Exercise 2
1.5. p75 (100): 23, 25, 27, 33, 35, 47, 48, 51, 53, 55, 65.
1.6. p88 (113): 13, 37, 39, 43, 46, 57, 59.
Chapter summary. p93 (118): 6, 17, 29, 31, 40, 45, 52, 55.
2.1. p113 (130): 19, 23, 37, 41, 43, 44, 45, 46, 49.
2.2. p128 (153): 24, 33, 52, 55, 57, 58, 59, 67, 77.
2.3. p142 (167): 13, 25, 46, 53, 55, 57, 69.
2.4. p156 (181): 17, 35, 38, 48, 49, 65, 72, 79, 84, 90.
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Extra works
1. Find derivative of the following functions:
a) 𝑦𝑦 = 𝑥𝑥 2 . (1 + 2√3𝑥𝑥)
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b) 𝑦𝑦 =
2
c) 𝑦𝑦 = �√𝑥𝑥 + √𝑥𝑥 �
e) 𝑦𝑦 = 5𝑡𝑡(𝑡𝑡 − 1)(2𝑡𝑡 + 3)
2. Find 𝑦𝑦′′:
a) 𝑦𝑦 = 𝑥𝑥 4 +
3
𝑥𝑥
𝑥𝑥 3 − 2𝑥𝑥+1
3−5𝑥𝑥
√
𝑒𝑒 𝑥𝑥 � 𝑥𝑥
d) 𝑦𝑦 =
√ − 1�
f) 𝑓𝑓 (𝑥𝑥) = 12(2𝑥𝑥 + 1)2/3 (3𝑥𝑥 − 4)5/4
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b) 𝑦𝑦 = �(𝑥𝑥 2 + 1)3
c) 𝑦𝑦 = (𝑥𝑥 3 − 2)(5𝑥𝑥 + 1)
3. For 𝑔𝑔(𝑥𝑥) = 6𝑥𝑥 5 + 2𝑥𝑥 4 − 4𝑥𝑥 3 + 7𝑥𝑥 2 − 8𝑥𝑥 + 3, find 𝑔𝑔(7) (𝑥𝑥).
4. Draw a graph that is continuous, but not differentiable, at 𝑥𝑥 = 3.
5. Draw a graph that has a horizontal tangent line at 𝑥𝑥 = 5.
6. Oxygen Concentration. Suppose a small pond normally contains 12 units of
dissolved oxygen in a fixed volume of water. Suppose also that at time 𝑡𝑡 = 0 a quantity
of organic waste is introduced into the pond, with the oxygen concentration 𝑡𝑡 weeks
later given by
12𝑡𝑡 2 − 15𝑡𝑡 + 12
𝑓𝑓(𝑡𝑡 ) =
.
𝑡𝑡 2 + 1
As time goes on, what will be the ultimate concentration of oxygen? Will it return to
12 units?
7. Consumer Demand. When the price of an essential commodity (such as gasoline)
rises rapidly, consumption drops slowly at first. If the price continues to rise, however,
a “tipping” point may be reached, at which consumption takes a sudden substantial
drop. Suppose the accompanying graph shows the consumption of gasoline 𝐺𝐺(𝑡𝑡), in
millions of gallons, in a certain area. We assume that the price is rising rapidly. Here 𝑡𝑡
is time in months after the price began rising. Use the graph to find the following.
a) lim 𝐺𝐺(𝑡𝑡)
𝑥𝑥→12
b) lim 𝐺𝐺(𝑡𝑡)
𝑥𝑥→16
d) The tipping point (in months)
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c) 𝐺𝐺(16)
8. Employee Productivity. A company training program has determined that, on the
average, a new employee produces 𝑃𝑃(𝑠𝑠) items per day after 𝑠𝑠 days of on-the-job
training, where
63𝑠𝑠
.
𝑃𝑃(𝑠𝑠) =
𝑠𝑠 + 8
Find and interpret lim 𝑃𝑃(𝑠𝑠).
𝑠𝑠→∞
9. Car Rental. Recently, a car rental firm charged $36 per day or portion of a day to
rent a car for a period of 1 to 5 days. Days 6 and 7 were then free, while the charge for
days 8 through 12 was again $36 per day. Let 𝐴𝐴(𝑡𝑡) represent the average cost to rent
the car for 𝑡𝑡 days, where 0 < 𝑡𝑡 ≤ 12. Find the average cost of a rental for the following
number of days.
a) 4
b) 5
c) 6
d) 7
e) 8
f) Find lim− 𝐴𝐴(𝑡𝑡) , lim+ 𝐴𝐴(𝑡𝑡)
g) Where is 𝐴𝐴 discontinuos on the given interval?
𝑡𝑡→5
𝑡𝑡→5
10. Use the following graph to find the average rate of change of the U.S. trade deficit
with Japan from 1990 to 1995, from 1995 to 2000, and from 2000 to 2009.
11. Utility. Utility is a type of function that occurs in economics. When a consumer
receives 𝑥𝑥 units of a certain product, a certain amount of pleasure, or utility 𝑈𝑈, is
derived. The following is a graph of a typical utility function
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a) Find the average rate of change of U as 𝑥𝑥 changes from 0 to 1; from 1 to 2; from 2
to 3; from 3 to 4.
b) Why do you think the average rates of change are decreasing as increases?
12. Memory. The total number of words, 𝑀𝑀(𝑡𝑡 ), that a person can memorize in t
minutes is shown in the following graph.
a) Find the average rate of change of 𝑀𝑀 as 𝑡𝑡 changes from 0 to 8; from 8 to 16; from
16 to 24; from 24 to 32; from 32 to 36.
b) Why do the average rates of change become 0 after 24 min?
13. Rising cost of college. Like the cost of most things, the cost of a college education
has gone up over the past 35 years. The graphs below display the yearly costs of 4year colleges in 2008 dollars—indicating that the costs prior to 2008 have been
adjusted for inflation.
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a) In what school year did the cost of a private 4-year college increase the most?
b) In what school year(s) did the cost of a public 4-year college increase the most?
c) Assuming an annual inflation rate of 3%, calculate the cost of a year at a public and
at a private 4-year college in 1975. Express the costs in 1975 dollars.
14. Business: growth of an investment. A company determines that the value of an
investment is 𝑉𝑉, in millions of dollars, after time 𝑡𝑡, in years, where 𝑉𝑉 is given by
𝑉𝑉 (𝑡𝑡 ) = 5𝑡𝑡 3 − 30𝑡𝑡 2 + 45𝑡𝑡 + 5√𝑡𝑡
a) Graph 𝑉𝑉 over the interval [0, 5].
b) Find the equation of the secant line passing through the points (1, 𝑉𝑉 (1)) and
�5, 𝑉𝑉 (5)�. Then graph this secant line using the same axes as in part (a).
c) Find the average rate of change of the investment between year 1 and year 5.
d) Approximate the rate at which the value of the investment is changing after 1 yr.
15. Baseball ticket prices. The average price, in dollars, of a ticket for a Major League
baseball game 𝑥𝑥 years after 1990 can be estimated by
𝑝𝑝(𝑥𝑥) = 9.41 − 0.19𝑥𝑥 + 0.09𝑥𝑥 2
a) Find the rate of change of the average ticket price with respect to the year, 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑.
b) What is the average ticket price in 2010?
c) What is the rate of change of the average ticket price in 2010?
16. Gross domestic product. The U.S. gross domestic product (in billions of dollars)
can be approximated using the function
𝑃𝑃(𝑡𝑡) = 567 + 𝑡𝑡(36𝑡𝑡 0.6 − 104),
where 𝑡𝑡 is the number of the years since 1960.
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a) Find 𝑃𝑃′ (𝑡𝑡 ).
b) Find 𝑃𝑃′ (45).
c) In words, explain what 𝑃𝑃′ (45) means.
17. Temperature during an illness. The temperature 𝑇𝑇 of a person during an illness is
given by
4𝑡𝑡
𝑇𝑇(𝑡𝑡 ) = 2
+ 98.6,
𝑡𝑡 + 1
where 𝑇𝑇 is the temperature, in degrees Fahrenheit, at time 𝑡𝑡, in hours.
a) Find the rate of change of the temperature with respect to time.
b) Find the temperature at 𝑡𝑡 = 2 hr.
c) Find the rate of change of the temperature at 𝑡𝑡 = 2 hr.
18. Utility. Utility is a type of function that occurs in economics. When a consumer
receives 𝑥𝑥 units of a product, a certain amount of pleasure, or utility, 𝑈𝑈, is derived.
Suppose that the utility related to the number of tickets 𝑥𝑥 for a ride at a county fair is
𝑈𝑈(𝑥𝑥) = 80 �
2𝑥𝑥 + 1
.
3𝑥𝑥 + 4
Find the rate at which the utility changes with respect to the number of tickets bought.
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19. Chemotherapy. The dosage for Carboplatin chemotherapy drugs depends on
several parameters of the particular drug as well as the age, weight, and sex of the
patient. For female patients, the formulas giving the dosage for such drugs are
𝐷𝐷 = 0.85𝐴𝐴(𝑐𝑐 + 25)
and
𝑤𝑤
𝑐𝑐 = (140 − 𝑦𝑦)
72𝑥𝑥
where 𝐴𝐴 and 𝑥𝑥 depend on which drug is used, 𝐷𝐷 is the dosage in milligrams (mg), 𝑐𝑐 is
called the creatine clearance, 𝑦𝑦 is the patient’s age in years, and 𝑤𝑤 is the patient’s
weight in kilograms (kg). (Source: U.S. Oncology.)
a) Suppose that a patient is a 45-year-old woman and the drug has parameters 𝐴𝐴 = 5
and 𝑥𝑥 = 0.6. Use this information to write formulas for 𝐷𝐷 and 𝑐𝑐 that give 𝐷𝐷 as a
function of 𝑐𝑐 and 𝑐𝑐 as a function of 𝑤𝑤.
b) Use your formulas from part (a) to compute 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑.
c) Use your formulas from part (a) to compute 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑.
d) Compute 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑.
e) Interpret the meaning of the derivative 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑.
20. Free Fall. When an object is dropped, the distance it falls in seconds, assuming that
air resistance is negligible, is given by
𝑠𝑠(𝑡𝑡 ) = 4.905𝑡𝑡 2 ,
where 𝑠𝑠(𝑡𝑡) is in meters (m). If a stone is dropped from a cliff, find each of the following,
assuming that air resistance is negligible:
a) how far it has traveled 5 sec after being dropped.
b) how fast it is traveling 5 sec after being dropped.
c) the stone’s acceleration after it has been falling for 5 sec.
21. The following graph describes a bicycle racer’s distance from a roadside television
camera.
a) When is the bicyclist’s velocity the greatest? How can you tell?
b) Is the bicyclist’s acceleration positive or negative? How can you tell?
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