Week #4 - The Derivative
Section 2.4
From “Calculus, Single Variable” by Hughes-Hallett, Gleason, McCallum et. al.
Copyright 2005 by John Wiley & Sons, Inc.
This material is used by permission of John Wiley & Sons, Inc.
SUGGESTED PROBLEMS
1. The temperature, H, in degrees Celsius, of a cup of coffee placed on the kitchen counter
, where t is in minutes since the coffee was put on the counter.
is given by
(a) Is
positive or negative? Give a reason for your answer.
(b) What are the units of
perature of the coffee?
? What is its practical meaning in terms of the tem-
(a) As the cup of coffee cools, the temperature decreases, so f ′ (t) is negative.
(b) Since f ′ (t) = dH/dt, the units are degrees Celsius per minute. The quantity f ′ (20)
represents the rate at which the coffee is cooling, in degrees per minute, 20 minutes
after the cup is put on the counter.
4. The time for a chemical reaction, T (in minutes), is a function of the amount of catalyst
.
present, a (in milliliters), so
(a) If
, what are the units of 5? What are the units of 18? What does this
statement tell us about the reaction?
(b) If
, what are the units of 5? What are the units of
statement tell us?
? What does this
(a) The statement f (5) = 18 means that when 5 milliliters of catalyst are present, the
reaction will take 18 minutes. Thus the units for 5 are ml while the units for 18 are
minutes.
(b) As in part (1), 5 is measures in ml. Since f ′ tells how fast T changes per unit a, we
have f ′ measured in minutes/ml. If the amount of catalyst increases by 1 ml (from
5 to 6 ml), the reaction time decreases by about 3 minutes.
be the elevation in feet of the Mississippi river x miles from its source. What
9. Let
? What can you say about the sign of
?
are the units of
The units of f ′ (x) are feet/mile. The derivative, f ′ (x), represents the rate of change of
elevation with distance from the source, so if the river is flowing downhill everywhere, the
elevation is always decreasing and f ′ (x) is always negative. (In fact, there may be some
stretches where the elevation is more and less constant, so f ′ (x) = 0.)
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11. A laboratory study investigating the relationship between diet and weight in adult humans
, of the average
found that the weight of a subject, W , in pounds, was a function,
number of Calories per day, c, consumed by the subject.
(a) Interpret the statements
of diet and weight.
,
(b) What are the units of
, and
in terms
?
(a) Since W = f (c) where W is weight in pounds and c is the number of Calories
consumed per day:
– f (1800) = 155 means that consuming 1800 Calories per day results in a weight
of 155 pounds
– f ′ (2000) = 0 means that consuming 2000 Calories per day causes neither weight
gain nor loss
– f −1 (162) = 2200 means that a weight of 162 is caused by a consumption of 2200
Calories per day
(b) The units of dW/dc are pounds/(Calories/day).
13. If t is the number of years since 1993, the population, P , of China, in billions, can be
approximated by the function
Estimate
and
population of China?
, giving units. What do these two numbers tell you about the
Since f (t) = 1.15(1.014)t , we have
f (6) = 1.15(1.014)6 = 1.25 billion people
Interpretation: in 1993 + 6 = 1999, the population of China will be approximately 1.25
billion people.
To estimate f ′ (6), we use a small interval around 6:
f (6.001) − f (6)
f.006 − 6
1.15(1.014)6.001 − 1.15(1.014)6
=
0.001
≃ 0.0174 billion people per year
f ′ (6) ≃
In 1999, the population of China is growing at roughly 0.0174 billion people per year, or
17.4 million people per year.
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16. Let
be the pressure in dynes per
on a diver at a depth of h meters below the
surface of the ocean. What do each of the following quantities mean to the diver? Give
units for the quantities.
(a)
(b) h such that
(c)
(d)
(e)
(f ) h such that
The pressure is in dynes/cm2 at a depth of 100 meters.
The depth of water in meters giving a pressure of 1.2 × 106 dynes/cm2 .
The pressure at a depth of h meters plus a pressure of 20 dynes/cm2 .
The pressure at a depth of 20 meters below the diver.
The rate of increase of pressure with respect to depth, at 100 meters, in units of
dynes/cm2 per meter. Approximately, p′ (100) represents the increase in pressure in
going from 100 meters to 101 meters.
(f) the depth, in meters, at which the rate of change of pressure with respect to depth
is 20 dynes/cm2 per meter.
(a)
(b)
(c)
(d)
(e)
is the fuel efficiency, in miles per gallon, of a car going at v miles per hour, what
17. If
are the units of
? What is the practical meaning of the statement
?
Units of g′ (55) are mpg/mph. The statement g′ (55) = −0.54 means that at 55 miles per
hour the fuel efficiency (in miles per gallon, or mpg) of the car decreases at a rate of
approximately one half mpg as the velocity increases by one mph.
21. When you breathe, a muscle (called the diaphragm) reduces the pressure around your
lungs and they expand to fill with air. The table shows the volume of a lung as a function
of the reduction in pressure from the diaphragm. Pulmonologists (lung doctors) define the
compliance of the lung as the derivative of this function.
(a) What are the units of compliance?
(b) Estimate the maximum compliance of the lung.
(c) Explain why the compliance gets small when the lung is nearly full (around 1 liter).
Pressure reduction Volume
(cm of water)
(liters)
0
0.20
5
0.29
10
0.49
15
0.70
20
0.86
25
0.95
30
1.00
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(a) The units of compliance are units of volume per units of pressure, or liters per
centimeter of water.
(b) The increase in volume for a 5 cm reduction in pressure is largest between 10 and
15 cm. Thus the compliance appears maximum between 10 and 15 cm of pressure
reduction. The derivative is given by the slope, so
Compliance ≃
0.70 − 0.49
= 0.042 liters per centimeter
15 − 10
(c) When the lung is nearly full, it cannot expand much more to accommodate more
air.
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