1-2 Identify the variables and parameters in the following situations,
give the units they might be measured in, and choose an appropriate
letter or symbol to represent each.
1.
A scientist measures the mass of fish over the course of 100 days, and
repeats the experiment at three different levels of salinity: 0%, 2% and
5%.
Get 1.2.1 exercise solution
2.
A scientist measures the body temperature of bandicoots every day
during the winter, and does so at three different altitudes: 500 m, 750
m, and 1000 m. Get 1.2.2 exercise solution
3-6 Compute the values of the following functions at the points
indicated and sketch a graph of the function.
3. f ( x ) = x + 5 at x = 0, x = 1, and x = 4.
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4. g ( y ) = 5 y at y = 0, y = 1, and y = 4.
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5.
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6. F ( r ) = r 2 + 5 at r = 0, r = 1, and r = 4.
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7-10 Graph the given points and say which point does not seem to fall on the graph of a simple function.
7. (0, -1), (1, 1), (2, 1), (3, 5), (4, 7).
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8. (0, 5), (1, 10), (2, 8), (3, 6), (4, 4).
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9. (0, 2), (1, 3), (2, 6), (3, 11), (4, 10).
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10. (0, 45), (1, 25), (2, 12), (3, 12.5), (4, 10). Get 1.2.10 exercise solution
11–14 Evaluate the following functions at the given algebraic arguments.
11. f ( x ) = x + 5 at x = a, x = a + 1, and x = 4 a .
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12. g ( y )= 5 y at y = x 2 , y = 2 x + 1, and y = 2 - x .
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13.
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14.
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15-16 Sketch graphs of the following relations. Is there a more convenient order for the arguments?
15.
A function whose argument is the name of a state and whose value is the
highest altitude in that state. State Highest Altitude (ft) California
14,491 Idaho 12,662 Nevada 13,143 Oregon 11,239 Utah 13,528 Washington
14,410
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16.
A function whose argument is the name of a bird and whose value is the
length of that bird. Bird Length Cooper’s hawk 50 cm Goshawk 66 cm
Sharp-shinned hawk 35 cm Get 1.2.16 exercise solution
17-20 For each of the following sums of functions, graph each component
piece. Compute the values at x = -2, x = -1, x = 0, x = 1, and x = 2 and
plot the sum.
17. f ( x ) 2 x + 3 and g ( x ) = 3 x - 5.
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18. f ( x ) = 2 x + 3 and h ( x ) = -3 x - 12.
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19. F ( x ) = x 2 + 1 and G ( x ) = x + 1.
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20.
F ( x )= x 2 + 1 and h ( x )= - x + 1. 21-24 For each of the following
products of functions, graph each component piece. Compute the value of
the product at x = -2, x = - 1, x = 0, x1, and x = 2 and graph the
result.
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21. f ( x ) = 2 x + 3 and g ( x ) = 3 x - 5.
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22. f ( x ) = 2 x + 3 and h ( x ) = -3 x - 12.
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23. F ( x ) = x 2 + 1 and G ( x ) = x + 1.
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24.
F ( x )= x 2 + 1 and h ( x )= - x + 1.
25-28 Find the inverses of each of the following functions. In each
case, compute the output of the original function at an input of 1.0,
and show that the inverse undoes the action of the function.
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25. f ( x ) = 2 x + 3.
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26. g ( x ) = 3 x - 5.
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27. G ( y )= 1/(2+ y ) for y = 0.
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28. F ( y )= y 2 + 1 for y = 0.
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29-32 Graph each of the following functions and its inverse. Mark the given point on the graph of each function.
29. f ( x ) = 2 x + 3. Mark the point (1, f (1)) on the graphs of f and f -1 (based on Exercise 25 ).
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30. g ( x ) = 3 x - 5. Mark the point (1, g (1)) on the graphs of g and g -1 (based on Exercise 26 ).
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31. G ( y ) = 1/(2 + y ). Mark the point (1, G (1)) on the graphs of G and G -1 (based on Exercise 27 ).
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32. F ( y )= y 2 + 1 for y = 0. Mark the point (1, F (1)) on the graphs of F and F -1 (based on Exercise 28 ).
33–36 Find the compositions of the given functions. Which pairs of functions commute?
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33. f ( x ) = 2 x + 3 and g ( x ) = 3 x - 5.
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34. f ( x ) = 2 x + 3 and h ( x ) = -3 x - 12.
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35. F ( x ) = x 2 + 1 and G ( x ) = x + 1.
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36. F ( x )= x 2 + 1 and h ( x ) = - x + 1. Get 1.2.36 exercise solution
Applications 37–40 Describe what is happening in the graphs shown.
37. A plot of cell volume against time in days.
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38. A plot of a Pacific salmon population against time in years.
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39. A plot of the average height of a population of trees plotted against age in years.
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40. A plot of an Internet stock price against time.
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41-44 Draw graphs based on the following descriptions.
41. A population of birds begins at a large value, decreases to a tiny value, and then increases again to an intermediate value.
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42.
The amount of DNA in an experiment increases rapidly from a very small
value and then levels out at a large value before declining rapidly to
0.
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43. Body temperature oscillates between high values during the day and low values at night.
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44.
Soil is wet at dawn, quickly dries out and stays dry during the day,
and then becomes gradually wetter again during the night.
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45-48 Evaluate the following functions over the suggested range, sketch a
graph of the function, and answer the biological question.
45.
The number of bees b found on a plant is given by b = 2 f + 1 where f
is the number of flowers, ranging from 0 to about 20. Explain what might
be happening when f = 0.
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46.
The number of cancerous cells c as a function of radiation dose r
(measured in rads) is c = r - 4 for r greater than or equal to 5, and is
zero for r less than 5. r ranges from 0 to 10. What is happening at r =
5 rads?
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47.
Insect development time A (in days) obeys represents temperature in °C
for 10 = T = 40. Which temperature leads to the more rapid development?
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48.
Tree height h (in meters) follows the formula where a represents the
age of the tree in years for 0 = a = 1000. How tall would this tree get
if it lived forever?
49–52 Consider the following data describing the growth of a tadpole.
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49. Graph length as a function of age.
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50. Graph tail length as a function of age.
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51. Graph tail length as a function of length.
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52. Graph mass as a function of length and then graph length as a function of mass. How do the two graphs compare?
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53-56 The following series of functional compositions describe connections between several measurements.
53.
The number of mosquitos ( M ) that end up in a room is a function of
how far the window is open ( W , in cm 2 ) according to M ( W ) = 5 W +
2. The number of bites ( B ) depends on the number of mosquitos
according to B ( M )= 0.5 M . Find the number of bites as a function of
how far the window is open. How many bites would you get if the window
was 10 cm 2 open?
Get 1.2.53 exercise solution
54.
The temperature of a room ( T ) is a function of how far the window is
open (W) according to T ( W ) = 40 - 0.2 W . How long you sleep ( S ,
measured in hours) is a function of the temperature according to . Find
how long you sleep as a function of how far the window is open. How long
would you sleep if the window was 10 cm 2 open?
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55.
The number of viruses ( V , measured in trillions or 10 12 ) that
infect a person is a function of the degree of immunosuppression ( I ,
the fraction of the immune system that is turned off by stress)
according to V ( I )= 5 I 2 . The fever ( F , measured in °C) associated
with an infection is a function of the number of viruses according to F
( V ) = 37 + 0.4 V . Find fever as a function of immunosuppression. How
high will the fever be if immunosuppression is complete ( I = 1)?
Get 1.2.55 exercise solution
56.
The length of an insect ( L , in mm) is a function of the temperature
during development ( T , measured in °C) according to The volume of the
insect ( V , in cubic mm) is a function of the length according to V ( L
)= 2 L 3 . The mass ( M in milligrams) depends on volume according to M
( V ) = 1.3 V . Find mass as a function of temperature. How much would
an insect weigh that developed at 25°C?
Get 1.2.56 exercise solution
57-58 Each of the following
measurements is the sum of two components. Find the formula for the sum.
Sketch a graph of each component and the total as functions of time for
0 = t = 3. Describe each component and the sum in words.
57.
A population of bacteria consists of two types a and b . The first
follows a ( t ) = 1 + t 2 , and the second follows b ( t )= 1 - 2 t + t 2
where populations are measured in millions and time is measured in
hours. The total population is P ( t )= a ( t )+ b ( t ).
Get 1.2.57 exercise solution
58.
The above-ground volume (stem and leaves) of a plant is , and the
below-ground volume (roots) is V b ( t ) = - 1.0 t + 40.0 where t is
measured in days and volumes are measured in cm 3 . The total volume is V
( t )= V a ( t ) + V b ( t ).
59–62 Consider the following data describing a plant.
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59. Graph M as a function of a . Does this function have an inverse? Could we use mass to figure out the age of the plant?
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60. Graph V as a function of a . Does this function have an inverse? Could we use volume to figure out the age of the plant?
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61.
Graph G as a function of a . Does this function have an inverse? Could
we use glucose production to figure out the age of the plant?
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62.
Graph G as a function of M . Does this function have an inverse? What
is strange about it? Could we use glucose production to figure out the
mass of the plant?
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63-66 The total mass of a population (in kg) as a
function of the time, t , is the product of the number of individuals, P
( t ), and the mass per person, W ( t ) (in kg). In each of the
following exercises, find the formula for the total mass, sketch graphs
of P ( t ), W ( t ), and the total mass as functions of time for 0 = t =
100, and describe the results in words.
63.
The population of people is P ( t ) = 2.0 × 10 6 + 2.0 × 10 4 t , and
the mass per person W ( t ) (in kg) is W ( t ) = 80 - 0.5 t .
Get 1.2.63 exercise solution
64. The population is P ( t ) = 2.0 × 10 6 - 2.0 × 10 4 t , and the mass per person W ( t ) is W ( t )= 80 + 0.5 t .
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65. The population is P ( t )= 2.0 × 10 6 + 1000 t 2 , and the mass per person W ( t ) is W ( t )= 80 - 0.5 t .
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66.
The population is P ( t )= 2.0 × 10 6 + 2.0 × 10 4 t , and the mass per
person W ( t ) is W ( t )= 80 - 0.005 t 2 .
Get 1.2.66 exercise solution
Computer Exercises
67-70 Have your graphics calculator or computer plot the following
functions. How would you describe them in words?
67.
a. f ( x ) = x 2 e - x for 0 = x = 20. b. g ( x )= 1.5 + e -0.1 x sin (
x ) for 0 = x = 20. c. h ( x ) = sin (5 x ) - cos (7 x ) for 0 = x = 5.
d. f ( x ) + h ( x ) for 0 = x = 20 (using the functions in parts a and
c ). e. g ( x )· h ( x ) for 0 = x = 20 (using the functions in parts b
and c ). f. h ( x )· h ( x ) for 0 = x = 20 (using the function in part
c ).
Get 1.2.67 exercise solution
68.
Have your computer plot the function h ( x )= e - x 2 - e -1000( x
-0.13) 2 - 0.2 for values of x between -10 and 10. a. How would you
describe the result in words? b. Blow up the graph by changing the range
to find all points where the value of the function is 0. For example,
one such value is between 1 and 2. Plot the function again for x between
1 and 2 to zoom in. c. If you only found 2 points where h ( x ) = 0,
blow up the region between 0 and 1 to try to find two more.
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69. Use your computer to find and plot the following functional compositions. a. b. c.
Get 1.2.69 exercise solution
70.
Have your graphics calculator or computer plot the following functions
for -2 < x < 2. Do they have inverses? a. h 2 ( x )= x 2 + 2 x .
b. h 3 ( x ) = x 3 + 2 x . c. h 4 ( x ) = x 4 + 2 x . d. h 5 ( x ) = x 5
+ 2 x . Have your computer try to find the formula for the inverses of
these functions and plot the results. Does the machine always succeed in
finding an inverse when there is one? Does it sometimes find an inverse
when there is none?