Mr. Orchard’s Math 142 WIR
Final Review
Week 15
1. A runner’s speed (measured in feet per second) was measured at 2-second intervals are
given in the table below. The runner steadily increases her speed the entire time.
t
v(t)
0
0
2
5.7
4
11.4
6
15.6
8
17.5
10
19.8
12
20
(a) Give an upper bound for the distance traveled by the runner during this time using
n = 3.
(b) Give a lower bound for the distance traveled by the runner during this time using
n = 3.
√
2. Evaluate g(x, y) = ln( x + 4y − 8) at the point(e2 , 2)
Mr. Orchard’s Math 142 WIR
Final Review
Week 15
3. Find the dimensions of a rectangular box with volume 8 ft3 that minimizes the surface
area.
4. State the limit definition of the derivative.
Mr. Orchard’s Math 142 WIR
5. f (x) = log7
(x2 +2)3
x4 (x−1)2
Final Review
Week 15
(a) Express f (x) as a sum and difference of logarithms in its simplest terms.
(b) Find f 0 (x).
6. The cost function for thingamajigs is C(x) = 2x3 − x2 + 1. Use the marginal cost to
approximate the cost of making the 12t h thingamajig.
7. Find the domain of g(x, y) = ln(x −
√
y − 10)
Mr. Orchard’s Math 142 WIR
Final Review
Week 15
8. Bank account A has an annual rate of 7.2% compounded monthly. Account B has an
annual rate of 6.9% compounded continuously.
(a) If $2000 was put into account A, how much money will be in the account 2 years
from now?
(b) There is currently $3000 in account B. If $1000 was put into it at the start, how
long ago did the account open?
(c) Use effective rates to decide which account is better for your money.
Mr. Orchard’s Math 142 WIR
Final Review
Week 15
√
3
x+20
9. The marginal cost function for dog sweaters is given by C 0 (x) = 20 √
and the total
3x
cost of making 27 dog sweaters is $900. What is the total cost of making 64 dog sweaters?
10. Two nonnegative numbers, x and y, have a sum of 84. What is the maximum value of
T = x2 + 2xy?
11. What is the average value of f (x) = x3 + 3x2 + 12 on the interval [−2, 2]?
Mr. Orchard’s Math 142 WIR
Final Review
12. A product has the price-demand function x =
√
Week 15
90 − 2p.
(a) What is the formula for the elasticity of demand?
(b) Is the demand elastic, inelastic, or unit elasticity when the price is $20?
(c) At what price should the items be sold in order to maximize the revenue? (Round
to the nearest cent if necessary.)
13. Find the exact value of
region.
R3 √
− 9 − x2 dx by finding the area of an appropriate geometric
0
Mr. Orchard’s Math 142 WIR
Final Review
Week 15
14. A manufacturer sells lunchboxes for $8 each. The manufacturer has a fixed cost of $300
per month, and a cost of $4 per lunchbox.
(a) What is the cost function for the manufacturer, assuming it is linear?
(b) What is the revenue function for the manufacturer, assuming it is linear?
(c) What is the profit function for the manufacturer?
(d) How many lunchboxes must the manufacturer sell in order to break even?
(e) What is the break even cost?
Mr. Orchard’s Math 142 WIR
Final Review
Week 15
15. Given is the graph of g(x). Where is g(x) differentiable on its domain [−3, 3]?
3
2
1
0
-1
-2
-3
-3
-2
-1
0
1
2
3
16. Find the exact values of the following integrals using The Fundamental Theorem of
Calculus.
R2
(a) −2 x5 dx
(b)
R1
0
(8x + 2)e2x
2 +x
dx
Mr. Orchard’s Math 142 WIR
Final Review
Week 15
−3x
17. f (x) = (x−2)
2 is defined everywhere except x = 2. Use calculus to answer the following
questions.
(a) Find the relative extrema of f .
i. Relative Maximums:
ii. Relative Minimums:
(b) Where is f increasing?
(c) Where is f concave down?
Mr. Orchard’s Math 142 WIR
Final Review
18. The supply and demand functions for a commodity are given by
D(x) = 300e−.4x
P (x) = 60e0.3x
(a) What is the market equilibrium point?
(b) What is the consumers’ surplus?
Week 15
Mr. Orchard’s Math 142 WIR
Final Review
Week 15
19. Classify functions as power, polynomial (state its degree and leading coefficient), rational,
exponential, or none of the above.
1
(a) 2x− 3
(b)
3x2 √
2x3 +13 x
(c) 8(2x )
(d)
9x+12
14x3 +22
(e) x3
Mr. Orchard’s Math 142 WIR
Final Review
Week 15
20. Below is the graph of H 0 (x).
8
6
4
2
0
-2
-4
-4
-3
-2
-1
(a) Find the critical values of H(x).
(b) Where is H(x) concave down?
21. Find the area below the x-axis and above y = (x − 2)2 − 4.
0
1
Mr. Orchard’s Math 142 WIR
22. Find the following limits.
5x2 −15x
x→3 x−3
(a) lim
6x4 −2x
4 +22x2
4x
x→∞
(b) lim
2
−9x
x→∞ 1+e
(c) lim
2
−9x
x→−∞ 1+e
(d) lim
Final Review
Week 15
Mr. Orchard’s Math 142 WIR
Final Review

 x + 2k x < 3
0
x=3
23. y(x) =
 3
x +k x>3
(a) Calculate lim+ y(x) in terms of k.
x→3
(b) Calculate lim− y(x) in terms of k.
x→3
(c) What value of k makes this limit exist?
(d) What is lim y(x) for this value of k?
x→3
(e) Calculate y(3).
(f) Is this function continuous? Why or why not?
Week 15