − 2)3 noting all relevant properties (intercepts,
relative extrema, inflection points).
5. Find all absolute extrema of the function
f (x) =
x−2
x2 + 5
on the closed interval [−2, 3].
6. During the Christmas season, a promotional company purchases cheap red felt stockings,
glues fake white fur and sequins onto them, and packages them for distribution. The total
cost of producing q cases of stockings is given by
c = 3q 2 + 50q − 18q ln q + 120.
Find the number of cases that should be processed in order to minimize the average cost
per case. Determine (to two decimal places) this minimum average cost.
7. Evaluate the integral
∫
(1 − ex )2
1
+ √ dx.
e2x
4x
8. The sole producer of a product has determined that the marginal-revenue function is
dr
= 100 − 3q 2 .
dq
Determine the demand function.
9. Evaluate the integral
∫
0
2
(ln(4x + 1))3
dx.
8x + 2
1
10. Evaluate the integral
∫
x5 (x2 + 1)3 dx.
11. Find the area of the region bounded between the curves y = (x − 2)3 and y = x − 2.
12. The demand equation for a product is p = (q −4)2 and the supply equation is p = q 2 +q +7
where p (in thousands of dollars) is the price per 100 units when q hundred units are
demanded or supplied. Determine consumers’ surplus and producers’ surplus under market
equilibrium.
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