MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
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The profit for Scott’s Scooters can be estimated by ( , ) = 0.4 + 0.1 + 0.3 + 2 + 1.5 hundred dollars,
where is the number of mopeds sold and is the number of dirt bikes sold. (Check: (3,2) = 14.3)
Use this to answer the next three questions.
1. Find and interpret (14,10).
a. When 10 mopeds and 14 dirt bikes are sold, the profit is $14,460.
b. When 14 mopeds and 10 dirt bikes are sold, the profit is $14,460.
c. When 14 mopeds and 10 dirt bikes are sold, the profit is $17,140.
d. When 10 mopeds and 14 dirt bikes are sold, the profit is $17,140.
2. How many dirt bikes must be sold if Scott wants to make a profit of $19,500 and has sold 10 mopeds?
a. 15
b. 20
c. 215
d. 41
3. How quickly is profit increasing with respect to the number of dirt bikes sold when he sells 2 mopeds and 6 dirt
bikes?
a. 3.8 hundred dollars per dirt bike
b. 4.9 hundred dollars per dirt bike
c. 3 hundred dollars per dirt bike
d. 8.1 hundred dollars per dirt bike
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4. Consider the function ( ) =
( )
(
Evaluate the definite integral ∫
a. 0.685
)
Check: (3) = 0.2746530722
.
( )
.
b. 0.796
c. 1.808
d. 1.848
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5. A scooter has been traveling at a rate of ( ) = 4 miles per hour, where
What distance did the scooter travel between 2 pm and 4 pm?
a. 8 miles
b. 16 miles
is the number of hours since noon.
c. 24 miles
d. 32 miles
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A major entertainment complex has determined that they will invest ( ) = 17.2 + 1.3 million dollars per year years
from now, where 0 ≤ ≤ 10. Assume a continuous income stream and an APR of 4%, compounded continuously. Use
this information to answer the next two questions.
6. Find the 10-year future value of the investment.
a. 198.897 million dollars
b. 286.092 million dollars
c. 296.720 million dollars
d. 353.562 million dollars
7. Find the 10-year present value of their investment.
a. 158.866 million dollars
b. 235.995 million dollars
c. 296.720 million dollars
d. 191.773 million dollars
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MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
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Use the function ( ) =
− 6 + 5 as graphed to answer
the next three questions. (Check: (3) = −4)
8. Find the total signed area of the shaded regions
of the graph of ( ).
a.
b.
c.
d.
13
−13
25/3
−25/3
9. The function ( ) is an accumulation function of ( ). If (0) = 4/3, what is the value of (5)?
a. −25/3
b. −7
c. 0
d. 43/3
10. Which of the following expressions gives the total area trapped between the graph of ( ) and the -axis between
= −1 and = 6.
a. ∫
( )
−∫
( )
+∫
( )
c. ∫
b. ∫
( )
+∫
( )
+∫
( )
d. − ∫
( )
( )
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11. It is estimated that it will require a continuous income stream of ( ) = 850(1.03 ) thousand dollars per year to
support a local homeless shelter years from now, where 0 ≤ ≤ 5. A very wealthy philanthropist is willing to
provide the funds to support the shelter for the next 5 years, but he intends to make a single, lump-sum, donation
instead of making a continuous stream of payments. Assuming his donation will be invested in an account earning
3% interest, compounded continuously, how large does his donation need to be? Round to the nearest dollar.
a. $3,942,149
b. $4,245,316
c. $4,932,353
d. $5,321,344
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The size of an insect population within a controlled
experiment is growing at a rate of
( ) = 10 ln(1.02) (1.02 ) hundred insects per day,
days after the experiment began, 0 ≤ ≤ 15.Let A
denote the average value of ( ) from 0 to 15, as
illustrated by the dotted line on the graph.
Use this information to answer the next two questions.
12. Calculate A.
a. 0.231
b. 0.232
c. 3.459
13. Which of the following correctly completes the interpretation of A?
“From day 0 to day 15, ____________________________.”
a.
b.
c.
d.
the average size of the insect population was A hundred insects
the size of the insect population increased by A hundred insects
the average size of the insect population increased by A hundred insects per day
the size of the insect population increased by an average of A hundred insects per day
d. 0.005
MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
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( , ) thousand students denotes the number of applicants to a major land-grant university when the tuition is thousand
dollars and the university won football games in the previous year. Some values associated with ( , ) are given:
(12,8) = 16.2,
= −1.4,
= 2.2 Use this information to answer the next five questions.
(
14. Interpret
a.
b.
c.
d.
(
, )
, )
(
, )
= −1.4 “When tuition is $12k and 8 football games were won in the previous year, _______.”
the number of applicants is decreasing by 1400 students per year.
the tuition is decreasing by 1400 dollar per football game won.
the number of applicants is decreasing by 1400 students per football game won.
the number of applicants is decreasing by 1400 students per thousand dollars of tuition.
15. Estimate the number of applicants to the university if the tuition is 14 thousand dollars and the number of football
games won in the previous year is 8.
a. 13,400 students
b. 14,800 students
c. 19,000 students
d. 20,600 students
16. Suppose the university wishes to keep the number of applicants fixed at 16,200 and it won 10 football games in
the previous year. How should the tuition be adjusted?
a. It should be decreased by about $3143.
b. It should be increased by about $3143.
17. Specify the units for
c. It should be decreased by about $1273.
d. It should be increased by about $1273.
.
a. thousand dollars per football game won
b. football games won per thousand dollars
c. thousand students per football game won
d. thousand students per thousand dollars
18. Based on the given information, which of the following is a true statement?
a. The point (12,8) is a relative extreme point (maximum or minimum) of ( , ).
b. The point (12,8) is a saddle point of ( , ).
c. The point (12,8) is a critical point of ( , ), but there is not enough information to determine type.
d. The point (12,8) is not a critical point of ( , ).
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( ) = 0.05 − 0.4 + 10.8 thousand gallons represents the amount of water available to a farmer for irrigation
purposes days from the beginning of growing season, 0 ≤ ≤ 90. Use this model to answer the next three questions.
19. Interpret
a.
b.
c.
d.
∫
( )
= 19.8. “During the first 30 days of the growing season, _________________.”
there was an average of 19.8 thousand gallons available for irrigation
there was enough water available to irrigate the crops for an average of 19.8 days
the average amount of water available for irrigation increased by 19.8 thousand gallons
the amount of water available for irrigation increased by an average of 19.8 thousand gallons per day
20. Find the average rate of change of
a. 4.1
21. Find the average value of
a. 4.1
( ) from
= 0 to
b. 369
( ) from
= 0 to
b. 94.05
= 90.
c. 379.68
d. -0.024
c. 127.8
d. 195.3
= 90.
MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
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Entomologists studied the female and male populations
of ladybugs in a botanical garden over an 8-month
period. They found that the female population was
changing at a rate of ( ) = 0.8 − 8 + 25 thousand
ladybugs per month and the male population was
changing at a rate of ( ) = −0.6 + 3.6 + 7.4
thousand ladybugs per month, where is the number of
months since they first began observing. The graph of
their finings is shown below. (check: (3) = 8.2 and
(3) = 12.8)
Use this graph and functions to answer the next five
questions.
22. Interpret ∫
a.
b.
c.
d.
( )
= 58.333. “In the first 5 months of observation, ____________________.”
the female population decreased by 58.333 thousand ladybugs
the female population increased by 58.333 thousand ladybugs
the female population decreased by 58.333 thousand ladybugs per month
the female population increased by 58.333 thousand ladybugs per month
23. Which of the following calculates the total combined area of the shaded regions marked B and C?
a. ∫
( )−
( )
c. ∫
b. ∫
( )+
( )
d. ∫
.
.
( )+
( )
−∫.
( )+
( )
( )− ( )
+∫.
( )−
( )
24. Find the area of the shaded region of the graph marked A.
a. 56.533
b. 36.133
c. 20.4
d. 15.733
25. Which of the following is the correct interpretation of area A found in the previous question?
a. The increase in the female population was A thousand ladybugs greater than the increase in the male
population during the first two months.
b. The increase in the male population was A thousand ladybugs greater than the increase in the female
population during the first two months.
c. The female population was A thousand ladybugs greater than the male population after two months.
d. The male population was A thousand ladybugs greater than the female population after two months.
26. Given that there were a total of 10 thousand ladybugs at the beginning of the observation, how many ladybugs
were there after 2 months?
a. 34.4 thousand ladybugs
c. 46.133 thousand ladybugs
b. 66.533 thousand ladybugs
d. 25.733 thousand ladybugs
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MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
( ) customers per hour gives the rate of change of the
number of customers in a shopping mall hours after
the mall opens. The graph of ( ) is shown.
Use this information to answer the next two questions.
27. How did the number of customers in the mall change, on average, over the first 3 hours of operation?
a.
b.
c.
d.
The number of customers decreased, on average, by 20 customers per hour.
The number of customers increased, on average, by 11 customers per hour.
The number of customers increased, on average, by 25 customers per hour.
Not enough information provided.
28. What was the average number of customers in the shopping mall during the first 3 hours of operation?
a. 20 customers
c. 42 customers
b. 25 customers
d. Not enough information provided
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29. The function ( , ) has a critical point at (−2,0) with a corresponding output of (−2,0) ≈ −7.333. The
second partial derivatives matrix of ( , ) is given below.
−2
−8
Use the determinant test to classify the critical point(−2,0), if possible.
−8
−8
a. The point (−2,0) is a relative maximum
c. The point (−2,0) is a saddle point
d. The determinant test is inconclusive
b. The point (−2,0) is a relative minimum
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The number of bacteria in a lab sample is increasing at a rate of ( ) = + .
− 1 million bacteria per day, days
after the beginning of an experiment, 1 ≤ ≤ 3. Check: (2) = 0.1243365465
Use this information to answer the next four questions.
When asked to find the area between ( ) and the horizontal axis from = 1 to = 3, rounded to 3 decimal places, a
student decided to draw rectangles and increase the number of rectangles each time. When he drew 10 midpoint
rectangles, the sum of their areas was 0.439. When he drew 20, the sum was 0.443.
30. If he drew only 4 midpoint rectangles, what would the sum of their areas be?
a. 0.430
b. 0.425
c. 0.420
d. 0.410
31. What would the area be if he drew an infinite number of rectangles?
a. 0.447
b. 0.446
c. 0.445
d. 0.443
32. What are the units for the value found in the previous question?
a. million bacteria per day
b. days
c. million bacteria per rectangle
d. million bacteria
33. What you found in question 32 is equivalent to which of the following?
a. ∫
( )
b. ∫
( )
c.
(3) − (1)
d.
(1) − (3)
MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
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At a price of dollars per wafer, the demand for Soylent Green is ( ) = 3(0.95 ) million wafers.
Use this model to answer the next five questions.
Check: (7) = 2.095011888
34. If Soylent Green is selling at a price of $12 per wafer, what is the consumers’ surplus?
a. 20.042 million wafers
b. 26.883 million wafers
c. 31.604 million wafers
d. 58.487 million wafers
35. Find the market price when 2.5 million wafers are demanded.
a. $2.64
36. Given that
b. $3.55
c. $7.04
d. $8.88
( ) = 3 ln(0.95) (0.95 ), find the price at which demand is unit elastic.
a. $0.05
b. $2.85
c. $19.50
d. $21.42
37. When the price of Soylent Green is $23, calculate the elasticity of demand.
a.
= 0.047
b.
= 0.922
c.
= 1.180
d.
= 2.824
38. Classify the elasticity of demand, if possible when the price of Soylent Green is $23.
a. Demand is elastic
c. Neither elastic nor inelastic
b. Demand is inelastic
d. Not enough information
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( , ) million dollars gives a delivery
company’s profit when the company employs
hundred drivers and hundred customer service
representatives. A contour graph of ( , ) is
shown below.
Use this information to answer the next four
questions.
39. If the company currently employs 5 hundred drivers and 1 hundred customer service representatives, which one of
the following would result in the greatest increase in profit?
a.
b.
c.
d.
decreasing the number of drivers by 50
decreasing the number of customer service representatives by 50
increasing the number of drivers by 50
increasing the number of customer service representatives by 50
40. If the company employs 3 hundred drivers and would like to see 50 million dollars in profit, how many customer
service representatives should they employ?
a. 1.1 hundred
b. 2.6 hundred
c. 3.3 hundred
d. not possible
MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
41. Which of the following best completes the interpretation of critical point A? “When the company employs 500
drivers and 225 customer service representatives, their profit is at a ________.”
a. relative minimum of -5 million dollars
b. relative minimum of 5 million dollars
c. relative maximum of -5 million dollars
d. relative maximum of 5 million dollars
42. How many critical points does the graph of ( , ) indicate?
a. two
b. three
c. four
d. five
Suppose the delivery company referred to in the
previous set of problems finds it necessary to
limit the number of employees according to
( , ) = −0.55 + = 0.1. The constraint
has been sketched on the graph below.
43. How many drivers and customer service representatives should the company employ under the constraint so as to
maximize profit?
a.
b.
c.
d.
490 drivers and 290 customer service representatives
150 drivers and 40 customer service representatives
100 drivers and 65 customer service representatives
100 drivers and 70 customer service representatives
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An MP3 song is downloaded from the internet, and a graph of the download rate for the first 10 seconds is given below.
Use this information to answer the next two questions.
44. Four seconds after the download starts, the amount of data downloaded is _____.
a.
b.
c.
d.
increasing least rapidly
increasing most rapidly
decreasing most rapidly
at a relative minimum
MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
45. The total amount of data downloaded is best represented by which of the following graphs?
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0
< 10
The producers of football jerseys will supply ( ) =
thousand jerseys when the jersey are
− 3 + 20
≥ 10
sold for dollars each. Check: (18) = 290
Use this model to answer the next two questions.
46. How many jerseys are supplied when the market price is $45 per jersey?
a. 6.72 thousand jerseys
b. 85.950 thousand jerseys
c. 213.59 thousand jerseys
d. 1910 thousand jerseys
47. Calculate the producer’s surplus when the price of a jersey is $45.
a. 27854.167 thousand dollars
c. 58095.833 thousand dollars
b. 28237.5 thousand dollars
d. 85950 thousand dollars
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The graph of a function ( ) is given below.
Let ( ) = ∫ ( ) define a particular accumulation
function of ( ). Answer the next three questions about
this accumulation function.
48. Which one of the following statements correctly describes the behavior of the accumulation function ( ) as you
move left to right across the interval 0 < < 2?
a.
b.
( ) decreases slower and slower
( ) decreases faster and faster
c.
d.
( ) increases slower and slower
( ) increases faster and faster
MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
49. Which of the following is NOT an accurate statement about the accumulation function?
a.
b.
( ) decreases on the interval 2 < < 5.
( ) has a relative minimum at = 2.
c.
d.
( ) has a zero at = 5.
( ) has no inflection points.
50. Given the graph of ( ), which one of the following is the graph of ( ) = ∫
( ) ?
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( ) . Check: (2) ≈ 0.714286
51. Given that ( ) = (
, use the numerical method to evaluate ∫
)
a. Diverges
b. 0
c. 1.180
d. 7.889
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The 550 contour curve for the function ( , ) = 2
+
is shown in the graph. Also shown is a line tangent to
the contour curve at the point (5,10). Use this information
to answer the next two questions.
52. Calculate the slope of the line tangent to the curve
at the point (5,10) . Round to 3 decimal places.
a.
b.
c.
d.
-0.262
-2.778
-3.750
-3.818
53. From the point (5,10), which of the following estimates how much
unit decrease in if the value of is to remain at 550?
a.
Δ ≈− ⋅
b.
( ,
)
Δ ≈−
( ,
)
c.
would need to change to compensate for a
Δ ≈− ⋅
( ,
)
d.
Δ ≈−
( ,
)
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MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
54. An oil well is producing oil at a rate of ( ) thousand barrels per year
∫ ( ) = 200.
years after production begins. Interpret
a. The well will continue to produce oil at the current rate for 200 years.
b. Eventually, the well will produce 200 thousand barrels of oil.
c. The well will produce 200 thousand barrels of oil per year forever.
d. The well will never produce 200 thousand barrels of oil.
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55. A farmer recently purchased a tract of land. He subdivided this land into ¼ acre plots, and measured the height in
feet above sea level at the midpoint of each plot. The data is given in the table below, where the rows and
columns are indicated with letters.
a. Draw and label the 5.0 and 10.0-foot contour curves on the data table. (Hint: There are 2 10.0-ft contours.)
b. The table indicates a relative __________________ in column ______ and row ______.
c. The table indicates a saddle point in column ______ and row ______.
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56. The number of rabbits living on the UVic campus was increasing at a rate of ( ) = 18.42(1.771 ) rabbits per
year, years after 1980, 0 ≤ ≤ 4.
Check: (4) = 181.2023687
a. Find ∫ ( )
=_____________________________
b. Use the fact that there were 60 rabbits in 1981 to find ( ),the specific anti-derivative of ( ). Complete the
model and show your work.
( ) =____________________________________
_____________ (units)
gives the _____________________________________________ years after 1980, 0 ≤ ≤ 4.
c. Show how each function can be used to find the change in the number of rabbits between 1981 and 1983.
(You only need to write the notation. It is not necessary to complete the calculation.)
Using ( ):
d. How many rabbits were on the campus in 1984?
Using ( ):
MATH 2070
57.
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
( , )℉ gives the average temperature in Clemson, SC, where
number of years after 2000.
is the month (Jan=1, Feb=2, etc.) and is the
a. Find a cross-sectional model for the average temperature in Clemson, SC as a function of in the year 2004.
Use appropriate notation and report your model with coefficients rounded to 3 decimal places. (Assume the
data is best represented by a quadratic model.)
b. Use the unrounded model found in part (a) to estimate the average temperature in July of 2004. Round your
answer to 3 decimal places and include units.
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58. Evaluate the following improper integral. If the integral diverges state ∞ or −∞, as appropriate, for your final
answer.
∫
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59. Given ( , ) =
+ 12 ⋅ ln( ) + 6 , find the following. It is not necessary to simplify your answer.
=________________________________
=________________________________
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60. Write a simplified value or algebraic expression for each of the following.
a.
∫ (5
b. ∫ 4 +6 ) MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
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61. The College of Western Idaho posted net profits of 125 million dollars in 2010. During the 5-year period from
January 1, 2011 to January 1, 2016, they will continuously invest some or all of their profits, as described below,
in an account bearing 2% interest, compounded continuously. Write the function, ( ), that describes the
continuous income stream for their investment years after the beginning of 2011 for each scenario. Include
units.
a. Assume the profit will decrease by 0.05 billion dollars per year and they will invest 60% of their profit.
( )=
b. Assume the profit will increase by 7% each year and they will invest all of their profit.
( )=
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62. Find each general anti-derivative. It is not necessary to simplify your answers.
a. ∫ √ + 4
b. ∫ 3
−
4 −8+
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63. The first partial derivatives of the function ( , ) are given by = 4
−
Use this information to find the following. It is not necessary to simplify.
a.
=_______________________________
b.
=_______________________________
and
=4
+
.
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64. At the end of 2006 a company began continuously investing 1.5 million dollars per year into an account returning
4% APR, compounded continuously. At the end of 2011 they will use the money from this investment account to
expand their company. How much money will they have available for the expansion? Show your work and
include units with your answer.
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MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
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65. The rate of change in the annual profit of a
major restaurant chain from 1970 through 2000
can be modeled by the equation ( ) =
−0.46 + 1.85 + 108.7 thousand dollars per
year, years after 1970.
a. On the graph provided, sketch and shade three right rectangles of equal width that can be used to estimate the
signed area between ( ) and the axis from = 0 to = 30.
b. Let refer to the width of each rectangle and ℎ , ℎ ,and ℎ refer to the signed height of each rectangle you
were asked to sketch (in order from left to right). Calculate each value and specify the units.
width units: _______________
height units: _________________________________
=_____________ ℎ =______________ ℎ =________________
ℎ =_______________
c. Use these three rectangles to estimate and interpret the signed area between ( ) and the
= 30.
axis from
= 0 to
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66. The quantity of CPUs that producers will supply is modeled by ( ) = 0.00087 − 0.01 + 24 thousand CPUs
when ≥ 50, 0 otherwise, where dollars per CPU is the price.
The quantity of CPUs demanded by consumers is modeled by ( ) = −0.0007
where dollars per CPU is the price.
− 0.07 + 54 thousand CPUs,
a. Find each price and quantity indicated below.
The market equilibrium price is $___________.
The market equilibrium quantity is ___________ thousand CPUs.
The price at which consumers will no longer purchase CPUs is $__________.
The minimum price which producers are willing to accept for CPUs is $__________.
b. Fill in each of the prices and quantity found above on the graph shown below.
MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
c. Find the amount producers are willing and able to receive when the market price is $80 per CPU. Show your
work. Shade the corresponding area on the graph below.
d. Find the amount consumers are willing and able to spend at market equilibrium. Show your work. Shade the
corresponding area on the graph below.
e. Find Total Social Gain. Show your work. Shade the corresponding area on the graph below.
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67. Use the algebraic method to find the EXACT value of the definite integral below. Clearly show each step in the
process. Use proper mathematical notation. Simplify your answer as much as possible without converting to
decimal form.
∫
−4
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MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
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68. Tailgate Essentials sells two types of tailgate tents, one plain tent and one with a team logo. The profit from the
sale of these tents is given by ( , ) = 83.45 + 176.32 − 0.17 − 0.22 − 0.34 − 11933.2 dollars,
when dollars is the price of a plain tent and dollars is the price of a tent with a team logo.
Check: (40,60) = 0
a. Find the first partial derivatives of ( , ).
=_______________________________
=_________________________________
b. Write out the system of equations that can be used to find the critical point of ( , ).
c. Solve the system to find the critical point of ( , ). Show all work. Round to the nearest cent.
Critical point:
= $__________
= $__________
= $________________
d. Find the second partials derivative matrix and the determinant value for the critical point of ( , ).
2nd Partial Derivatives Matrix:
Determinant Value = _____________
e. Classify the critical point as a relative maximum, relative minimum, or saddle point. Show how your answers
to part (c) support your conclusion.
MATH 2070
Final Exam Mixed Practice (MISSING 6.5 & 6.6)
69. Refer back to the profit model for Tailgate Essentials. Suppose Tailgate Essentials wants to restrict the price
difference between tents to that logo tent price is $80 more than the price of a plain tent. The price constraint is
given by ( , ) = − = 80.
a. Set up the system of equations that can be used to find the optimal point of ( , ) subject to the constraint.
b. Solve the system of equations to find the new optimal point. Show all work. Round to the nearest cent.
=_______________
=_______________
( , ) =__________________
=_____________
c. Use the close point test to classify the optimal point as either a maximum or a minim. The complete the table
and the sentence that follows.
( , )
Close point 1
125
Optimal Point
Close point 2
140
If Tailgate Essentials requires that the price of a logo tent be $80 more than the price of a plain tent, they will
have a _______________ (maximum or minimum) profit of $________________ when they sell plain tents
for $____________ each and logo tents for $____________ each.
d. Estimate the optimal profit earned by Tailgate Essentials if they increase the price difference they are willing
to allow between the two tents by $10. (Do NOT re-solve the problem. Estimate using previous solution.)
__________________________________________________________________________________________________
Note:
1. There may be questions on the final that are not included in this review.
2. Section 6.5 and 6.6 are not in the review, but will be on the final exam.
3. This review is longer than the length of your final exam.
4. This is the only document that has material from Chapters 5 – 8 mixed up together.
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