Mr. Orchard’s Math 142 WIR
Final Exam Review
1. Find the domain of the following functions:
(a) f (x) = ln(3 − 6x)
(b) f (x) =
√ 8x
2x+18
(c) f (x, y) = ln(3y − 6x)
(d) f (x, y) =
1
xy
−
p
3x2 + y 2
Week 15
Mr. Orchard’s Math 142 WIR
Final Exam Review
Week 15
2. The supply function for a particular commodity is p = S(x) = 5x + 10. p is price per
unit, and x is thousands of units demanded. Furthermore, the demand is linear, and
when the price is $5 per unit, 2000 are demanded. When the price is $15 per unit, 1000
are demanded.
(a) Find the equilibrium price.
(b) Find the equilibrium quantity.
(c) Find the producers’ surplus.
(d) Find the consumers’ surplus.
Mr. Orchard’s Math 142 WIR
Final Exam Review
Week 15
3. The fixed costs for a company are $42, and the variable cost is $10. The revenue for the
company is given by the formula R(x) = −2x2 + 30x. Find the break even point(s) for
the company.
4. $1000 is invested at an annual rate of 9% compounded monthly.
(a) How much is in the account after 10 years?
(b) At what rate is the account growing at the end of the third year?
(c) How long will it take for the account to triple in value?
5. Write log8 (x) − 3 log8 (2y) + 2 log8 (z) using only one logarithm.
Mr. Orchard’s Math 142 WIR
Final Exam Review
 2
x<2
 x −2
4
x = 2 Find the following limits.
6. f (x) =

−2x3 + 14 x > 2
(a) lim+ f (x)
x→2
(b) lim− f (x)
x→2
(c) lim f (x)
x→−∞
(d) lim f (x)
x→−∞
Week 15
Mr. Orchard’s Math 142 WIR
Final Exam Review
Week 15

 Kx2 + 4x + K x < 4
50
x = 4 continuous at
7. Find the value(s) of K that makes f (x) =

Kx − K + 23 x > 4
x = 4.
8. The position of a particle is given by d(t) = 20t2 , which is measured in meters and t is
measured in seconds. Find the average velocity of the particle on the interval [3, 5].
9. Use the limit definition of the derivative to find the derivative of f (x) = x2 − 2.
Mr. Orchard’s Math 142 WIR
Final Exam Review
10. Find the derivatives of the following functions:
(a) f (x) = 3ex − 4x
(b) f (x) =
√ x
xe
(c) f (x) =
x3 −ln x
2e2x +3x
(d) f (x, y) = xy +
x
y2
with respect to x
(e) f (x, y) = xy +
x
y2
with respect to y
Week 15
Mr. Orchard’s Math 142 WIR
Final Exam Review
Week 15
11. The demand function is given by x = 82 − p2 .
(a) Find the elasticity of demand if p = 9. Is the demand elastic, inelastic, or at unit
elasticity?
(b) Use the elasticity of demand to determine the price where revenue is maximized.
12. Find f 00 (x) given f (x) below:
(a) f (x) = e2x
2
(b) f (x) = ln(3x2 + 2x + 5)
Mr. Orchard’s Math 142 WIR
13. f (x) =
1
.
3x+1
Final Exam Review
Use calculus to determine the following:
(a) Where is f (x) increasing?
(b) Where is f (x) decreasing?
(c) Where does f (x) have local extrema?
(d) Where is f (x) concave up?
(e) Where is f (x) concave down?
(f) Where does f (x) have inflection points?
Week 15
Mr. Orchard’s Math 142 WIR
Final Exam Review
Week 15
14. Below is given information about f (x), f 0 (x), and f 00 (x). Use it to sketch a graph of
f (x).
f (x) is continuous everywhere.
f 0 (−10) = f 0 (15) = 0
f 00 (−5) = f 00 (5) = 0
0
f (x) < 0 on (−∞, −10) ∪ (0, 15) ∪ (15, ∞)
f 00 (x) > 0 on (−∞, −5) ∪ (5, ∞)
20
15
10
5
0
-5
-10
-15
-20
-20
-15
-10
-5
0
5
10
15
20
Mr. Orchard’s Math 142 WIR
Final Exam Review
Week 15
15. Find the absolute extrema of the following functions on the given intervals.
(a) f (x) = x2 − 4x + 1 on [−1, 1]
(b) f (x) = x2 − 4x + 1 on (0, 4]
(c) f (x) = 3x + x−3 on (−∞, 0)
16. Operating costs for company cars based on average speed is given by C(v) = 6 + 118
+
v
0.0002v 2 , measured in thousands of dollars. Find the average speed at which operating
costs are minimized and find the minimum cost.
Mr. Orchard’s Math 142 WIR
Final Exam Review
Week 15
17. A company sells two products x and y. x sells at p = 1100 − 5x − 2y dollars per unit
and y sells at q = 852 − 4x − 3y dollars per unit. What is the maximum revenue of the
company?
18. A box with a square base must have a volume of 32,000 cm3 . Find the dimensions of
the box that minimize the amount of material used.
Mr. Orchard’s Math 142 WIR
Final Exam Review
Week 15
19. Solve for the following antiderivatives.
R
(a)
6x2 + 4x − ex + x1 dx
(b)
R
t
(t2 +1)2
+ 1 dt
R8
20. Estimate 0 xe−x dx using n = 4 equal subintervals and a right hand sum. Round to
three decimal places.
21. The velocity of a decelerating object is measured at equal intervals, given in the table
below. Use the information to find an upper and lower bound on the distance the object
traveled.
0
1
2
3
4 5
t
v(t) 50 40 35 25 10 5
Mr. Orchard’s Math 142 WIR
Final Exam Review
Week 15
22. Evaluate the following definite integrals exactly.
R4 1
(a) 1 8x
dx
(b)
R0
7
dx
−1 (x−3)4
23. Find the average value of f (x) = 18x + 3x2 on the interval [0, 3].
24. A continuously compounded bank account is opened with $500 at an annual rate of 5.5%
per year. What is the average amount of money in the account over the first 7 years?
Mr. Orchard’s Math 142 WIR
Final Exam Review
25. Find the area between f (x) = −x and g(x) = 2 − x2 on the interval [−1, 3].
26. f (x, y) = x5 y 5 + 3x5 y + x4 + 10y 2
(a) Find all first order partial derivatives of f .
(b) Find all second order partial derivatives of f .
27. Find all critical points of f (x, y) = x7 + y 7 − 7xy and classify them.
Week 15