1-10 Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so.
1. 43.2 0
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2. 43.2 1
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3. 43.2 -1
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4. 43.2 -0.5 + 43.2 0.5
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5. 43.2 7.2 /43.2
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6. 2 6 43.2 0.23 · 43.2 0.77
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7. (3 4 ) 0.5
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8. (43.2 -1/8 ) 16
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9. 2 2 3 · 2 2 2
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10.
4 2 · 2 4
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11-22 Use the laws of logs to rewrite the following if possible. If no
law of logs applies or the quantity is not defined, say so.
11. ln(1)
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12. ln(-6.5)
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13. log 432 43.2
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14. log 10 (3.5 + 6.5)
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15. log 10 (5) + log 10 (20)
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16. log 10 (0.5) + log 10 (0.2)
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17. log 10 (500) - log 10 (50)
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18. log 43.2 (5· 43.2 2 ) - log 43.2 (5)
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19. log 432 (43.2 7 )
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20. log 432 (43.2 7 ) 4
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21. Using the fact that log 7 43.2 = 1.935, find log 7
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22. Using the fact that log 7 43.2 = 1.935, find log 7 (43.2) 3 .
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23-26 Solve the following equations for x and check your answer.
23. 7 e 3 x = 21.
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24. 4 e 2 x +1 = 20.
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25. 4 e -2 x +1 = 7 e 3 x .
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26.
4 e 2 x +3 = 7 e 3 x -2 .
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27-30 Sketch graphs of the following exponential functions. For each,
find the value of x where it is equal to 7.0. For the increasing
functions, find the doubling time, and for the decreasing functions,
find the half-life. For what value of x is the value of the function
3.5? For what value of x is the value of the function 14.0?
27. e 2 x
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28. e -3 x
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29. 5e 0.2 x
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30. 0.1 e -0.2 x
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31-32 Sketch graphs of the following updating functions over the given range and mark the equilibria.
31. h ( z = e -z for 0 = z = 2.
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32. F ( x ) = ln( x ) + 1 for 0 = x = 2. (Although this cannot be solved algebraically, you can guess the answer.)
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33-36 Find the equations of the lines after transforming the variables to create semilog or double-log plots.
33. Suppose M ( t = 43.2 e 5.1 t . Find the slope and intercept of ln( M ( t )).
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34. Suppose L ( t = 0.72 e -2.34 t . Find the slope and intercept of ln( L ( t )).
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35.
Suppose M ( t ) = 43.2 e 5.1 t and S ( t )= 18.2 e 4.3 t . Find the
slope and intercept of ln ( M ( t )) as a function of ln (S( t )).
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36.
Suppose L ( t )= 0.72 e -2.34 t and K ( t )= 4.23 e 0.91 t . Find the
slope and intercept of ln ( L ( t )) as a function of ln ( K ( t )).
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37-40 Find the solution of each discrete-time dynamical system, express
it in exponential notation, and solve for the time when the value
reaches the given target. Sketch a graph of the solution.
37.
A population follows the discrete-time dynamical system b t +1 = rb t
with r = 1.5 and b 0 = 1.0 x 10 6 . When will the population reach 1.0 x
10 7 ?
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38.
A population follows the discrete-time dynamical system b t +1 = rb t
with r = 0.7 and b 0 = 5.0 × 10 5 . When will the population reach 1.0 ×
10 5 ?
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39.
Cell volume follows the discrete-time dynamical system v t +1 = 1.5 v t
with initial volume of 1350 µ m 3 (as in Exercise 37 ). When will the
volume reach 3250 µ m 3 ?
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40.
Gnat number follows the discrete-time dynamical system n t +1 = 0.5 n t
with an initial population of 5.5 × 10 4 . When will the population
reach 1.5 × 10 3 ?
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41-44 Suppose the size of an organism at time t is given by S ( t )= S 0
e at where S 0 is the initial size. Find the time it takes for the
organism to double or quadruple in size in the following circumstances.
41. S 0 = 1.0 cm and a = 1.0/day.
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42. S 0 = 2.0 cm and a = 1.0/day.
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43. S 0 = 2.0 cm and a = 0.1/hour.
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44.
S 0 = 2.0 cm and a= 0.0/hour
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45-48 The amount of carbon-14 (14 c ) left t years after the death of an
organism is given by Q ( t = Q 0 e -0000122 t where Q 0 is the amount
left at the time of death. Suppose Q 0 = 6.0 × 10 10 14 c atoms.
45. How much is left after 50,000 years? What fraction is this of the original amount?
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46. How much is left after 100,000 years? What fraction is this of the original amount?
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47. Find the half-life of 14 c .
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48.
About how many half-lives will occur in 50,000 years? Roughly what
fraction will be left? How does this compare with the answer of Exercise
45 ?
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49-52 Suppose a population has a doubling time of 24 years and an
initial size of 500.
49. What is the population in 48 years?
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50. What is the population in 12 years?
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51. Find the equation for population size P ( t ) as a function of time.
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52.
Find the one-year discrete-time dynamical system for this population
(figure out the factor multiplying the population in one year).
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53-56
Suppose a population is dying with a half-life of 43 years. The initial
size is 1600.
53. How long will it take to reach 200?
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54. Find the population in 86 years.
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55. Find the equation for population size P ( t ) as a function of time.
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56.
Find the one year discrete-time dynamical system for this population
(figure out the factor multiplying the population in one year).
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57-60 Plot semilog graphs of the values.
57.
The growing organism in Exercise 41 for 0 = t = 10. Mark where the
organism has doubled in size and when it has quadrupled in size.
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58. The carbon- 14 in Exercise 45 for 0 = t = 20, 000. Mark where the amount of carbon has gone down by half.
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59. The population in Exercise 49 for 0 = t = 100. Mark where the population has doubled.
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60.
The population in Exercise 53 for 0 = t = 100. Mark where the
population has gone down by half.
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61-64 The following pairs of measurements can be described by ordinary,
semilog, and double-log graphs. a. Graph each measurement as a function
of time on both ordinary and semilog graphs. b. Graph the second
measurement as a function of the first on both ordinary and double-log
graphs.
61.
The antler size A ( t ) in centimeters of an elk increases with age t
in years according to A( t ) = 53.2 e 0.17 t and its shoulder height L (
t ) increases according to L ( t ) = 88.5 e 0.1 t .
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62.
Suppose a population of viruses in an infected person grows according
to V ( t ) = 2.0 e 2.0 t and that the immune response (described by the
number of antibodies) increases according to I ( t ) = 0.01 e 3.0 t
during the first week of an infection. When will the number of
antibodies equal the number of viruses?
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63.
The growth of a fly in an egg can be described allometri-cally (see H.
F.Nijhout and D. E. Wheeler, 1996). During growth, two imaginal disks
(the first later becomes the wing and the second becomes the haltere)
expand according to S 1 ( t ) = 0.007 e 0.1 t and S 2 ( t ) = 0.007 e
0.4 t where size is measured in mm 3 and time is measured in days.
Development takes about 5 days.
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64.
While the imaginal disks are growing ( Exercise 63 ), the yolk of the
egg is shrinking according to Y ( t ) = 4.0 e -1.2 t . Create graphs
comparing S 1 ( t ) and Y ( t ).
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65-66 For each of the given shapes, find the constant c in the power
relationship S = cV 2/3 between the surface area S and volume V . By how
much is does c exceed the value (36 p ) 1/3 = 4.836 for the sphere
(which is in fact the minimum for any shape).
65. For a cube with side length w .
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66.
For a cylinder with radius r and height 3 r .
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67-68 Many measurements in biology are related by power functions. For
each of the following, graph the second measurement as a function of the
first on both ordinary and double-log graphs.
67.
The -3/2 law of self-thinning in plants argues that the mean weight W
of surviving trees in a stand increases while their number N decreases,
related by W = cN -3/2 . Suppose 10 4 trees start out with mass of 0.001
kg. Graph the relationship,and find how heavy the trees would be when
only 100 remain alive, and again when only 1 remains alive. Is the total
mass larger or smaller than when it started?
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68.
Suppose that the population density D of a species of mammal is a
decreasing function of its body mass M according to the relationship D =
cM -3/4 . Suppose that an unlikely 1 g mammal would have a density of
10 4 per hectare. What is the predicted density of species with mass of
1000 g? A species with a mass of 100 kg? According to the metabolic
scaling law ( Example 1.7.24 ), which species will use the most energy?
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69.
Use your computer to find the following. Plot the graphs to check. a.
The doubling time of S 1 ( t ) = 3.4 e 0.2 t . b. The doubling time of S
2 ( t ) = 0.2 e 3.4 t . c. The half-life of H 1 ( t ) = 3.4 e -0.2 t .
d. The half-life of H 2 ( t ) = 0.2 e -3.4 t .
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70.
Solve for the times when the following hold. Plot the graphs to check
your answer. a. S 1 ( t ) = S 2 ( t ) with S 1 and S 2 from the previous
problem. b. H 1 ( t ) = 2 H 2 ( t ) with H 1 and H 2 from the previous
problem. c. H 1 ( t ) = 0.5 H 2 ( t ) with H 1 and H 2 from the previous
problem.
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71.
Plot the following functions. a. ln( x ) for 10 = x = 100,000. b.
ln[ln( x )] for 10 = x = 100,000. c. ln[ln[ln( x )]] for 10 = x =
100,000. d. e x for 0 = x = 2. e. e e x for 0 = x = 2. f. e e e x for 0 =
x = 2. Will your machine let you do it? Can you compute the value of e e
e 2 ?
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72. Compute the following. Does this give you any idea why e is special? a. 20.001 b. 10 0.001 c. 0.5 0.001 d. e 0.001
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73.
Suppose that the antler size A ( t ) in centimeters of an elk increases
with age in years during the first five years of growth according to
the exponential function A ( t ) = 53.2 e 0.17 t and that the shoulder
height L ( t ) increases according to L ( t ) = 88.5 e 0.1 t . a. Plot A
and L as functions of t on ordinary and on semilog graphs. b. Plot A as
a function of L with an ordinary and with a semilog graph. c. Find when
the antler size would exceed the shoulder height of the elk.
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Modeling dynamic of life. Calculus and probabilities for life scientists - Adler - Chapter 1.6 Solutions
1-2 The following steps are used to build a cobweb diagram. Follow them
for the given discrete-time dynamical system based on bacterial
populations. a. Graph the updating function. b. Use your graph of the
updating function to find the point ( b 0 , b 1 ). c. Reflect it off the
diagonal to find the point ( b 1 , b 1 ). d. Use the graph of the
updating function to find ( b 1 , b 2 ). e. Reflect off the diagonal to
find the point ( b 2 , b 2 ). f. Use the graph of the updating function
to find ( b 2 , b 3 ). g. Sketch the solution as a function of time.
1. The discrete-time dynamical system b t +1 = 2.0b t with b 0 = 1.0.
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2. The discrete-time dynamical system n t +1 = 0.5 n t with n 0 = 1.0.
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3-6 Cobweb the following discrete-time dynamical systems for three steps starting from the given initial condition. Compare with the solution found earlier.
3. v t+1 = 1.5 v t , starting from v 0 = 1220 µm 3 (as in Section 1.5, Exercise 5 ).
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4. l t +1 = l t - 1.7, starting from l 0 = 13.1 cm (as in Section 1.5, Exercise 6 ).
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5. n t +1 = 0.5 n t , starting from n 0 = 1200 (as in Section 1.5, Exercise 7 ).
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6. M t +1 = 0.75 M t + 2.0 starting from the initial condition M 0 = 16.0 (as in Section 1.5, Exercise 8 ).
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7-12 Graph the updating functions associated with the following discrete-time dynamical systems, and cobweb for five steps starting from the given initial condition.
7. x t +1 = 2 x t - 1, starting from x 0 = 2.
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8. Z t +1 = 0.9 z t + 1, starting from z 0 = 3.
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9. w t +1 = -0.5 w t + 3, starting from w 0 = 0.
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10. x t +1 = 4 - x t , starting from x 0 = 1 (as in Section 1.5, Exercise 25 ).
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11. starting from x 0 = 1 (as in Section 1.5, Exercise 23 ).
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12. for x t > 1, starting from x 0 = 3 (as in Section 1.5, Exercise 26 ).
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13-16 Find the equilibria of the following discrete-time dynamical system from the graphs of their updating functions Label the coordinates of the equilibria.
13.
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14.
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15.
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16.
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17-22 Sketch graphs of the following updating functions over the given range and mark the equilibria. Find the equilibria algebraically if possible.
17. f ( x ) = x 2 for 0 = x = 2.
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18. g ( y )= y 2 - 1 for 0 = y = 2. 19-22 Graph the following discrete-time dynamical systems. Solve for the equilibria algebraically, and identify equilibria and the regions where the updating function lies above the diagonal on your graph.
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19. c t +1 = 0.5 c t + 8.0, for 0 = c t = 30.
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20. b t +1 = 3 b t , for 0 = b t = 10.
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21. b t +1 =0.3 b t , for 0 = b t = 10.
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22. b t +1 = 2.0 b t - 5.0, for 0 = b t = 10.
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23-30 Find the equilibria of the following discrete-time dynamical systems. Compare with the results of your cobweb diagram from the earlier problem.
23. v t +1 = 1.5 v t (as in Section 1.5, Exercise 5 ).
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24. l t+1 = l t - 1.7 (as in Section 1.5, Exercise 6 ).
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25. x t+1 = 2 x t - 1 (as in Exercise 7 ).
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26. Z t +1 = 0.9 z t + 1 (as in Exercise 8 ).
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27. w t +1 = -0.5 w t + 3 (as in Exercise 9 ).
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28. x t +1 = 4 - x t (as in Exercise 10 ).
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29. (as in Exercise 11 ).
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30. (as in Exercise 12 ).
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31-34 Find the equilibria of the following discrete-time dynamical systems that include parameters. Identify values of the parameter for which there is no equilibrium, for which the equilibrium is negative, and for which there is more than one equilibrium.
31. w t +1 = aw t + 3.
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32. x t +1 = b - x t .
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33.
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34.
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35-40 Cobweb the following discrete-time dynamical systems for five steps starting from the given initial condition.
35. An alternative tree growth discrete-time dynamical system with form h t +1 = h t + 5.0 with initial condition h 0 = 10.
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36. The lizard-mite system ( Example 1.5.3 ) x t +1 = 2 x t + 30 with initial condition x 0 = 0.
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37. The model defined in Section 1.5, Exercise 37 starting from an initial volume of 1420.
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38. The model defined in Section 1.5, Exercise 38 starting from an initial mass of 13.1.
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39. The model defined in Section 1.5, Exercise 39 starting from an initial population of 800.
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40. The model defined in Section 1.5, Exercise 40 starting from an initial yield of 20.
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41-42 Reconsider the data describing the levels of a medication in the blood of two patients over the course of several days (measured in mg per liter), used in Section 1.5, Exercises 53 and 54 .
41. For the first patient, graph the updating function and cobweb starting from the initial condition on day 0. Find the equilibrium.
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42. For the second patient, graph the updating function and cobweb starting from the initial condition on day 0. Find the equilibrium.
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43-44 Cobweb and find the equilibrium of the following discrete-time dynamical system.
43. Consider a bacterial population that doubles every hour, but 1.0 × 10 6 individuals are removed after reproduction ( Section 1.5, Exercise 57 ). Cobweb starting from b 0 = 3.0 × 10 6 bacteria.
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44. Consider a bacterial population that doubles every hour, but 1.0 × 10 6 individuals are removed before reproduction ( Section 1.5, Exercise 58 ). Cobweb starting from b 0 = 3.0 × 10 6 bacteria.
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45-46 Consider the following general models for bacterial populations with harvest.
45. Consider a bacterial population that doubles every hour, but h individuals are removed after reproduction. Find the equilibrium. Does it make sense?
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46. Consider a bacterial population that increases by a factor of r every hour, but 1.0 × 10 6 individuals are removed after reproduction. Find the equilibrium. What values of r produce a positive equilibrium?
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47-48 Consider the general model M t +1 = (1 - a ) M t + S for medication ( Example 1.6.11 ). Find the loading dose ( Example 1.6.7 ) in the following cases.
47. a = 0.2, S = 2.
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48. a = 0.8, S = 4.
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49. Use your computer (it may have a special feature for this) to find and graph the first 10 points on the solutions of the following discrete-time dynamical systems. The first two describe populations with reproduction and immigration of 100 individuals per generation, and the last two describe populations that have 100 individuals harvested or removed each generation. a. b t +1 = 0.5 b t + 100 starting from b 0 = 100. b. b t +1 = 1.5 b t + 100 starting from b 0 = 100. c. b t +1 = 1.5 b t - 100 starting from b 0 = 201. d. b t +1 = 1.5 b t - 100 starting from b 0 = 199. e. What happens if you run the last one for 15 steps? What is wrong with the model?
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50. Compose the medication discrete-time dynamical system M t +1 = 0.5 M t + 1.0 with itself 10 times. Plot the resulting function. Use this composition to find the concentration after 10 days starting from concentrations of 1.0, 5.0, and 18.0 milligrams per liter. If the goal is to reach a stable concentration of 2.0 milligrams per liter, do you think this is a good therapy?
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1. The discrete-time dynamical system b t +1 = 2.0b t with b 0 = 1.0.
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2. The discrete-time dynamical system n t +1 = 0.5 n t with n 0 = 1.0.
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3-6 Cobweb the following discrete-time dynamical systems for three steps starting from the given initial condition. Compare with the solution found earlier.
3. v t+1 = 1.5 v t , starting from v 0 = 1220 µm 3 (as in Section 1.5, Exercise 5 ).
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4. l t +1 = l t - 1.7, starting from l 0 = 13.1 cm (as in Section 1.5, Exercise 6 ).
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5. n t +1 = 0.5 n t , starting from n 0 = 1200 (as in Section 1.5, Exercise 7 ).
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6. M t +1 = 0.75 M t + 2.0 starting from the initial condition M 0 = 16.0 (as in Section 1.5, Exercise 8 ).
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7-12 Graph the updating functions associated with the following discrete-time dynamical systems, and cobweb for five steps starting from the given initial condition.
7. x t +1 = 2 x t - 1, starting from x 0 = 2.
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8. Z t +1 = 0.9 z t + 1, starting from z 0 = 3.
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9. w t +1 = -0.5 w t + 3, starting from w 0 = 0.
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10. x t +1 = 4 - x t , starting from x 0 = 1 (as in Section 1.5, Exercise 25 ).
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11. starting from x 0 = 1 (as in Section 1.5, Exercise 23 ).
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12. for x t > 1, starting from x 0 = 3 (as in Section 1.5, Exercise 26 ).
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13-16 Find the equilibria of the following discrete-time dynamical system from the graphs of their updating functions Label the coordinates of the equilibria.
13.
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14.
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15.
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16.
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17-22 Sketch graphs of the following updating functions over the given range and mark the equilibria. Find the equilibria algebraically if possible.
17. f ( x ) = x 2 for 0 = x = 2.
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18. g ( y )= y 2 - 1 for 0 = y = 2. 19-22 Graph the following discrete-time dynamical systems. Solve for the equilibria algebraically, and identify equilibria and the regions where the updating function lies above the diagonal on your graph.
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19. c t +1 = 0.5 c t + 8.0, for 0 = c t = 30.
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20. b t +1 = 3 b t , for 0 = b t = 10.
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21. b t +1 =0.3 b t , for 0 = b t = 10.
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22. b t +1 = 2.0 b t - 5.0, for 0 = b t = 10.
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23-30 Find the equilibria of the following discrete-time dynamical systems. Compare with the results of your cobweb diagram from the earlier problem.
23. v t +1 = 1.5 v t (as in Section 1.5, Exercise 5 ).
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24. l t+1 = l t - 1.7 (as in Section 1.5, Exercise 6 ).
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25. x t+1 = 2 x t - 1 (as in Exercise 7 ).
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26. Z t +1 = 0.9 z t + 1 (as in Exercise 8 ).
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27. w t +1 = -0.5 w t + 3 (as in Exercise 9 ).
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28. x t +1 = 4 - x t (as in Exercise 10 ).
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29. (as in Exercise 11 ).
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30. (as in Exercise 12 ).
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31-34 Find the equilibria of the following discrete-time dynamical systems that include parameters. Identify values of the parameter for which there is no equilibrium, for which the equilibrium is negative, and for which there is more than one equilibrium.
31. w t +1 = aw t + 3.
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32. x t +1 = b - x t .
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33.
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34.
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35-40 Cobweb the following discrete-time dynamical systems for five steps starting from the given initial condition.
35. An alternative tree growth discrete-time dynamical system with form h t +1 = h t + 5.0 with initial condition h 0 = 10.
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36. The lizard-mite system ( Example 1.5.3 ) x t +1 = 2 x t + 30 with initial condition x 0 = 0.
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37. The model defined in Section 1.5, Exercise 37 starting from an initial volume of 1420.
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38. The model defined in Section 1.5, Exercise 38 starting from an initial mass of 13.1.
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39. The model defined in Section 1.5, Exercise 39 starting from an initial population of 800.
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40. The model defined in Section 1.5, Exercise 40 starting from an initial yield of 20.
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41-42 Reconsider the data describing the levels of a medication in the blood of two patients over the course of several days (measured in mg per liter), used in Section 1.5, Exercises 53 and 54 .
41. For the first patient, graph the updating function and cobweb starting from the initial condition on day 0. Find the equilibrium.
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42. For the second patient, graph the updating function and cobweb starting from the initial condition on day 0. Find the equilibrium.
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43-44 Cobweb and find the equilibrium of the following discrete-time dynamical system.
43. Consider a bacterial population that doubles every hour, but 1.0 × 10 6 individuals are removed after reproduction ( Section 1.5, Exercise 57 ). Cobweb starting from b 0 = 3.0 × 10 6 bacteria.
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44. Consider a bacterial population that doubles every hour, but 1.0 × 10 6 individuals are removed before reproduction ( Section 1.5, Exercise 58 ). Cobweb starting from b 0 = 3.0 × 10 6 bacteria.
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45-46 Consider the following general models for bacterial populations with harvest.
45. Consider a bacterial population that doubles every hour, but h individuals are removed after reproduction. Find the equilibrium. Does it make sense?
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46. Consider a bacterial population that increases by a factor of r every hour, but 1.0 × 10 6 individuals are removed after reproduction. Find the equilibrium. What values of r produce a positive equilibrium?
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47-48 Consider the general model M t +1 = (1 - a ) M t + S for medication ( Example 1.6.11 ). Find the loading dose ( Example 1.6.7 ) in the following cases.
47. a = 0.2, S = 2.
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48. a = 0.8, S = 4.
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49. Use your computer (it may have a special feature for this) to find and graph the first 10 points on the solutions of the following discrete-time dynamical systems. The first two describe populations with reproduction and immigration of 100 individuals per generation, and the last two describe populations that have 100 individuals harvested or removed each generation. a. b t +1 = 0.5 b t + 100 starting from b 0 = 100. b. b t +1 = 1.5 b t + 100 starting from b 0 = 100. c. b t +1 = 1.5 b t - 100 starting from b 0 = 201. d. b t +1 = 1.5 b t - 100 starting from b 0 = 199. e. What happens if you run the last one for 15 steps? What is wrong with the model?
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50. Compose the medication discrete-time dynamical system M t +1 = 0.5 M t + 1.0 with itself 10 times. Plot the resulting function. Use this composition to find the concentration after 10 days starting from concentrations of 1.0, 5.0, and 18.0 milligrams per liter. If the goal is to reach a stable concentration of 2.0 milligrams per liter, do you think this is a good therapy?
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Modeling dynamic of life. Calculus and probabilities for life scientists - Adler - Chapter 1.5 Solutions
1-4 Write the updating function associated with each of the following
discrete-time dynamical systems and evaluate it at the given arguments.
Which are linear?
1. p t +1 = p t - 2, evaluate at p t = 5, p t = 10, and p t = 15.
Get 1.5.1 exercise solution
2. evaluate at ? t = 4, ? t = 8, and ? t = 12.
Get 1.5.2 exercise solution
3. , evaluate at x t = 0, x t = 2, and x t = 4.
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4. , evaluate at Q t = 0, Q t = 1, and Q t = 2.
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5-8 Compose the updating function associated with each discrete-time dynamical system with itself. Find the two-step discrete-time dynamical system. Check that the result of applying the original discrete-time dynamical system twice to the given initial condition matches the result of applying the new discrete-time dynamical system to the given initial condition once.
5. Volume follows v t +1 = 1.5 v t , with v 0 = 1220 µ m 3 .
Get 1.5.5 exercise solution
6. Length obeys l t +1 = l t - 1.7, with l 0 = 13.1 cm.
Get 1.5.6 exercise solution
7. Population size follows n t+1 = 0.5n t , with n 0 = 1200.
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8. Medication concentration obeys M t+ 1 = 0.75 M t + 2.0 with M 0 = 16.0.
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9-12 Find the backwards discrete-time dynamical system associated with each discrete-time dynamical system. Use it to find the value at the previous time.
9. v t+ 1 = 1.5 v t . Find v 0 if v 1 = 1220 µ m 3 (as in Exercise 5 ).
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10. l t + 1 = l t - 1.7. Find l 0 if l 1 = 13.1cm (as in Exercise 6 ).
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11. n t +1 = 0.5 n t . Find n 0 if n 1 = 1200 (as in Exercise 7 ).
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12. M t+ 1 = 0.75 M t + 2.0. Find M 0 if M 1 = 16.0 (as in Exercise 8 ).
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13-14 Find the composition of each of the following mathematically elegant updating functions with itself, and find the inverse function.
13. The updating function . Put things over a common denominator to simplify the composition.
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14. The updating function ???. Put things over a common denominator to simplify the composition.
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15-18 Find and graph the solutions of the following discrete-time dynamical systems for five steps with the given initial condition. Compare the graph of the solution with the graph of the updating function.
15. v t +1 = 1.5 v t , with v 0 = 1220 µ m 3 .
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16. l t + 1 = l t - 1.7, with l 0 = 13.1 cm.
Get 1.5.16 exercise solution
17. n t +1 = 0.5 n t , with n 0 = 1200.
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18. M t +1 = 0.75 M t + 2.0 with M 0 = 16.0. 19-22 Using a formula for the solution, you can project far into the future without computing all the intermediate values. Find the following, and say whether the results are reasonable.
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19. Find a formula for v t for the discrete-time dynamical system in Exercise 15 , and use it to find the volume at t = 20.
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20. Find a formula for v t for the discrete-time dynamical system in Exercise 16 , and use it to find the length at t = 20.
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21. Find a formula for v t for the discrete-time dynamical system in Exercise 17 , and use it to find the number at t = 20.
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22. Find a formula for v t for the discrete-time dynamical system in Exercise 18 , and use it to find the concentration at t = 20 (use the method in Example 1.5.14 after finding the value it seems to be approaching). 23-26 Experiment with the following mathematically elegant updating functions and try to find the solution.
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23. Consider the updating function from Exercise 13 . Starting from an initial condition of x 0 = 1, compute x 1 , x 2 , x 3 , and x 4 , and try to spot the pattern.
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24. Use the updating function in Exercise 23 but start from the initial condition x 0 = 2.
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25. Consider the updating function g ( x ) = 4 - x . Start from initial condition of x 0 = 1, and try to spot the pattern. Experiment with a couple of other initial conditions. How would you describe your results in words?
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26. Consider the updating function from Exercise 14 . Start from initial condition of x 0 = 3, and try to spot the pattern. Experiment with a couple of other initial conditions. How would you describe your results in words?
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27-30 Consider the following actions. Which of them commute (produce the same answer when done in either order)?
27. A population doubles in size; 10 individuals are removed from a population. Try starting with 100 individuals, and then try to figure out what happens in general.
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28. A population doubles in size; population size is divided by 4. Try starting with 100 individuals, and then try to figure out what happens in general.
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29. An organism grows by 2.0 cm; an organism shrinks by 1.0 cm.
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30. A person loses half his money. A person gains $10.
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31-34 Use the formula for the solution to find the following, and say whether the results are reasonable.
31. Using the solution for tree height h t = 10.0 + t m ( Example 1.5.13 ), find the tree height after 20 years.
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32. Using the solution for tree height h t = 10.0 + t m ( Example 1.5.13 ), find the tree height after 100 years.
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33. Using the solution for bacterial population number b t = 2.0 t · 1.0 ( Equation 1.5.2 ), find the bacterial population after 20 hours. If an individual bacterium weighs about 10 -12 grams, how much will the whole population weigh?
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34. Using the solution for bacterial population number b t = 2.0 t · 1.0 ( Equation 1.5.2 ), find the bacterial population after 40 hours. How much would this population weigh?
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35-36 Find a formula for the solution of the given discrete-time dynamical system.
35. Find the pattern in the number of mites on a lizard with x 0 = 10 and following the discrete-time dynamical system x t +1 = 2 x t + 30. ( Hint: Add 30 to the number of mites.)
Get 1.5.35 exercise solution
36. Find the pattern in the number of mites on a lizard with x 0 = 10 and following the discrete-time dynamical system x t+1 = 2 x t + 20.
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37-40 The following tables display data from four experiments: a. Cell volume after 10 minutes in a watery bath b. Fish mass after 1 week in a chilly tank c. Gnat population size after 3 days without food d. Yield of several varieties of soybean before and after fertilization For each, graph the new value as a function of the initial value, write the discrete-time dynamical system, and fill in the missing value in the table.
37.
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38.
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39.
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40.
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41–44 Recall the data used for Exercises 49 - 52 in Section 1.2. These data define several discrete-time dynamical systems. For example, between the first measurement (on day 0.5) and the second (on day 1.0), the length increases by 1.5 cm. Between the second measurement (on day 1.0) and the third (on day 1.5), the length again increases by 1.5 cm.
41. Graph the length at the second measurement as a function of length at the first, the length at the third measurement as a function of length at the second, and so forth. Find the discrete-time dynamical system that reproduces the results.
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42. Find and graph the discrete-time dynamical system for tail length.
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43. Find and graph the discrete-time dynamical system for mass.
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44. Find and graph the discrete-time dynamical system for age.
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45-48 Suppose students are permitted to take a test again and again until they get a perfect score of 100. We wish to write a discrete-time dynamical system describing these dynamics.
45. In words, what is the argument of the updating function? What is the value?
Get 1.5.45 exercise solution
46. What are the domain and range of the updating function? What value do you expect if the argument is 100?
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47. Sketch a possible graph of the updating function.
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48. Based on your graph, how would a student do on her second try if she scored 20 on her first try?
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49-50 Consider the discrete-time dynamical system b t +1 = 2.0 b t for a bacterial population ( Example 1.5.1 ).
49. Write a discrete-time dynamical system for the total volume of bacteria (suppose each bacterium takes up 10 4 µ m 3 ).
Get 1.5.49 exercise solution
50. Write a discrete-time dynamical system for the total area taken up by the bacteria (suppose the thickness is 20 µ m).
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51-52 Recall the equation h t +1 = h t + 1.0 for tree height.
51. Write a discrete-time dynamical system for the total volume of the cylindrical trees in Section 1.3, Exercise 27 .
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52. Write a discrete-time dynamical system for the total volume of a spherical tree (this is kind of tricky).
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53-54 Consider the following data describing the level of medication in the blood of two patients over the course of several days.
53. Graph three points on the updating function for the first patient. Find the discrete-time dynamical system for the first patient.
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54. Graph three points on the updating function for the second patient and find the discrete-time dynamical system.
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55-56 For the following discrete-time dynamical systems, compute solutions with the given initial condition. Then find the difference between the solutions as a function of time, and the ratio of the solutions as a function of time. In which cases is the difference constant, and in which cases is the ratio constant? Can you explain why?
55. Two bacterial populations follow the discrete-time dynamical system b t +1 = 2.0 b t , but the first starts with initial condition b 0 = 1.0 × 10 6 and the second starts with initial condition b 0 = 3.0 × 10 5 .
Get 1.5.55 exercise solution
56. Two trees follow the discrete-time dynamical system h t +1 = h t + 1.0, but the first starts with initial condition h 0 = 10.0 m and the second starts with initial condition h 0 = 2.0 m.
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57-60 Follow the steps to derive discrete-time dynamical systems describing the following contrasting situations.
57. A population of bacteria doubles every hour, but 1.0 × 10 6 individuals are removed after reproduction to be converted into valuable biological by-products. The population begins with b 0 = 3.0 × 10 6 bacteria. a. Find the population after 1, 2, and 3 hours. b. How many bacteria were harvested? c. Write the discrete-time dynamical system. d. Suppose you waited to harvest bacteria until the end of 3 hours. How many could you remove and still match the population b 3 found in part a ? Where did all the extra bacteria come from?
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58. Suppose a population of bacteria doubles every hour, but that 1.0 × 10 6 individuals are removed before reproduction to be converted into valuable biological by-products. Suppose the population begins with b 0 = 3.0 × 10 6 bacteria. a. Find the population after 1, 2, and 3 hours. b. Write the discrete-time dynamical system. c. How does the population compare with that in the previous problem? Why is it doing worse?
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59. Suppose the fraction of individuals with some superior gene increases by 10% each generation. a. Write the discrete-time dynamical system for the fraction of organisms with the gene (denote the fraction at time t by f t and figure out the formula for f t +1 ). b. Write the solution with f 0 = 0.0001. c. Will the fraction reach 1.0? Does the discrete-time dynamical system make sense for all values of f t ?
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60. The Weber-Fechner law describes how human beings perceive differences. Suppose, for example, that a person first hears a tone with a frequency of 400 hertz (cycles per second). He is then tested with higher tones until he can hear the difference. The ratio between these values describes how well this person can hear differences. a. Suppose the next tone he can distinguish has a frequency of 404 hertz. What is the ratio? b. According to the Weber-Fechner law, the next higher tone will be greater than 404 by the same ratio. Find this tone. c. Write the discrete-time dynamical system for this person. Find the fifth tone he can distinguish. d. Suppose the experiment is repeated on a musician, and she manages to distinguish 400.5 hertz from 400 hertz. What is the fifth tone she can distinguish?
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61-62 The total mass of a population of bacteria will change if either the number of bacteria changes, the mass per bacterium changes, or both. The following problems derive discrete-time dynamical systems when both change.
61. The number of bacteria doubles each hour, and the mass of each bacterium triples during the same time.
Get 1.5.61 exercise solution
62. The number of bacteria doubles each hour, and the mass of each bacterium increases by 1.0 × 10 -9 g. What seems to go wrong with this calculation? Can you explain why?
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1. p t +1 = p t - 2, evaluate at p t = 5, p t = 10, and p t = 15.
Get 1.5.1 exercise solution
2. evaluate at ? t = 4, ? t = 8, and ? t = 12.
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3. , evaluate at x t = 0, x t = 2, and x t = 4.
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4. , evaluate at Q t = 0, Q t = 1, and Q t = 2.
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5-8 Compose the updating function associated with each discrete-time dynamical system with itself. Find the two-step discrete-time dynamical system. Check that the result of applying the original discrete-time dynamical system twice to the given initial condition matches the result of applying the new discrete-time dynamical system to the given initial condition once.
5. Volume follows v t +1 = 1.5 v t , with v 0 = 1220 µ m 3 .
Get 1.5.5 exercise solution
6. Length obeys l t +1 = l t - 1.7, with l 0 = 13.1 cm.
Get 1.5.6 exercise solution
7. Population size follows n t+1 = 0.5n t , with n 0 = 1200.
Get 1.5.7 exercise solution
8. Medication concentration obeys M t+ 1 = 0.75 M t + 2.0 with M 0 = 16.0.
Get 1.5.8 exercise solution
9-12 Find the backwards discrete-time dynamical system associated with each discrete-time dynamical system. Use it to find the value at the previous time.
9. v t+ 1 = 1.5 v t . Find v 0 if v 1 = 1220 µ m 3 (as in Exercise 5 ).
Get 1.5.9 exercise solution
10. l t + 1 = l t - 1.7. Find l 0 if l 1 = 13.1cm (as in Exercise 6 ).
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11. n t +1 = 0.5 n t . Find n 0 if n 1 = 1200 (as in Exercise 7 ).
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12. M t+ 1 = 0.75 M t + 2.0. Find M 0 if M 1 = 16.0 (as in Exercise 8 ).
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13-14 Find the composition of each of the following mathematically elegant updating functions with itself, and find the inverse function.
13. The updating function . Put things over a common denominator to simplify the composition.
Get 1.5.13 exercise solution
14. The updating function ???. Put things over a common denominator to simplify the composition.
Get 1.5.14 exercise solution
15-18 Find and graph the solutions of the following discrete-time dynamical systems for five steps with the given initial condition. Compare the graph of the solution with the graph of the updating function.
15. v t +1 = 1.5 v t , with v 0 = 1220 µ m 3 .
Get 1.5.15 exercise solution
16. l t + 1 = l t - 1.7, with l 0 = 13.1 cm.
Get 1.5.16 exercise solution
17. n t +1 = 0.5 n t , with n 0 = 1200.
Get 1.5.17 exercise solution
18. M t +1 = 0.75 M t + 2.0 with M 0 = 16.0. 19-22 Using a formula for the solution, you can project far into the future without computing all the intermediate values. Find the following, and say whether the results are reasonable.
Get 1.5.18 exercise solution
19. Find a formula for v t for the discrete-time dynamical system in Exercise 15 , and use it to find the volume at t = 20.
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20. Find a formula for v t for the discrete-time dynamical system in Exercise 16 , and use it to find the length at t = 20.
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21. Find a formula for v t for the discrete-time dynamical system in Exercise 17 , and use it to find the number at t = 20.
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22. Find a formula for v t for the discrete-time dynamical system in Exercise 18 , and use it to find the concentration at t = 20 (use the method in Example 1.5.14 after finding the value it seems to be approaching). 23-26 Experiment with the following mathematically elegant updating functions and try to find the solution.
Get 1.5.22 exercise solution
23. Consider the updating function from Exercise 13 . Starting from an initial condition of x 0 = 1, compute x 1 , x 2 , x 3 , and x 4 , and try to spot the pattern.
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24. Use the updating function in Exercise 23 but start from the initial condition x 0 = 2.
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25. Consider the updating function g ( x ) = 4 - x . Start from initial condition of x 0 = 1, and try to spot the pattern. Experiment with a couple of other initial conditions. How would you describe your results in words?
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26. Consider the updating function from Exercise 14 . Start from initial condition of x 0 = 3, and try to spot the pattern. Experiment with a couple of other initial conditions. How would you describe your results in words?
Get 1.5.26 exercise solution
27-30 Consider the following actions. Which of them commute (produce the same answer when done in either order)?
27. A population doubles in size; 10 individuals are removed from a population. Try starting with 100 individuals, and then try to figure out what happens in general.
Get 1.5.27 exercise solution
28. A population doubles in size; population size is divided by 4. Try starting with 100 individuals, and then try to figure out what happens in general.
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29. An organism grows by 2.0 cm; an organism shrinks by 1.0 cm.
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30. A person loses half his money. A person gains $10.
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31-34 Use the formula for the solution to find the following, and say whether the results are reasonable.
31. Using the solution for tree height h t = 10.0 + t m ( Example 1.5.13 ), find the tree height after 20 years.
Get 1.5.31 exercise solution
32. Using the solution for tree height h t = 10.0 + t m ( Example 1.5.13 ), find the tree height after 100 years.
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33. Using the solution for bacterial population number b t = 2.0 t · 1.0 ( Equation 1.5.2 ), find the bacterial population after 20 hours. If an individual bacterium weighs about 10 -12 grams, how much will the whole population weigh?
Get 1.5.33 exercise solution
34. Using the solution for bacterial population number b t = 2.0 t · 1.0 ( Equation 1.5.2 ), find the bacterial population after 40 hours. How much would this population weigh?
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35-36 Find a formula for the solution of the given discrete-time dynamical system.
35. Find the pattern in the number of mites on a lizard with x 0 = 10 and following the discrete-time dynamical system x t +1 = 2 x t + 30. ( Hint: Add 30 to the number of mites.)
Get 1.5.35 exercise solution
36. Find the pattern in the number of mites on a lizard with x 0 = 10 and following the discrete-time dynamical system x t+1 = 2 x t + 20.
Get 1.5.36 exercise solution
37-40 The following tables display data from four experiments: a. Cell volume after 10 minutes in a watery bath b. Fish mass after 1 week in a chilly tank c. Gnat population size after 3 days without food d. Yield of several varieties of soybean before and after fertilization For each, graph the new value as a function of the initial value, write the discrete-time dynamical system, and fill in the missing value in the table.
37.
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38.
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39.
Get 1.5.39 exercise solution
40.
Get 1.5.40 exercise solution
41–44 Recall the data used for Exercises 49 - 52 in Section 1.2. These data define several discrete-time dynamical systems. For example, between the first measurement (on day 0.5) and the second (on day 1.0), the length increases by 1.5 cm. Between the second measurement (on day 1.0) and the third (on day 1.5), the length again increases by 1.5 cm.
41. Graph the length at the second measurement as a function of length at the first, the length at the third measurement as a function of length at the second, and so forth. Find the discrete-time dynamical system that reproduces the results.
Get 1.5.41 exercise solution
42. Find and graph the discrete-time dynamical system for tail length.
Get 1.5.42 exercise solution
43. Find and graph the discrete-time dynamical system for mass.
Get 1.5.43 exercise solution
44. Find and graph the discrete-time dynamical system for age.
Get 1.5.44 exercise solution
45-48 Suppose students are permitted to take a test again and again until they get a perfect score of 100. We wish to write a discrete-time dynamical system describing these dynamics.
45. In words, what is the argument of the updating function? What is the value?
Get 1.5.45 exercise solution
46. What are the domain and range of the updating function? What value do you expect if the argument is 100?
Get 1.5.46 exercise solution
47. Sketch a possible graph of the updating function.
Get 1.5.47 exercise solution
48. Based on your graph, how would a student do on her second try if she scored 20 on her first try?
Get 1.5.48 exercise solution
49-50 Consider the discrete-time dynamical system b t +1 = 2.0 b t for a bacterial population ( Example 1.5.1 ).
49. Write a discrete-time dynamical system for the total volume of bacteria (suppose each bacterium takes up 10 4 µ m 3 ).
Get 1.5.49 exercise solution
50. Write a discrete-time dynamical system for the total area taken up by the bacteria (suppose the thickness is 20 µ m).
Get 1.5.50 exercise solution
51-52 Recall the equation h t +1 = h t + 1.0 for tree height.
51. Write a discrete-time dynamical system for the total volume of the cylindrical trees in Section 1.3, Exercise 27 .
Get 1.5.51 exercise solution
52. Write a discrete-time dynamical system for the total volume of a spherical tree (this is kind of tricky).
Get 1.5.52 exercise solution
53-54 Consider the following data describing the level of medication in the blood of two patients over the course of several days.
53. Graph three points on the updating function for the first patient. Find the discrete-time dynamical system for the first patient.
Get 1.5.53 exercise solution
54. Graph three points on the updating function for the second patient and find the discrete-time dynamical system.
Get 1.5.54 exercise solution
55-56 For the following discrete-time dynamical systems, compute solutions with the given initial condition. Then find the difference between the solutions as a function of time, and the ratio of the solutions as a function of time. In which cases is the difference constant, and in which cases is the ratio constant? Can you explain why?
55. Two bacterial populations follow the discrete-time dynamical system b t +1 = 2.0 b t , but the first starts with initial condition b 0 = 1.0 × 10 6 and the second starts with initial condition b 0 = 3.0 × 10 5 .
Get 1.5.55 exercise solution
56. Two trees follow the discrete-time dynamical system h t +1 = h t + 1.0, but the first starts with initial condition h 0 = 10.0 m and the second starts with initial condition h 0 = 2.0 m.
Get 1.5.56 exercise solution
57-60 Follow the steps to derive discrete-time dynamical systems describing the following contrasting situations.
57. A population of bacteria doubles every hour, but 1.0 × 10 6 individuals are removed after reproduction to be converted into valuable biological by-products. The population begins with b 0 = 3.0 × 10 6 bacteria. a. Find the population after 1, 2, and 3 hours. b. How many bacteria were harvested? c. Write the discrete-time dynamical system. d. Suppose you waited to harvest bacteria until the end of 3 hours. How many could you remove and still match the population b 3 found in part a ? Where did all the extra bacteria come from?
Get 1.5.57 exercise solution
58. Suppose a population of bacteria doubles every hour, but that 1.0 × 10 6 individuals are removed before reproduction to be converted into valuable biological by-products. Suppose the population begins with b 0 = 3.0 × 10 6 bacteria. a. Find the population after 1, 2, and 3 hours. b. Write the discrete-time dynamical system. c. How does the population compare with that in the previous problem? Why is it doing worse?
Get 1.5.58 exercise solution
59. Suppose the fraction of individuals with some superior gene increases by 10% each generation. a. Write the discrete-time dynamical system for the fraction of organisms with the gene (denote the fraction at time t by f t and figure out the formula for f t +1 ). b. Write the solution with f 0 = 0.0001. c. Will the fraction reach 1.0? Does the discrete-time dynamical system make sense for all values of f t ?
Get 1.5.59 exercise solution
60. The Weber-Fechner law describes how human beings perceive differences. Suppose, for example, that a person first hears a tone with a frequency of 400 hertz (cycles per second). He is then tested with higher tones until he can hear the difference. The ratio between these values describes how well this person can hear differences. a. Suppose the next tone he can distinguish has a frequency of 404 hertz. What is the ratio? b. According to the Weber-Fechner law, the next higher tone will be greater than 404 by the same ratio. Find this tone. c. Write the discrete-time dynamical system for this person. Find the fifth tone he can distinguish. d. Suppose the experiment is repeated on a musician, and she manages to distinguish 400.5 hertz from 400 hertz. What is the fifth tone she can distinguish?
Get 1.5.60 exercise solution
61-62 The total mass of a population of bacteria will change if either the number of bacteria changes, the mass per bacterium changes, or both. The following problems derive discrete-time dynamical systems when both change.
61. The number of bacteria doubles each hour, and the mass of each bacterium triples during the same time.
Get 1.5.61 exercise solution
62. The number of bacteria doubles each hour, and the mass of each bacterium increases by 1.0 × 10 -9 g. What seems to go wrong with this calculation? Can you explain why?
Get 1.5.62 exercise solution
Modeling dynamic of life. Calculus and probabilities for life scientists - Adler - Chapter 1.4 Solutions
1–4 For the following lines, find the slopes between the two given
points by finding the change in output divided by the change in input.
What is the ratio of the output to the input at each of the points?
Which are proportional relations? Which are increasing and which are
decreasing? Sketch a graph.
1. y = 2 x + 3, using points with x = 1 and x = 3.
Get 1.4.1 exercise solution
2. z = -5 w , using points with w = 1 and w = 3.
Get 1.4.2 exercise solution
3. z = 5( w - 2) + 8, using points with w = 1 and w = 3.
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4. y - 5 = -3( x + 2) - 6, using points with x = 1 and x = 3.
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5–6 Check that the point indicated lies on the line and find the equation of the line in point-slope form using the given point. Multiply out to check that the point-slope form matches the original equation.
5. The line f ( x ) = 2 x + 3 and the point (2, 7).
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6. The line g( y ) = -2 y + 7 and the point (3, 1).
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7–12 Find equations in slope-intercept form for the following lines. Sketch a graph indicating the original point from point-slope form.
7. The line f ( x ) = 2( x - 1) + 3.
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8. The line g ( z ) = -3( z + 1) - 3.
Get 1.4.8 exercise solution
9. A line passing through the point (1, 6) with slope -2.
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10. A line passing through the point (-1, 6) with slope 4.
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11. A line passing through the points (1, 6) and (4, 3).
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12. A line passing through the points (6, 1) and (3, 4).
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13–16 Check whether the following are linear functions.
13.
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14. F ( r ) = r 2 + 5.
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15. P ( q ) = 8(3 q + 2) - 6.
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16. Q ( w ) = 8(3 q + 2) - 6( q + 4).
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17–18 Check that the following curves do not have constant slope by computing the slopes between the points indicated. Compare with the graphs in Exercises 5 and 6 in Section 1.2 .
17. at z = 1, z = 2, and z = 4, as in Exercise 5 . Find the slope between z = 1 and z = 2, and the slope between z = 2 and z = 4.
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18. F ( r ) = r 2 + 5 at r = 0, r = 1, and r = 4, as in Exercise 6 . Find the slope between r = 0 and r = 1, and the slope between r = 1 and r = 4.
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19–24 Solve the following equations. Check your answer by plugging in the value you found.
19. 2 x + 3 = 7.
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20.
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21. 2 x + 3 = 3 x + 7.
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22. -3 y + 5 = 8 + 2 y .
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23. 2(5 ( x - 1) + 3 = 5(2( x - 2) + 5).
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24. 2(4( x - 1) + 3 = 5(2( x - 2) + 5).
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25-28 Solve the following equations for the given variable, treating the other letters as constant parameters.
25. Solve 2 x + b = 7 for x .
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26. Solve mx + 3 = 7 for x .
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27. Solve 2 x + b = mx + 7 for x . Are there any values of b or m for which this has no solution?
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28. Solve mx + b = 3 x + 7 for x . Are there any values of b or m for which this has no solution? 29-32 Most unit conversions are proportional relations. Find the slope and graph the relations between the following units.
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29. Graph the length of a fish in inches on the horizontal axis and centimeters on the vertical axis. Use the fact that 1 in. = 2.54 cm. Mark the point corresponding to a length of 1 in.
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30. Graph the length of a fish in centimeters on the horizontal axis and inches on the vertical axis. Use the fact that 1 in. = 2.54 cm. Mark the point corresponding to a length of 1 in.
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31. Graph the mass of a fish in grams on the horizontal axis and its weight in pounds on the vertical axis. Use the identity 1 lb = 453.6 g. Mark the point corresponding to a weight of 1 lb.
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32. Graph the weight of a fish in pounds on the horizontal axis and its mass in grams on the vertical axis. Use the identity 1 lb = 453.6 g. Mark the point corresponding to a weight of 1 lb.
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33–34 Not very many functions commute with each other. The following problems ask you to find all linear functions that commute with the given linear function.
33. Find all functions of the form g ( x ) = mx + b that commute with the function f ( x ) = x + 1. Can you explain your answer in words?
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34. Find all functions of the form g ( x ) = mx + b that commute with the function f ( x ) = 2 x . Can you explain your answer in words?
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35-38 Many fundamental relations express a proportional relation between two measurements with different dimensions. Find the slopes and the equations of the relations between the following quantities.
35. Volume = area × thickness. Find the volume V as a function of the area A if the thickness is 1.0 cm.
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36. Volume = area × thickness. Find the volume V as a function of thickness T if the area is 7.0 cm 2 .
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37. Total mass = mass per bacterium × number of bacteria. Find total mass M as a function of the number of bacteria b if the mass per bacterium is 5.0 × 10 -9 g.
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38. Total mass = mass per bacterium × number of bacteria. Find total mass M as a function of mass per bacterium m if the total number is 10 6 .
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39–42 A ski slope has a slope of -0.2. You start at an altitude of 10,000 ft.
39. Write the equation giving altitude a as a function of horizontal distance moved d .
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40. Write the equation of the line in meters.
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41. What will be your altitude when you have gone 2000 ft horizontally?
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42. The ski run ends at an altitude of 8000 ft. How far will you have gone horizontally?
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43–46 The following data give the elevation of the surface of the Great Salt Lake in Utah.
43. Graph these data.
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44. During which periods is the surface elevation changing linearly?
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45. What was the slope between 1965 and 1975? What would the surface elevation have been in 1990 if things had continued as they began? How different is this from the actual depth?
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46. What was the slope during between 1985 and 1995? What would the surface elevation have been in 1965 if things had always followed this trend? How different is this from the actual depth?
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47–50 Graph the following relations between measurements of a growing plant, checking that the points lie on a line. Find the equations in both point-slope and slope-intercept form.
47. Mass as a function of age. Find the mass on day 1.75.
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48. Volume as a function of age. Find the volume on day 2.75.
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49. Glucose production as a function of mass. Estimate glucose production when the mass reaches 20.0 g.
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50. Volume as a function of mass. Estimate the volume when the mass reaches 30.0 g. How will the density at that time compare with the density when a = 0.5?
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51–44 Consider the data in the following table (adapted from Parasitoids by H. C. F. Godfray), describing the number of wasps that can develop inside caterpillars of different weights.
51. Graph these data. Which point does not lie on the line?
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52. Find the equation of the line connecting the first two points.
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53. How many wasps does the function predict would develop in a caterpillar weighing 0.72 g?
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54. How many wasps does the function predict would develop in a caterpillar weighing 0.0 g? Does this make sense? How many would you really expect?
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55–58 The world record times for various races are decreasing at roughly linear rates (adapted from Guinness Book of Records , 1990).
55. The men’s Olympic record for the 1500 meters was 3:36.8 in 1972 and 3:35.9 in 1988. Find and graph the line connecting these. (Don’t forget to convert everything into seconds.)
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56. The women’s Olympic record for the 1500 meters was 4:01.4 in 1972 and 3:53.9 in 1988. Find and graph the line connecting these.
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57. If things continue at this rate, when will women finish the race in exactly no time? What might happen before that date?
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58. If things continue at this rate, when will women be running this race faster than men?
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59. Try Exercise 58 on the computer. Compute the year when the times will reach 0. Give your best guess of the times in the year 1900.
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60. Graph the ratio of temperature measured in Fahrenheit to temperature measured in Celsius for -273 = °C < 200. What happens near °C = 0? What happens for large and small values of °C? How would the results differ if the zero for Fahrenheit were changed to match that of Celsius?
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1. y = 2 x + 3, using points with x = 1 and x = 3.
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2. z = -5 w , using points with w = 1 and w = 3.
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3. z = 5( w - 2) + 8, using points with w = 1 and w = 3.
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4. y - 5 = -3( x + 2) - 6, using points with x = 1 and x = 3.
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5–6 Check that the point indicated lies on the line and find the equation of the line in point-slope form using the given point. Multiply out to check that the point-slope form matches the original equation.
5. The line f ( x ) = 2 x + 3 and the point (2, 7).
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6. The line g( y ) = -2 y + 7 and the point (3, 1).
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7–12 Find equations in slope-intercept form for the following lines. Sketch a graph indicating the original point from point-slope form.
7. The line f ( x ) = 2( x - 1) + 3.
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8. The line g ( z ) = -3( z + 1) - 3.
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9. A line passing through the point (1, 6) with slope -2.
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10. A line passing through the point (-1, 6) with slope 4.
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11. A line passing through the points (1, 6) and (4, 3).
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12. A line passing through the points (6, 1) and (3, 4).
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13–16 Check whether the following are linear functions.
13.
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14. F ( r ) = r 2 + 5.
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15. P ( q ) = 8(3 q + 2) - 6.
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16. Q ( w ) = 8(3 q + 2) - 6( q + 4).
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17–18 Check that the following curves do not have constant slope by computing the slopes between the points indicated. Compare with the graphs in Exercises 5 and 6 in Section 1.2 .
17. at z = 1, z = 2, and z = 4, as in Exercise 5 . Find the slope between z = 1 and z = 2, and the slope between z = 2 and z = 4.
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18. F ( r ) = r 2 + 5 at r = 0, r = 1, and r = 4, as in Exercise 6 . Find the slope between r = 0 and r = 1, and the slope between r = 1 and r = 4.
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19–24 Solve the following equations. Check your answer by plugging in the value you found.
19. 2 x + 3 = 7.
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20.
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21. 2 x + 3 = 3 x + 7.
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22. -3 y + 5 = 8 + 2 y .
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23. 2(5 ( x - 1) + 3 = 5(2( x - 2) + 5).
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24. 2(4( x - 1) + 3 = 5(2( x - 2) + 5).
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25-28 Solve the following equations for the given variable, treating the other letters as constant parameters.
25. Solve 2 x + b = 7 for x .
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26. Solve mx + 3 = 7 for x .
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27. Solve 2 x + b = mx + 7 for x . Are there any values of b or m for which this has no solution?
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28. Solve mx + b = 3 x + 7 for x . Are there any values of b or m for which this has no solution? 29-32 Most unit conversions are proportional relations. Find the slope and graph the relations between the following units.
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29. Graph the length of a fish in inches on the horizontal axis and centimeters on the vertical axis. Use the fact that 1 in. = 2.54 cm. Mark the point corresponding to a length of 1 in.
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30. Graph the length of a fish in centimeters on the horizontal axis and inches on the vertical axis. Use the fact that 1 in. = 2.54 cm. Mark the point corresponding to a length of 1 in.
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31. Graph the mass of a fish in grams on the horizontal axis and its weight in pounds on the vertical axis. Use the identity 1 lb = 453.6 g. Mark the point corresponding to a weight of 1 lb.
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32. Graph the weight of a fish in pounds on the horizontal axis and its mass in grams on the vertical axis. Use the identity 1 lb = 453.6 g. Mark the point corresponding to a weight of 1 lb.
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33–34 Not very many functions commute with each other. The following problems ask you to find all linear functions that commute with the given linear function.
33. Find all functions of the form g ( x ) = mx + b that commute with the function f ( x ) = x + 1. Can you explain your answer in words?
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34. Find all functions of the form g ( x ) = mx + b that commute with the function f ( x ) = 2 x . Can you explain your answer in words?
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35-38 Many fundamental relations express a proportional relation between two measurements with different dimensions. Find the slopes and the equations of the relations between the following quantities.
35. Volume = area × thickness. Find the volume V as a function of the area A if the thickness is 1.0 cm.
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36. Volume = area × thickness. Find the volume V as a function of thickness T if the area is 7.0 cm 2 .
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37. Total mass = mass per bacterium × number of bacteria. Find total mass M as a function of the number of bacteria b if the mass per bacterium is 5.0 × 10 -9 g.
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38. Total mass = mass per bacterium × number of bacteria. Find total mass M as a function of mass per bacterium m if the total number is 10 6 .
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39–42 A ski slope has a slope of -0.2. You start at an altitude of 10,000 ft.
39. Write the equation giving altitude a as a function of horizontal distance moved d .
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40. Write the equation of the line in meters.
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41. What will be your altitude when you have gone 2000 ft horizontally?
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42. The ski run ends at an altitude of 8000 ft. How far will you have gone horizontally?
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43–46 The following data give the elevation of the surface of the Great Salt Lake in Utah.
43. Graph these data.
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44. During which periods is the surface elevation changing linearly?
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45. What was the slope between 1965 and 1975? What would the surface elevation have been in 1990 if things had continued as they began? How different is this from the actual depth?
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46. What was the slope during between 1985 and 1995? What would the surface elevation have been in 1965 if things had always followed this trend? How different is this from the actual depth?
Get 1.4.46 exercise solution
47–50 Graph the following relations between measurements of a growing plant, checking that the points lie on a line. Find the equations in both point-slope and slope-intercept form.
47. Mass as a function of age. Find the mass on day 1.75.
Get 1.4.47 exercise solution
48. Volume as a function of age. Find the volume on day 2.75.
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49. Glucose production as a function of mass. Estimate glucose production when the mass reaches 20.0 g.
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50. Volume as a function of mass. Estimate the volume when the mass reaches 30.0 g. How will the density at that time compare with the density when a = 0.5?
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51–44 Consider the data in the following table (adapted from Parasitoids by H. C. F. Godfray), describing the number of wasps that can develop inside caterpillars of different weights.
51. Graph these data. Which point does not lie on the line?
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52. Find the equation of the line connecting the first two points.
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53. How many wasps does the function predict would develop in a caterpillar weighing 0.72 g?
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54. How many wasps does the function predict would develop in a caterpillar weighing 0.0 g? Does this make sense? How many would you really expect?
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55–58 The world record times for various races are decreasing at roughly linear rates (adapted from Guinness Book of Records , 1990).
55. The men’s Olympic record for the 1500 meters was 3:36.8 in 1972 and 3:35.9 in 1988. Find and graph the line connecting these. (Don’t forget to convert everything into seconds.)
Get 1.4.55 exercise solution
56. The women’s Olympic record for the 1500 meters was 4:01.4 in 1972 and 3:53.9 in 1988. Find and graph the line connecting these.
Get 1.4.56 exercise solution
57. If things continue at this rate, when will women finish the race in exactly no time? What might happen before that date?
Get 1.4.57 exercise solution
58. If things continue at this rate, when will women be running this race faster than men?
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59. Try Exercise 58 on the computer. Compute the year when the times will reach 0. Give your best guess of the times in the year 1900.
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60. Graph the ratio of temperature measured in Fahrenheit to temperature measured in Celsius for -273 = °C < 200. What happens near °C = 0? What happens for large and small values of °C? How would the results differ if the zero for Fahrenheit were changed to match that of Celsius?
Get 1.4.60 exercise solution
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