Math 103 Review Final Exam
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math103practicev01student.pdf
Math 103 Review for Final Exam
v01
1. Please simplify your answers as far as possible without a calculator. You may leave your answers in terms of exponential and logarithmic expressions.
(a) John invests $5,000 in an account with semi-annual compounding and an annual interest rate of 4%. How much money will he have after 35 years?
Answer with units:
(b) $7000 was deposited in a bank account earning 5.5% interest compounded continuously. How long will it take for the amount in the account to double?
Answer with units:
(c) Assume that the yearly rate of price inflation for housing in the the next ten years is 3.5% and assume that this is compounded annually. A house costing $780,000 today will cost more in ten years. How much will it cost in ten years?
Answer with units:
2. (a) What is the yield on a investment earning 10% compounded continuously? One of the following calculations will help you answer the question. Circle the calculation that helps.
Yield (as a percentage):
(1.10)2 = 1.2100 (1.05)2 = 1.1025 (1.025)4 = 1.1038
e0.10 = 1.1052 e0.05 = 1.0513 e1.10 = 3.0042
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(b) What is the present value of a cash flow of $1000 per year at a rate of 4%?
Present value (with units):
(c) What is the yield on a investment earning 8% compounded quarterly? One of the following calculations will help you answer the question. Circle the calculation that helps.
Yield (as a percentage):
(1.08)4 = 1.3605 (1.04)2 = 1.0816 (1.02)4 = 1.0824
e0.08 = 1.0833 4e.02 = 4.0808 e1.08 = 2.9447
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(d) What is the yearly cash flow from a perpetuity with a present value of $200,000. The annual rate of return is 8%.
Yearly cash flow (with units):
3. Grandpa Bill wants to buy an outdoor playground set with a swing and slide for his grandkids. That set costs $550 now. If he waits 2 years to buy it, how much will it cost if the yearly inflation rate for these playground sets is 5%?
Summary:
4
4. The cost (in dollars) to produce x floor tiles is
C(x) = 1500 + 2.50x.
Graph the cost function. Your graph should show the missing ticks marks and the y-intercept.
0 100 200 300 400 500 600 700 800 900 1000 Number of tiles0
?
?
?
?
?
3000
?
? Cost in dollars
5. The graph of f(x) = −2x2 −x + 4 is a parabola. Answer the following questions:
a. Does the parabola opens up or down?
b. What are the coordinates of the vertex?
coordinates of the vertex:
c. The function f(x) has a minimum � maximum � (check one box).
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6. The revenue from selling x file cabinets is given by a function R(x). Revenue is in dollars.
(a) Give a summary statement for R(10) = 92. Be sure to include the proper units in your statement. Summary:
(b) Give a summary statement for R′(10) = 5. Be sure to include the proper units in your statement. Summary:
7. The revenue, in dollars, from selling x electric guitars is given by a function
R(x) = 100x−x2
(a) What ist the change in revenue if the sales level changes from 10 to 12 guitars?
Answer with units:
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(b) What is the average rate of change in revenue if the sales level changes from 10 to 12 guitars?
Answer with units:
(c) Find the instantaneous rate of change in revenue at a sales level of 5 guitars.
Answer with units:
8. Widget General reports their monthly revenue, R(x), for sales of mini-widgets, where x is in hundreds of widgets and R is in dollars.
(a) Interpret this statement: R(20) = 860
Summary:
7
(b) Write the following statement in symbolic form:
At a sales level of 2000 widgets, the revenue is increasing at a rate of $3 per 100 widgets.
Symbolic form:
(c) On the basis of these results, what would be your estimate for the revenue of producing 2100 widgets?
Summary:
9. Let f(x) = x2 + 4x− 5.
a. Simplify the expression f(x + h) −f(x)
h
Simplified expression:
8
b. Find the limit lim h→0
f(x + h) −f(x) h
lim h→0
f(x + h) −f(x) h
=
10. Find the derivatives of the following functions.
(a) f(x) = 3x2 + 5x + 100
(b) g(x) = √
2x3 −x (Do not simplify.)
(c) r(x) = 3x− 2 2x + 5
(Simplify.)
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(d) What is the slope of the line tangent to the graph of h(x) = −x2 + 3x at the point (1, h(1))?
11. The average cost, in dollars, to produce x smoke detectors is
C̄(x) = 1000 + 5x
x .
a. What is the average cost if 100 smoke detectors are produced?
Average cost for 100 detectors, with units:
b. Graph the average cost function, C̄(x). Mark the horizontal asymptote and the point corre- sponding to a production level of 100 detectors. (The point corresponding to a production level of 200 detectors is shown on the graph.)
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0 100 200 300 400 500 600 Number of smoke detectors0
10
20
30
40
50 Average Cost
12. Dunder Mifflin manufactures paper. Producing x bales of paper has a fixed cost of $140 and a variable cost of $16 per bale. The price demand function is expressed by p(x) = 190−2x, where p(x) is the price (in dollars) at which x bales of paper can be sold.
(a) Write a formula for the revenue function, R(x).
R(x) =
(b) Write a formula for the cost function, C(x).
C(x) =
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13. BronKo grows and sells bales of hay for horses. The graphs of the revenue and cost functions are shown below.
(a) Do each of the following:
i. Mark the break-even points.
ii. Find the approximate coordinates of the break-even points.
iii. Mark a vertical line segment on the graph whose length represents the maximum profit and estimate the maximum profit.
0 10 20 30 40 Number of bales0
200
400
600
800
1000 Revenue, Cost
Coordinates of the break-even points:
(b) On the graph below, graph the cost function if fixed costs rise by $100.
0 10 20 30 40 Number of bales0
200
400
600
800
1000 Revenue, Cost
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14. A car was purchased for $15,000 and is assumed to have a value of $7,000 after 10 years. Use linear interpolation to estimate the value of the car after 6 years.
Value after 6 years:
15. The price-demand equation for selling telephones is
x + 0.1p = 500,
where x is the number of phones that can be sold at a price of p dollars.
(a) How many telephones can be sold at a price of $40 per telephone?
Answer with units:
(b) Find a formula for the revenue R(p) in terms of the price p.
R(p) =
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16. Amazon Airlines charges passengers for baggage. The charge for bags weighing 40 pounds or less is $50. The charge for bags weighing more than 40 pounds but less than or equal to 80 pounds is $100. The charge for bags weighing more than 80 pounds is $125 plus $1 for each pound over 80. The graph below shows the charge, H(x) that a passenger pays for a bag weighing x pounds.
ç
ç
æ
æ
0 40 80 120 160 Baggage weight0
25
50
75
100
125
150
175
200 Charge
a. What is the charge for a bag weighing 80 pounds?
Charge for bag weighing 80 pounds:
b. What is the charge for a bag weighing 100 pounds?
Charge for bag weighing 100 pounds::
c. For which value(s) of x is the payoff function discontinuous?
Discontinuous:
d. Find the limit limx→40 H(x) if it exists. If it does not exist, write DNE in the answer box.
limx→40 H(x) =
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17. Matador Wireless offers a data plan that charges $10.00 a month for access, with no additional charge if usage is at or below 75 MB. If a customer exceeds 75 MB of data, they are also charged 10 cents for each MB above 75. If a customer exceeds 125 MB of data, they are charged 40 cents for each MB above 125.
(a) What is the monthly cost to store 125 MB of data?
Answer with units:
(b) Write the formulas for the piecewise definition of the cost function C(x) for storing x MB of data a month. Fill in the boxes for the piecewise definition for the cost C(x) of storing x MB of data a month.
C(x) =
if 0 ≤ x ≤
if < x ≤
if x >
(c) Graph the cost function.
0 50 100 150 200 MB0
10
20
30
40 Monthly cost
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18. AnselPix is an on-line company that sells photographs of National Parks. The profit from selling x prints of a scene in Arches National Park is P (x) dollars. AnselPix believes that the profit from selling 100 photos will be $22,000. Assume that the marginal profit is P ′(100) = −1500.
(a) Use the marginal profit to estimate the profit if the number of prints changes from 100 to 101 prints.
Profit with units:
(b) As AnselPix’s financial advisor, would you recommend that they sell more or fewer than 100 prints? Why?
Why?
19. Vance Refrigeration company manufactures refrigerators. The revenue function is
R(x) = 2000x− 100x2,
where R(x) is measured in dollars, and x is the weekly demand for refrigerators. The cost of weekly production and delivery of x refrigerators is
C(x) = 150 + 1200x
where C(x) is measured in dollars.
(a) What is the profit if Vance produces and sells 3 refrigerators in a week?
Profit with units:
16
(b) Find the weekly production/sales level that maximizes profit.
Answer with units:
(c) If the production costs increase by $200 per refrigerator, then the weekly production/sales level that maximizes profit will � increase � decrease � remain the same.
20. Consider the revenue function R(x) = 250x−x2 for selling x widgets. Revenue is in dollars.
a. Find the change in revenue when the sales level changes from x = 10 to x = 20.
Change in revenue (include units) =
b. Find the average rate of change of revenue for this change in sales levels.
Average rate of change in revenue (include units) =
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21. The number of miles that a new car can travel on x gallons of gas is M(x). What are the units for the derivative M ′(x)?
Units:
22. The price-demand equation for hamburgers at a fast-food restaurant is x + 400p = 2000, where x is the number of hamburgers sold, and p is the price in dollars.
(a) Find the elasticity of demand E(p). E(p) =
(b) Find E(2). E(2) =
(c) When the price is p = $2, is the demand elastic, inelastic, or unitary?
� elastic � inelastic � unitary
(d) For what price p is the elasticity of demand unitary?
Price with units:
18
(e) Suppose the price increases by 1.5% from $2 per hamburger. Use the elasticity of demand to estimate how the demand changes. Express your answer as a percentage and state whether demand increases or decreases.
Summary:
19
23. Perform the following matrix calculation:[ 2 3 0 2
] + (−2)
[ 1 4 1 0
] =
24. Solve the following system of equations using any method. To receive credit, you must get the right answer, show your work, and show that your answer checks.
5x + 2y = 22 −2x + y = 2
x =
y =
20
25. Solve the following system of equations using any method. To receive credit, you must get the right answer, show your work, and show that your answer checks.
4a + 3b − c + 2d = 4 3b − c + d = −11 b + c − 2d = 0
6d = −6
a =
b =
c =
d =
26. DizzyLand and MountainMagic are theme parks that operate several roller coasters. To run their rides, they hire behind-the-scenes Technicians that test the tracks and coasters, ride Operators that secure passengers at the take-off point, as well as line Coordinators that assist guests’ transition to the ride while waiting in line. Here are the labor-hour and wage requirements for operating two roller coasters, Olympus Bobsleys and Space Hill, that each park happens to feature:
Labor Requirements (in hours)
Technicians Operators Coordinators Olympus Bobsleys 1 8 4
Space Hill 1 4 6
Wage Requirements (in dollars per hour)
DizzyLand MountainMagic Technicians 40 30 Operators 15 10
Coordinators 12 8
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The labor-hours and wage information is summarized in the following matrices:
M =
[ 1 8 4 1 4 6
] N =
40 3015 10
12 8
(a) Compute the missing entries of the matrix product MN.
MN =
208
172
(b) What is the (2, 1)-entry of matrix MN?
(MN)21 =
(c) The (1, 1) entry of MN is 208. Interpret the meaning of this in a summary statement.
Summary:
22
