Modeling dynamic of life. Calculus and probabilities for life scientists - Adler - Chapter 1.2 Solutions

1-2 Identify the variables and parameters in the following situations, give the units they might be measured in, and choose an appropriate letter or symbol to represent each.

1. A scientist measures the mass of fish over the course of 100 days, and repeats the experiment at three different levels of salinity: 0%, 2% and 5%. Get 1.2.1 exercise solution

2. A scientist measures the body temperature of bandicoots every day during the winter, and does so at three different altitudes: 500 m, 750 m, and 1000 m. Get 1.2.2 exercise solution

 3-6 Compute the values of the following functions at the points indicated and sketch a graph of the function.

3. f ( x ) = x + 5 at x = 0, x = 1, and x = 4. Get 1.2.3 exercise solution

4. g ( y ) = 5 y at y = 0, y = 1, and y = 4. Get 1.2.4 exercise solution

5. Get 1.2.5 exercise solution

6. F ( r ) = r 2 + 5 at r = 0, r = 1, and r = 4.  Get 1.2.6 exercise solution

7-10 Graph the given points and say which point does not seem to fall on the graph of a simple function.

7. (0, -1), (1, 1), (2, 1), (3, 5), (4, 7). Get 1.2.7 exercise solution

8. (0, 5), (1, 10), (2, 8), (3, 6), (4, 4). Get 1.2.8 exercise solution

9. (0, 2), (1, 3), (2, 6), (3, 11), (4, 10). Get 1.2.9 exercise solution

10. (0, 45), (1, 25), (2, 12), (3, 12.5), (4, 10).  Get 1.2.10 exercise solution

11–14 Evaluate the following functions at the given algebraic arguments.

11. f ( x ) = x + 5 at x = a, x = a + 1, and x = 4 a . Get 1.2.11 exercise solution

12. g ( y )= 5 y at y = x 2 , y = 2 x + 1, and y = 2 - x . Get 1.2.12 exercise solution

13. Get 1.2.13 exercise solution

14.  Get 1.2.14 exercise solution


15-16 Sketch graphs of the following relations. Is there a more convenient order for the arguments?

15. A function whose argument is the name of a state and whose value is the highest altitude in that state. State Highest Altitude (ft) California 14,491 Idaho 12,662 Nevada 13,143 Oregon 11,239 Utah 13,528 Washington 14,410 Get 1.2.15 exercise solution

16. A function whose argument is the name of a bird and whose value is the length of that bird. Bird Length Cooper’s hawk 50 cm Goshawk 66 cm Sharp-shinned hawk 35 cm Get 1.2.16 exercise solution


17-20 For each of the following sums of functions, graph each component piece. Compute the values at x = -2, x = -1, x = 0, x = 1, and x = 2 and plot the sum.

17. f ( x ) 2 x + 3 and g ( x ) = 3 x - 5. Get 1.2.17 exercise solution

18. f ( x ) = 2 x + 3 and h ( x ) = -3 x - 12. Get 1.2.18 exercise solution

19. F ( x ) = x 2 + 1 and G ( x ) = x + 1. Get 1.2.19 exercise solution

20. F ( x )= x 2 + 1 and h ( x )= - x + 1. 21-24 For each of the following products of functions, graph each component piece. Compute the value of the product at x = -2, x = - 1, x = 0, x1, and x = 2 and graph the result. Get 1.2.20 exercise solution

21. f ( x ) = 2 x + 3 and g ( x ) = 3 x - 5. Get 1.2.21 exercise solution

22. f ( x ) = 2 x + 3 and h ( x ) = -3 x - 12. Get 1.2.22 exercise solution

23. F ( x ) = x 2 + 1 and G ( x ) = x + 1. Get 1.2.23 exercise solution

24. F ( x )= x 2 + 1 and h ( x )= - x + 1. 25-28 Find the inverses of each of the following functions. In each case, compute the output of the original function at an input of 1.0, and show that the inverse undoes the action of the function. Get 1.2.24 exercise solution

25. f ( x ) = 2 x + 3. Get 1.2.25 exercise solution

26. g ( x ) = 3 x - 5. Get 1.2.26 exercise solution

27. G ( y )= 1/(2+ y ) for y = 0. Get 1.2.27 exercise solution

28. F ( y )= y 2 + 1 for y = 0.
  Get 1.2.28 exercise solution

29-32 Graph each of the following functions and its inverse. Mark the given point on the graph of each function.

29. f ( x ) = 2 x + 3. Mark the point (1, f (1)) on the graphs of f and f -1 (based on Exercise 25 ). Get 1.2.29 exercise solution

30. g ( x ) = 3 x - 5. Mark the point (1, g (1)) on the graphs of g and g -1 (based on Exercise 26 ). Get 1.2.30 exercise solution

31. G ( y ) = 1/(2 + y ). Mark the point (1, G (1)) on the graphs of G and G -1 (based on Exercise 27 ). Get 1.2.31 exercise solution

32. F ( y )= y 2 + 1 for y = 0. Mark the point (1, F (1)) on the graphs of F and F -1 (based on Exercise 28 ). 33–36 Find the compositions of the given functions. Which pairs of functions commute? Get 1.2.32 exercise solution

33. f ( x ) = 2 x + 3 and g ( x ) = 3 x - 5. Get 1.2.33 exercise solution

34. f ( x ) = 2 x + 3 and h ( x ) = -3 x - 12. Get 1.2.34 exercise solution

35. F ( x ) = x 2 + 1 and G ( x ) = x + 1. Get 1.2.35 exercise solution

36. F ( x )= x 2 + 1 and h ( x ) = - x + 1.  Get 1.2.36 exercise solution


Applications 37–40 Describe what is happening in the graphs shown.

37. A plot of cell volume against time in days. Get 1.2.37 exercise solution

38. A plot of a Pacific salmon population against time in years. Get 1.2.38 exercise solution

39. A plot of the average height of a population of trees plotted against age in years. Get 1.2.39 exercise solution

40. A plot of an Internet stock price against time.  Get 1.2.40 exercise solution


 41-44 Draw graphs based on the following descriptions.

41. A population of birds begins at a large value, decreases to a tiny value, and then increases again to an intermediate value. Get 1.2.41 exercise solution

42. The amount of DNA in an experiment increases rapidly from a very small value and then levels out at a large value before declining rapidly to 0. Get 1.2.42 exercise solution

43. Body temperature oscillates between high values during the day and low values at night. Get 1.2.43 exercise solution

44. Soil is wet at dawn, quickly dries out and stays dry during the day, and then becomes gradually wetter again during the night.  Get 1.2.44 exercise solution


45-48 Evaluate the following functions over the suggested range, sketch a graph of the function, and answer the biological question.

45. The number of bees b found on a plant is given by b = 2 f + 1 where f is the number of flowers, ranging from 0 to about 20. Explain what might be happening when f = 0. Get 1.2.45 exercise solution

46. The number of cancerous cells c as a function of radiation dose r (measured in rads) is c = r - 4 for r greater than or equal to 5, and is zero for r less than 5. r ranges from 0 to 10. What is happening at r = 5 rads? Get 1.2.46 exercise solution

47. Insect development time A (in days) obeys represents temperature in °C for 10 = T = 40. Which temperature leads to the more rapid development? Get 1.2.47 exercise solution

48. Tree height h (in meters) follows the formula where a represents the age of the tree in years for 0 = a = 1000. How tall would this tree get if it lived forever? 49–52 Consider the following data describing the growth of a tadpole. Get 1.2.48 exercise solution

49. Graph length as a function of age. Get 1.2.49 exercise solution

50. Graph tail length as a function of age. Get 1.2.50 exercise solution

51. Graph tail length as a function of length. Get 1.2.51 exercise solution

52. Graph mass as a function of length and then graph length as a function of mass. How do the two graphs compare?
  Get 1.2.52 exercise solution

53-56 The following series of functional compositions describe connections between several measurements.

53. The number of mosquitos ( M ) that end up in a room is a function of how far the window is open ( W , in cm 2 ) according to M ( W ) = 5 W + 2. The number of bites ( B ) depends on the number of mosquitos according to B ( M )= 0.5 M . Find the number of bites as a function of how far the window is open. How many bites would you get if the window was 10 cm 2 open? Get 1.2.53 exercise solution

54. The temperature of a room ( T ) is a function of how far the window is open (W) according to T ( W ) = 40 - 0.2 W . How long you sleep ( S , measured in hours) is a function of the temperature according to . Find how long you sleep as a function of how far the window is open. How long would you sleep if the window was 10 cm 2 open? Get 1.2.54 exercise solution

55. The number of viruses ( V , measured in trillions or 10 12 ) that infect a person is a function of the degree of immunosuppression ( I , the fraction of the immune system that is turned off by stress) according to V ( I )= 5 I 2 . The fever ( F , measured in °C) associated with an infection is a function of the number of viruses according to F ( V ) = 37 + 0.4 V . Find fever as a function of immunosuppression. How high will the fever be if immunosuppression is complete ( I = 1)? Get 1.2.55 exercise solution

56. The length of an insect ( L , in mm) is a function of the temperature during development ( T , measured in °C) according to The volume of the insect ( V , in cubic mm) is a function of the length according to V ( L )= 2 L 3 . The mass ( M in milligrams) depends on volume according to M ( V ) = 1.3 V . Find mass as a function of temperature. How much would an insect weigh that developed at 25°C?
  Get 1.2.56 exercise solution

57-58 Each of the following measurements is the sum of two components. Find the formula for the sum. Sketch a graph of each component and the total as functions of time for 0 = t = 3. Describe each component and the sum in words.

57. A population of bacteria consists of two types a and b . The first follows a ( t ) = 1 + t 2 , and the second follows b ( t )= 1 - 2 t + t 2 where populations are measured in millions and time is measured in hours. The total population is P ( t )= a ( t )+ b ( t ). Get 1.2.57 exercise solution

58. The above-ground volume (stem and leaves) of a plant is , and the below-ground volume (roots) is V b ( t ) = - 1.0 t + 40.0 where t is measured in days and volumes are measured in cm 3 . The total volume is V ( t )= V a ( t ) + V b ( t ). 59–62 Consider the following data describing a plant. Get 1.2.58 exercise solution

59. Graph M as a function of a . Does this function have an inverse? Could we use mass to figure out the age of the plant? Get 1.2.59 exercise solution

60. Graph V as a function of a . Does this function have an inverse? Could we use volume to figure out the age of the plant? Get 1.2.60 exercise solution

61. Graph G as a function of a . Does this function have an inverse? Could we use glucose production to figure out the age of the plant? Get 1.2.61 exercise solution

62. Graph G as a function of M . Does this function have an inverse? What is strange about it? Could we use glucose production to figure out the mass of the plant?  Get 1.2.62 exercise solution


63-66 The total mass of a population (in kg) as a function of the time, t , is the product of the number of individuals, P ( t ), and the mass per person, W ( t ) (in kg). In each of the following exercises, find the formula for the total mass, sketch graphs of P ( t ), W ( t ), and the total mass as functions of time for 0 = t = 100, and describe the results in words.

63. The population of people is P ( t ) = 2.0 × 10 6 + 2.0 × 10 4 t , and the mass per person W ( t ) (in kg) is W ( t ) = 80 - 0.5 t . Get 1.2.63 exercise solution

64. The population is P ( t ) = 2.0 × 10 6 - 2.0 × 10 4 t , and the mass per person W ( t ) is W ( t )= 80 + 0.5 t . Get 1.2.64 exercise solution

65. The population is P ( t )= 2.0 × 10 6 + 1000 t 2 , and the mass per person W ( t ) is W ( t )= 80 - 0.5 t . Get 1.2.65 exercise solution

66. The population is P ( t )= 2.0 × 10 6 + 2.0 × 10 4 t , and the mass per person W ( t ) is W ( t )= 80 - 0.005 t 2 .  Get 1.2.66 exercise solution


Computer Exercises
67-70 Have your graphics calculator or computer plot the following functions. How would you describe them in words?

67. a. f ( x ) = x 2 e - x for 0 = x = 20. b. g ( x )= 1.5 + e -0.1 x sin ( x ) for 0 = x = 20. c. h ( x ) = sin (5 x ) - cos (7 x ) for 0 = x = 5. d. f ( x ) + h ( x ) for 0 = x = 20 (using the functions in parts a and c ). e. g ( x )· h ( x ) for 0 = x = 20 (using the functions in parts b and c ). f. h ( x )· h ( x ) for 0 = x = 20 (using the function in part c ). Get 1.2.67 exercise solution

68. Have your computer plot the function h ( x )= e - x 2 - e -1000( x -0.13) 2 - 0.2 for values of x between -10 and 10. a. How would you describe the result in words? b. Blow up the graph by changing the range to find all points where the value of the function is 0. For example, one such value is between 1 and 2. Plot the function again for x between 1 and 2 to zoom in. c. If you only found 2 points where h ( x ) = 0, blow up the region between 0 and 1 to try to find two more. Get 1.2.68 exercise solution

69. Use your computer to find and plot the following functional compositions. a. b. c. Get 1.2.69 exercise solution

70. Have your graphics calculator or computer plot the following functions for -2 < x < 2. Do they have inverses? a. h 2 ( x )= x 2 + 2 x . b. h 3 ( x ) = x 3 + 2 x . c. h 4 ( x ) = x 4 + 2 x . d. h 5 ( x ) = x 5 + 2 x . Have your computer try to find the formula for the inverses of these functions and plot the results. Does the machine always succeed in finding an inverse when there is one? Does it sometimes find an inverse when there is none?