Modeling dynamic of life. Calculus and probabilities for life scientists - Adler - Chapter 1.7 Solutions

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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