Review for Final EXAM
MATH 142
Drost-Spring 2010
8.3 Maxima and Minima
Find the local extrema for each of the following in #1-#5:
1. f (x, y) = x2 + y 2 + 8x + 4y + 10
2. f (x, y) = 2x2 − xy + y 2 − x − 8y + 5
3. f (x, y) = exy
2
2
4. f (x, y) = 2x − 4x y + 6y
2
5. f (x, y) = 2y 3 − 6xy − x2
6. A store sells two brands of color print film. The
store pays $2 for each roll of brand A film, and
$3 for each roll of brand B film. A consulting
firm has estimated the daily demand equations
for these two competitive products to be x =
75 − 40p + 25q, y = 80 + 20p − 30q where x is
the demand equation for brand A, and y is the
demand equation for brand B, p is the selling
price for brand A, and q is the selling price for
brand B.
a. Determine the demands x and y when
p = $10 and q = $12.
b. What price for each brand of film will
maximize daily profits?
7. A rectangular box with no top and 1 parallel partition, is to be made to hold a volume of 48 cubic
inches. Find the dimensions that will require the
least amount of material.
11. Twenty t-shirts are sold when the price per shirt
is $10. One hundred t-shirts are sold if the price
is $5 each. Mark is willing to sell 40 t-shirts
if the price is $5 per shirt and is willing to sell
200 when the price is $10 per shirt. Find the
equilibrium price.
12. Find the domain and range of f (x) = −2(x −
3)2 + 5
13. Find the domain and range of g(x) =
5−x
4x − 28
14. Find the intercepts of y = 4x2 − 20x − 56.
15. Samantha deposits $800 in a savings account
that compounds quarterly at 3.75%. How much
money will she have in 12 years, assuming no
further deposits or withdrawals.
16. Solve for x, 5log3 (x + 2) = 20
17. What
is the domain of f (x)
=
x2 − 25
?
(x − 5)(2x + 5)(3x − 4)
Find the horizontal and vertical asymptotes
for part a.
18. A drug company has found N (t)
=
500t2 − 5t + 100
represents the number of
4t2 + 8
people infected by a disease in a small town in t
days. Find N ′ (4) and interpret the meaning.
19. The revenue function for a company is R(x) =
3x + 10
· x. find the marginal average revenue.
120 − 5x
√
20. f (x) = (3x4 − 5x2 + 4)( x + 5x1/2 Find the
derivative, and do not simplify.
21. Michael plans to sell birdhouses. It costs him $35
for supplies for each house and the tools needed
cost $850. He has studied the demand for birdhouses and determines p = −.2x + 50 Find the
marginal profit function.
8.5 Method of Least Squares
22. At what values of x is the function f (x) =
x2 − 10x
discontinuous?
x− 100
Find the least squares line for problems #8-#9.
23. Evaluate:
4x − 6x2
x→0 10x − 15x2
lim
8.
x 1 2 3 4
y 20 14 11 3
9.
x −5 0 5 10 15
y 60 50 30 20 15
24. What type of discontinuity occurs at x = 3 if
x− 9
?
f (x) = 3
x − 5x2 + 6x
Review for the Final EXAM
25. Find the derivative of f (x) = 6x4 −4x 2 −
1
10. Find the break even point(s) if R(x) = x(36 −
.02x) and C(x) = 10x + 6000.
6
+4.
x2
26. Where does f (x) have inflection points given
f ”(x) = −3x(x + 2)2 (x + 3), if the domain of
f (x) is ℜ.
27. Find the absolute extrema for the function
f (x) = x4 − 6x2 + 16.
28. A pecan grower in Las Cruces, NM estimates
from past seasons that when 20 trees are planted
per acre, each tree will average 50 pounds of nuts
per year. If for each additional tree planted (up
to 15), the average yield drops 2 pounds. How
many trees should be planted per acre to maximize the total nut yield?
47. Mark and Kevin are designing a new pizza box
to hold their amazing pizza rolls, and the box
is shown below with the lid open. Find the dimensions of the box which minimize construction
costs (at 0.12 /square inch) if the box must hold
216 cubic inches of food.
29. Find where f (x) is increasing or decreasing given
(x − 2)(x + 3)
.
f ′ (x) =
x(x − 4)(x + 1)
30. Find the interval(s) over which f (x) is concave
up and increasing.
f (x) = −x3 − 9x2 − 24x + 2
31. Find the absolute minimum value of f (x) =
8
16x2 + on the interval [0, 2].
x
32. Solve for x, 150 =
33. Find the derivative of f (x) = ln |x3 − x2 − 25|
34. What type of elasticity do you have when x =
2p2 − p and p = 0.8?
35. Find the relative extrema for f (x) = e8x
3
−3x4 +6
5x + 3
x2 − 6x
37. Using a right
Z 10hand sum with 4 rectangles, ap(3x2 + 5x + 10) dx
proximate
38.
Z
−2
√
5
(3x5 − 5x + 4 3 x + ) dx
x
39. f ”(x) = −3x + 10, f ′ (1) = 4, f (2) = 8. Find
f (x).
Z 3
Z
40.
3x(x − 2)3 dx,
b.
3x(x − 2)3 dx
−2
41.
Z
49. True or False: The derivative of ex and the integral of ex have the same answer.
50. True or False: A critical value occurs at x = a,
if f ′ (a) = 0
30
1 − 2e2x
36. Find the derivative of f (x) = ln
48. State the rule for the derivative of a rational
this
.
function, f (x) =
that
4x + 5
dx
8x2 + 20x
42. Find the area between y = −0.01x3 + x2 − 3x −
600 and the x−axis.
43. Find the consumers’ surplus if the equilibrium
price is $20 and the demand equation is p =
−5x2 + 100.
44. Find fx when f (x, y) = (3x2 + 3xy 2 − 5y)3
45. Find fxy when f (x, y) = 5x2 +2xy−4y 3 +3x2 y +
4y − 2
46. Find all four second order partial derivatives for
2
f (x) = 3x2 − 6xy 2 + 2(xy 3 + x2 y)2 + e2xy