MATH 142 Business Math II, Week In Review
Spring, 2015, Problem Set 11 (Final Review)
JoungDong Kim
1. Locate and classify all the critical points of the function.
f (x, y) = 16x2 + 2y 2 − 2y 2x − 1
2. Find the second partial derivatives of the function.
f (x, y) = ex
3 +y 5
3. Determine the consumers’ surplus and the producers’ surplus at the equilibrium price level for
the following demand and supply functions.
p = D(x) = 53.4 − 0.8x
p = S(x) = 0.2x2 + 15
1
4. Determine the area that is bounded by the graphs of the following equations.
y = −2x2 ,
5. If f (1) = 14, f is continuous, and
′
Z
y = x3 − 24x
3
f ′ (x) dx = 18, what is the value of f (3)?
1
6. Find the average value of the function f on the given interval.
f (x) = 3e2x ,
2
[4, 6]
7. Use a right sum with rectangles of equal width for the given value of n to approximate the integral.
Z 7
x2 ln(x) dx, n = 2
1
8. Evaluate the integral.
Z
x5
dx
x6 + 6
9. A box with a square base and open top must have a volume of 4,000 cm3 . Find the dimensions
of the box that minimize the amount of material used.
3
10. Find the absolute maximum and absolute minimum values of f on the given interval.
f (x) = 6x2 − 72x + 1000,
[−1, 15)
11. Given function f (x) = (2 − x)e−x . Find the intervals of increase or decrease, and concavity. Find
the point of inflection.
12. Given function f (x) = 6x3 + 9x2 − 1620x. Find the x-coordinate(s) of all local extrema.
4
13. The consumer demand curve for Professor Stefan Schwarzenegger’s dumbbells is given by x =
(84 − 4p)2 , 0 < p < 21, where p is the price per dumbbell, and x is the demand in weekly sales.
Find the price Professor Schwarzenegger should charge for his dumbbells in order to maximize
revenue.
14. Find the derivative of f (x) = ln((2x + 1)3 (5x − 1)4 )
15. Determine the equation of the line tangent to the following function at x = 3.
f (x) = (x − 2)(5 − x)4
5
16. Consider the function below. Use it to find the following limits.



 29 − x2
if x ≤ 5
f (x) =


 x−5
if x > 5
(a) lim− f (x)
x→5
(b) lim+ f (x)
x→5
(c) lim f (x)
x→5
17. Solve for x:
6(1.084x+1 ) = 8
18. Solve for x:
2
3x ·
6
1
= 318
33x
19. Your college newspaper, The Collegiate Investigator, has fixed production costs of $76 per edition
and marginal printing and distribution costs of $0.5 per copy. The Collegiate Investigator sells
for $0.75 per copy. How many copies should be sold in order to break even?
7