Math 141 Final Exam Review
Problems appearing on your in-class final will be similar to those here but will have numbers and
functions changed.
1. (2.1) Use the graph below to find the following:
a)
b)
lim 𝑓(𝑥) =
f)
lim 𝑓(𝑥) =
g)
𝑥→4−
𝑥→4+
lim 𝑓(𝑥) =
𝑥→3−
lim 𝑓(𝑥) =
𝑥→3+
c) lim 𝑓(𝑥) =
h) lim 𝑓(𝑥) =
d) 𝑓(4) =
i) 𝑓(3) =
𝑥→4
e)
Is 𝑓 continuous at 𝑥 = 4?
Math 141 Final Review, Dec. 18, 2015
𝑥→3
j) Is 𝑓 continuous at 𝑥 = 3?
2. (2.1)Given that lim 𝑓(𝑥) = 5 and lim 𝑔(𝑥) = −4 , find the limits below.
𝑥→1
a) lim(−5𝑓(𝑥))
𝑥→1
b) lim(3𝑔(𝑥) + 6)
𝑥→1
2 − 𝑓(𝑥)
𝑥→1 𝑥 + 𝑔(𝑥)
c) lim
3. (2.1) If
 x 2  6 x  1, if x  1

4,
if x  1

Find the following:
a)
b)
c)
lim 𝑓(𝑥) =
𝑥→−1−
lim 𝑓(𝑥) =
𝑥→−1+
lim 𝑓(𝑥) =
𝑥→−1
d) 𝑓(−1) =
e) Is f continuous at 𝑥 = −1?
Math 141 Final Review, Dec. 18, 2015
𝑥→1
4. (2.1) If 𝑓(𝑥) = 4𝑥 2 − 5𝑥 + 3, then find the limit of the following difference quotient:
lim
ℎ→0
𝑓(6 + ℎ) − 𝑓(6)
.
ℎ
5. (2.2) Find all vertical and horizontal asymptotes of the function
f ( x) 
2 x2  5
x2  5x  4
6. (2.3) Solve the inequality given below. Use both intervals notations and inequality notations
for your answer.
𝑥+1
≤0
𝑥 2 − 3𝑥
7. (2.4) If an object moves along a line so that it is at 𝑦 = 𝑓(𝑥) = 4𝑥 2 − 2𝑥 at time 𝑥 (in
seconds), find the instantaneous velocity function 𝑣 = 𝑓′(𝑥) and find the velocity at times 𝑥 =
1, 3, and 5 seconds (𝑦 is measured in feet).
8. (2.5) Find 𝑓 ′ (𝑥)
1
if 𝑓(𝑥) = 2√𝑥 + 8𝑥10 − 6𝑥 + 𝑥 5 − 5.
9. (2.6) Find 𝑑𝑦 and 𝛥𝑦 for 𝑦 = 2𝑥(𝑥 − 8) when 𝑥 = 3 and 𝑑𝑥 = 𝛥𝑥 = −1.
10. (2.7) The total cost and the total revenue (in dollars) for the production of 𝑥 alarm clock
radios are given by the following functions:
𝐶(𝑥) = 5𝑥 + 3,500 and 𝑅(𝑥) = 45𝑥 − 0.1𝑥 2
a)
b)
c)
d)
e)
0 ≤ 𝑥 ≤ 450.
Find the profit function 𝑃(𝑥).
Find the marginal profit at a production level of 150 alarm clock radios.
Find the average profit per unit at a production level of 150 alarm clock radios.
Find the marginal average profit at a production level of 150 alarm clock radios.
Use the results from part c) and d) to estimate the average profit per unit at a production
level of 151 alarm clock radios.
Math 141 Final Review, Dec. 18, 2015
11. (2.7) The total cost (in dollars) of producing x food processors is
C ( x)  2,000  50 x  0.5x2
(a) Find the exact cost of producing the 21st food processor.
(b) Use marginal cost to approximate the cost of producing the 21st food processor.
12. (2.6) A sphere with a radius of 5 centimeters is coated with ice 0.1 centimeter thick. Use
4
differentials to estimate the volume of the ice. [Recall that V   r 3 . ]
3
13. (3.1) A couple paid $10,000 for a house in 1978. They sold the house in 2015 for $450,000.
If interest is compounded continuously, what annual nominal rate of interest did the original
$10,000 earn?
14. (3.2) The model T  87.97  34.96ln p  7.91 p relates the temperature T, in °F at which
water boils at pressure p in pounds per square inch. Find the rate of change of temperature
when pressure is 70 pounds per square inch.
15. (3.1, 3.2, 3.4) The drug concentration in the bloodstream t hours after injection is given
approximately by
C (t )  5e0.3t , where C (t ) is concentration in milligrams per millimeter.
a. What is the rate change of concentration after 5 hours?
b. When is the rate of change of concentration  0.8 mg per milliliter per hour?
c. When is the drug concentration 1.5 mg per milliliter?
16. (3.2, 3.3, 3.4) Find
dy
for the following functions.
dx
a.
1
y  6e x  ln x  log x
3
b.
y  e3 x ln(5x)
c.
y
3  4x
x3  5
Math 141 Final Review, Dec. 18, 2015

5
3
d. y  ln 4 x  5x  9
e.

y   x6  4 x 4 
9
17. (3.4) Ecologists estimate that the average level of carbon monoxide in the air above a city is
given by L  10  0.4 x  0.001x where L is in parts per million and x is the population of
the city in thousands. The population x is estimated as a function of t years from the
2
present by x  752  53t  0.5t
a. Find
dL
dx
b. Find
dL
dt
2
c. How fast is the carbon monoxide level changing at time t  3 years?
18. (3.5) Consider the equation y e x  y 2  2 x  0
a. Use implicit derivative and find y
b. Find the equation of the tangent line at the point  0,1
19. (3.7) Given the demand equation x  f ( p)  8,100  9 p 2 ,
a. Find E ( p) , the elasticity of demand. Recall: E ( p)  
p f ( p)
.
f ( p)
b. Determine whether the demand is elastic, inelastic or has unit elasticity at p  25 .
20. (3.2) The function R( x)  80,000 1.1522  models the amount of plastic carry-out bags
x
recycled, in tons, in the United States where x is the number of years since 1996.
a. Find the amount of plastic carry-out bags recycled in 2015
b. Find the rate of growth in recycled plastic carry-out bags (in tons per year) in 2015
Math 141 Final Review, Dec. 18, 2015
21. (3.3) When a company produces and sells x thousands units per week, its total weekly
200 x
profit in thousands of dollars is given by P 
.
100  x 2
a. Find P(36) and P  36 
b. How fast is the total weekly profit of the company changing when x  36 thousand units?
22. (3.6) A 41 foot long ladder is leaning against a vertical wall of a house. The bottom of the
ladder is pulled away from the house wall at a constant rate of 1.6 feet per second. How fast
is the top of the ladder sliding down the wall when the foot of the ladder is 9 feet from the
wall?
23. (3.6)Suppose that for a company manufacturing calculators, the cost, revenue and profits
equations are given by C  90,000  30 x , R  300 x 
x2
and
30
P  R  C where the production
output in one week is x calculators.
If production is increasing at the rate of 300 calculators per week when the production output
is 4,000 calculators, find the rate of increase (or decrease) in
a. the cost,
b. the revenue and
c. the profit.
24. (3.7) Given the demand equation p  0.01x  25
a. Express the demand x as a function of a price p and find its domain.
b. Express the revenue R as a function of the price p and find its domain.
c. Find the elasticity of demand E ( p) . Recall: E ( p)  
d. For which values of p is demand inelastic?
e. For which values of p is revenue decreasing?
Math 141 Final Review, Dec. 18, 2015
p f ( p)
f ( p)
25. (4.1) Use the given graph of y  f ( x) to find the intervals on which f is increasing, the
intervals on which f is decreasing, and the x coordinates of the local extrema of f . Sketch
a possible graph of y  f ( x).
26. (4.1, 4.2) Use the given information to sketch the graph of
f . Assume that f is continuous on its
domain and that all intercepts are included in the information given.
27. (4.2) A company estimates that it will sell N  x  units of a product after spending $ x
thousand on advertising, as given by
N  x   0.25x 4  11x3  108x 2  3, 000,
9  x  24
When is the rate of change of sales increasing and when is it decreasing? What is the point of
diminishing returns and the maximum rate of change of sales? Graph N and N  on the same
coordinate system.
Math 141 Final Review, Dec. 18, 2015
28. (4.3) Find each limit. Note that L’Hôpital’s rule does not apply to every problem, and some
problems will require more than one application of L’Hôpital’s rule.
e3 x  1
x 0
x
ln(1  x)
(c) lim
x 0
x2
e4 x
(e) lim 2
x  x  1
(a) lim
1  x 1
x
ln(1  6 x)
(i) lim
x  ln(1  3 x)
(g) lim
x 0
x2  5x  6
x 2 x 2  x  6
ln(1  x)
(d) lim
x 0
1 x
x
e  e x  2
(f) lim
x 0
x2
(b) lim
ln x
x5
ln(1  6 x)
(j) lim
x  0 ln(1  3 x)
(h) lim
x 
29. (4.1, 4.2, 4.4) Use the given information to sketch the graph of f . Assume that f is
continuous on its domain and that all intercepts are included in the information given.
30. (4.5, 4.6) A company manufactures and sells x e-book readers per month. The monthly cost
and price-demand equations are, respectively,
C ( x)  350 x  50,000
p  500  0.025 x,
0  x  20,000
(a) Find the maximum revenue.
(b) How many readers should the company manufacture each month to maximize its profit?
What is the maximum monthly profit? How much should the company charge for each
reader?
Math 141 Final Review, Dec. 18, 2015
(c) If the government decides to tax the company $20 for each reader it produces, how many
readers should the company manufacture each month to maximize monthly profit? How
much should the company charge for each reader?
31. (4.5, 4.6) A fence is to be built to enclose a rectangular area. The fence along three sides is to
be made of material that costs $5 per foot. The material for the fourth side costs $15 per foot.
(a) If the area is 5,000 square feet, find the dimensions of the rectangle that will allow for the
most economical fence.
(b) If $3,000 is available for the fencing, find the dimensions of the rectangle that will
enclose the most area.
32. (4.5, 4.6) A 200-room hotel in Reno is filled to capacity every night at a rate of $40 per
room. For each $1 increase in the nightly rate, 4 fewer rooms are rented. If each rented room
costs $8 a day to service, how much should the management charge per room in order to
maximize gross profit? What is the maximum gross profit?
33. (4.5, 4.6) The price-demand equation for a GPS device is
p( x)  1,000e0.02 x
where x is the monthly demand and p is the price in dollars. Find the production level and
price per unit that produce the maximum revenue. What is the maximum revenue?
34. (4.1 and 4.4) Nicole owns a company that makes luxurious velvet robes. Her total cost to
make x robes can be modeled by the function C ( x)  1500  3x2 , x  0 .
(a) Find the average cost function.
(b) How many robes must be produced for the average cost to be minimized?
(c) What is the minimum average cost?
Math 141 Final Review, Dec. 18, 2015
35. (5.1) Find each indefinite integrals.
(a)  4 x3dx
2
dw
w
x2 3 3
(e)   2  dx
3 x
x
(b) 
(d)  3x  x 2  1 dx
(c)  e x dx
(f) 
t2  t
dt
t
36. (5.1) The marginal profit from the sales of x items is given by P  x   0.01x  450. Find
P  x  if
P(100)  2500.
37. (5.1) The rate of change of the monthly sales of a newly released football game is given by
S   t   500t1/4 ,
S  0  0
where t is the number of months since the game was released and S  t  is the number of
games sold each month. Find S  t  . When will monthly sales reach 20,000 games?
38. (5.2) Find each indefinite integral and check the result by differentiating.
x2
(a)  6 x5  x 6  1 dx
(b) 
(d)  7 e7 x dx
(e)   x  1 e x
3
(g) 
1
 3t  4 
2
dx
Math 141 Final Review, Dec. 18, 2015
5  x 
3 4
2
dx
2 x
(h)  x x  1 dx
dx
(c)  10 x  x 2  1 dx
5
x
dx
x 2
x
(i) 
dx
x7
(f) 
2
39. (5.2) The market research department for an automobile company estimates that sales (in
millions of dollars) of a new electric car will increase at the monthly rate of
S   t   4e0.08t
0  t  24
t months after the introduction of the car. What will be the total sales S  t  t months after
the car is introduced if we assume that there were 0 sales at the time the car entered the
marketplace? What are the estimated total sales during the first 12 months after the
introduction of the car? How long will it take for the total sales to each $40 million?
40. (5.3) The area of a healing skin wound changes at a rate given approximately by
dA
 5t 2
dt
1 t  5
where t is time in days and A 1  5 square centimeters. What will be the area of the
wound in 5 days?
dp
at x units of supply per day is proportional to the price p .
dx
There is no weekly demand at a price of $300 per unit [ p  0   300 ], and there is a weekly
41. (5.3) The marginal price
demand of 10 units at a price of $250 per unit [ p 10   250 ].
(a) Find the price-demand equation.
(b) At a demand of 20 units per week, what is the price?
(c) Graph the price-demand equation for 0  x  50 .
42. (5.5) The total cost (in dollars) of making x music boxes is given by C  x   12, 000  40 x.
(a) Find the average cost per unit if 200 music boxes are produced.
(b) Find the average value of the cost function on the interval [0, 200].
(c) Explain the difference in the meaning of the values found in part (a) and (b).
Math 141 Final Review, Dec. 18, 2015
43. (5.5) A company produces a printer that also scans documents. The research department
x
produced the marginal cost function C   x   200  where C  x  is the total cost (in
5
dollars) and x is the number of printers produced in a month. Compute the increase in cost
going from a production level of 100 printers per month to 500 printers per month. Set up a
definite integral and evaluate.
44. (5.5) The total accumulated costs C  t  and revenues R  t  (in thousands of dollars),
respectively, for a coal mine satisfy
C  t   3
and
R  t   20e0.1t
where t is the number of years that the mine has been in operation. Find (a) the useful life of
the mine, to the nearest year, recall the value of t for witch C(t )  R(t ) is called the useful
life. (b) What is the total profit accumulated during the useful life of mine?
Answer Keys:
1.
a) -1. b) -1. c) -1. d) -1. e) yes. f) -5. g) -3. h) does not exist. i) -5. j) no.
2.
a) -25. b) -6. c) 1.
3.
a) -6. b) 4. c) does not exist. d) 4. e) no.
4. 43
5. Vertical asymptotes: The line x  1 and the line x  4 . The horizontal asymptote is the line:
y2 .
6. Interval notations:(−∞, −1] ∪ (0, 3). Inequality notations: x  1 or 0  x  3 .
7. 6ft/sec; 22ft/sec; 38ft/sec
Math 141 Final Review, Dec. 18, 2015
8.
1
√𝑥
+ 80𝑥 9 − 6 −
5
𝑥6
9. 𝑑𝑦 = 4; ∆𝑦 = 6
10.
a) −0.1𝑥 2 + 40𝑥 − 3,500
b) $10 per alarm clock radio
c) $1.67 per alarm clock radio
d) The average profit is increasing at the rate of $0.06 per alarm clock radio
e) About $1.73 per alarm clock radio.
11. (a) $29.50 (b) $30.
12.
31.4 cm3
13. 10.3%
14.
0.972 F per pound per square inch
15. a. 0.335 mg/mL
16. a. 6e 
x
b. 2.10 hours
1
1

3x (ln10) x

e. 54 x5  144 x3
 x
6
b.
e
3 x 
x
c. 4.01 hours
 3e
3 x 
ln  5 x  c.
3
 5
2
20 x 4  15 x 2
4 x5  5 x3  9
8

2
 53  t 
per million per year
18. a.
2  ye x
b. y  3x  1
2 y  ex
19. a.
2 p2
b. elastic
900  p 2
21. a.
x
d.
 4x4 
17. a. 0.4  0.002x b. 1.904  0.106t  0.001t
20. a. 1,180,645.5 tons
8 x3  9 x 2  20
b. 167,265.8 tons/year
P(36)  5.16 and P(36)  0.1227
b. Decreasing at the rate of 12 cents per unit
Math 141 Final Review, Dec. 18, 2015
c. increasing at the rate of 124.9 parts
22. 0.36 ft/s
23. a.
$9,000 /week b. $10,000 /week c. $1,000 /week
24. a. x  2500  100 p ,
c. E ( p) 
0  p  25 b. R( x)  2500 p 100 p2 , 0  p  25
p
d.  0,12.5 e. 12.5, 25
25  p
25. Increasing on (,  3)and (1, ) ; decreasing on (3,1) ; local maximum at x  3 ; local
minimum at x  1 .
26.
27.
28. (a) 3 (b) -1/5 (c)
 (d) 0 (e)  (f) 1 (g) 0 (h) 0 (i) 1 (j) 2
Math 141 Final Review, Dec. 18, 2015
29.
30. (a) Max R( x)  R(10,000)  $2,500,000 .(b) Maximum profit is $175,000 when 3,000
readers are manufactured and sold for $425 each. (c) Maximum profit is $119,000 when 2,600
readers are manufactured and sold for $435 each.
31. (a) The expensive side is 50 ft; the other side is 100 ft. (b) The expensive side is 75 ft; the
other side is 150 ft.
32. $49; $6,724.
33. A maximum revenue of $18,394 is realized at a production level of 50 units at $367.88 each.
34. (a) C ( x)  1500 x 1  3x, (b) 22 robes, (c) $134.16.
35. (a) x  C (b) 2 ln w  C (c) e  C (d)
4
x
3x 4 3x 2
x3 3

 C (e)
  3ln x  C (f)
4
2
9 x
t2
t C
2
36. P  x   0.005x2  450 x  42, 450
37. S (t )  400t 5/4 , 50
38. (a)
x
6
 1
4
4/5
 23 mo.
4
 C (b)
1
9 5  x

3 3
 C (c)
5  x 2  1
6
6
 C (d) e7x  C (e) 1 e x
1
1
2
2
 C (h)  x  15/2   x  13/2  C
ln x 2  2  C (g) 
3  3t  4 
2
5
3
Math 141 Final Review, Dec. 18, 2015
2
2
2 x
 C (f)
(i)
2
3/2
1/2
 x  7   14  x  7   C
3
39. S  t   50  50e0.08t ; 50  50e0.96  $31million;   ln 0.2  / 0.08  20 mo.
40. 1cm2 .
41. (a) p  300e0.018 x (b) $209.30
(c) The graph of p( x) in the window [0, 50] by [-10, 310]
42. (a) $100 (b) $16,000 (c) The average cost function yields the average cost per music box.
The average value of the cost function yields the average total cost of each production level
between 0 and 240.
500
43.

x
  200  5  dx ;
$56,000
100
44. (a) Useful life = 10 ln
20
 19 years; (b) Total profit = 143  200e1.9  113.086 or $113,086 .
3
Math 141 Final Review, Dec. 18, 2015