Biocalculus: Calculus for Life Sciences 1st Edition solutions

Biocalculus: Calculus for Life Sciences 1
(a) What is a sequence? (b) What does it mean to say that limnl` an 8? (c) What does it mean to say that limnl

Biocalculus: Calculus for Life Sciences 1
(a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence? Give two examples.

Biocalculus: Calculus for Life Sciences 1
World record sprint times The graph plots the sequence of the world record times for the mens 100-meter sprint every five years t. Do you think that this sequence has a nonzero limit as tl`? What would that mean for this sporting event?

Biocalculus: Calculus for Life Sciences 1
World record hammer throws The graph plots the sequence of the world record distances for the womens hammer throw by year t. (a) Explain what it would mean for this sporting event if the sequence does not have a limit as tl`. (b) Do you think this sequence is convergent or divergent? Explain.

Biocalculus: Calculus for Life Sciences 1
58 Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. 5. an n2 2n 1 3n2 6. an 4 2 2 n 1 3 n2 7. an 3 1 (22 3) n 8. an n sn 1 1

Biocalculus: Calculus for Life Sciences 1
58 Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. 5. an n2 2n 1 3n2 6. an 4 2 2 n 1 3 n2 7. an 3 1 (22 3) n 8. an n sn 1 2

Biocalculus: Calculus for Life Sciences 1
58 Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. 5. an n2 2n 1 3n2 6. an 4 2 2 n 1 3 n2 7. an 3 1 (22 3) n 8. an n sn 1 3

Biocalculus: Calculus for Life Sciences 1
58 Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. 5. an n2 2n 1 3n2 6. an 4 2 2 n 1 3 n2 7. an 3 1 (22 3) n 8. an n sn 1 4

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
926 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit. 9. an 1 3n4 10. an 5 3n 11. an 2n2 1 n 2 1 n2 12. an n3 2 1 n 13. an 3 1 5n 2 1 7n 14. an n3 2 1 n3 1 1 15. an 1 2 s0.2d n 16. an 2 2n 1 6 2n 17. an n2 sn3 1 4n 18. an sinsny2d 19. an cossn

Biocalculus: Calculus for Life Sciences 1
2734 Calculate, to four decimal places, the first eight terms of the recursive sequence. Does it appear to be convergent? If so, guess the value of the limit. Then assume the limit exists and determine its exact value. 27. a1 1, an11 1 2an 1 1 28. a1 2, an11 1 2 1 3an 29. a1 2, an11 2an 2 1 30.

Biocalculus: Calculus for Life Sciences 1
2734 Calculate, to four decimal places, the first eight terms of the recursive sequence. Does it appear to be convergent? If so, guess the value of the limit. Then assume the limit exists and determine its exact value. 27. a1 1, an11 1 2an 1 1 28. a1 2, an11 1 2 1 3an 29. a1 2, an11 2an 2 1 30.

Biocalculus: Calculus for Life Sciences 1
2734 Calculate, to four decimal places, the first eight terms of the recursive sequence. Does it appear to be convergent? If so, guess the value of the limit. Then assume the limit exists and determine its exact value. 27. a1 1, an11 1 2an 1 1 28. a1 2, an11 1 2 1 3an 29. a1 2, an11 2an 2 1 30.

Biocalculus: Calculus for Life Sciences 1
2734 Calculate, to four decimal places, the first eight terms of the recursive sequence. Does it appear to be convergent? If so, guess the value of the limit. Then assume the limit exists and determine its exact value. 27. a1 1, an11 1 2an 1 1 28. a1 2, an11 1 2 1 3an 29. a1 2, an11 2an 2 1 30.

Biocalculus: Calculus for Life Sciences 1
2734 Calculate, to four decimal places, the first eight terms of the recursive sequence. Does it appear to be convergent? If so, guess the value of the limit. Then assume the limit exists and determine its exact value. 27. a1 1, an11 1 2an 1 1 28. a1 2, an11 1 2 1 3an 29. a1 2, an11 2an 2 1 30.

Biocalculus: Calculus for Life Sciences 1
2734 Calculate, to four decimal places, the first eight terms of the recursive sequence. Does it appear to be convergent? If so, guess the value of the limit. Then assume the limit exists and determine its exact value. 27. a1 1, an11 1 2an 1 1 28. a1 2, an11 1 2 1 3an 29. a1 2, an11 2an 2 1 30.

Biocalculus: Calculus for Life Sciences 1
2734 Calculate, to four decimal places, the first eight terms of the recursive sequence. Does it appear to be convergent? If so, guess the value of the limit. Then assume the limit exists and determine its exact value 33. a1 1, an11 s2 1 an 34. a1 100, an11 1 2 San 1 25 an D

Biocalculus: Calculus for Life Sciences 1
2734 Calculate, to four decimal places, the first eight terms of the recursive sequence. Does it appear to be convergent? If so, guess the value of the limit. Then assume the limit exists and determine its exact value 33. a1 1, an11 s2 1 an 34. a1 100, an11 1 2 San 1 25 an D

Biocalculus: Calculus for Life Sciences 1
Antibiotic pharmacokinetics A doctor prescribes a 100-mg antibiotic tablet to be taken every eight hours. Just before each tablet is taken, 20% of the drug present in the preceding time step remains in the body. (a) How much of the drug is in the body just after the second tablet is taken? After the

Biocalculus: Calculus for Life Sciences 1
Drug pharmacokinetics A patient is injected with a drug every 12 hours. Immediately before each injection the concentration of the drug has been reduced by 90% and the new dose increases the concentration by 1.5 mgymL. (a) What is the concentration after three doses? (b) If Cn is the concentration af

Biocalculus: Calculus for Life Sciences 1
Drug pharmacokinetics A patient takes 150 mg of a drug at the same time every day. Just before each tablet is taken, 5% of the drug present in the preceding time step remains in the body. (a) What quantity of the drug is in the body after the third tablet? After the nth tablet? (b) What quantity of t

Biocalculus: Calculus for Life Sciences 1
Insulin injection After injection of a dose D of insulin, the concentration of insulin in a patients system decays exponentially and so it can be written as De2at, where t represents time in hours and a is a positive constant. (a) If a dose D is injected every T hours, write an expression for the sum

Biocalculus: Calculus for Life Sciences 1
Let x 0.99999 . . . . (a) Do you think that x , 1 or x 1? (b) Sum a geometric series to find the value of x. (c) How many decimal representations does the number 1 have? (d) Which numbers have more than one decimal representation?

Biocalculus: Calculus for Life Sciences 1
A sequence is defined recursively by an s5 2 ndan21, a1 1. Find the sum of all the terms of the sequence.

Biocalculus: Calculus for Life Sciences 1
4146 Express the number as a ratio of integers. 41. 0.8 0.8888 . . . 42. 0.46 0.46464646 . . . 43. 2.516 2.516516516 . . . 44. 10.135 10.135353535 . . . 45. 1.5342 46. 7.12345

Biocalculus: Calculus for Life Sciences 1
4146 Express the number as a ratio of integers. 41. 0.8 0.8888 . . . 42. 0.46 0.46464646 . . . 43. 2.516 2.516516516 . . . 44. 10.135 10.135353535 . . . 45. 1.5342 46. 7.12346

Biocalculus: Calculus for Life Sciences 1
4146 Express the number as a ratio of integers. 41. 0.8 0.8888 . . . 42. 0.46 0.46464646 . . . 43. 2.516 2.516516516 . . . 44. 10.135 10.135353535 . . . 45. 1.5342 46. 7.12347

Biocalculus: Calculus for Life Sciences 1
4146 Express the number as a ratio of integers. 41. 0.8 0.8888 . . . 42. 0.46 0.46464646 . . . 43. 2.516 2.516516516 . . . 44. 10.135 10.135353535 . . . 45. 1.5342 46. 7.12348

Biocalculus: Calculus for Life Sciences 1
4146 Express the number as a ratio of integers. 41. 0.8 0.8888 . . . 42. 0.46 0.46464646 . . . 43. 2.516 2.516516516 . . . 44. 10.135 10.135353535 . . . 45. 1.5342 46. 7.12349

Biocalculus: Calculus for Life Sciences 1
4146 Express the number as a ratio of integers. 41. 0.8 0.8888 . . . 42. 0.46 0.46464646 . . . 43. 2.516 2.516516516 . . . 44. 10.135 10.135353535 . . . 45. 1.5342 46. 7.12350

Biocalculus: Calculus for Life Sciences 1
4752 Logistic equation Plot enough terms of the logistic difference equation xt11 cxts1 2 xtd to see how the terms behave. Does the sequence appear to be convergent? If so, estimate the limit and then, assuming the limit exists, calculate its exact value. If not, describe the behavior of the terms.

Biocalculus: Calculus for Life Sciences 1
4752 Logistic equation Plot enough terms of the logistic difference equation xt11 cxts1 2 xtd to see how the terms behave. Does the sequence appear to be convergent? If so, estimate the limit and then, assuming the limit exists, calculate its exact value. If not, describe the behavior of the terms.

Biocalculus: Calculus for Life Sciences 1
4752 Logistic equation Plot enough terms of the logistic difference equation xt11 cxts1 2 xtd to see how the terms behave. Does the sequence appear to be convergent? If so, estimate the limit and then, assuming the limit exists, calculate its exact value. If not, describe the behavior of the terms.

Biocalculus: Calculus for Life Sciences 1
4752 Logistic equation Plot enough terms of the logistic difference equation xt11 cxts1 2 xtd to see how the terms behave. Does the sequence appear to be convergent? If so, estimate the limit and then, assuming the limit exists, calculate its exact value. If not, describe the behavior of the terms.

Biocalculus: Calculus for Life Sciences 1
4752 Logistic equation Plot enough terms of the logistic difference equation xt11 cxts1 2 xtd to see how the terms behave. Does the sequence appear to be convergent? If so, estimate the limit and then, assuming the limit exists, calculate its exact value. If not, describe the behavior of the terms.

Biocalculus: Calculus for Life Sciences 1
4752 Logistic equation Plot enough terms of the logistic difference equation xt11 cxts1 2 xtd to see how the terms behave. Does the sequence appear to be convergent? If so, estimate the limit and then, assuming the limit exists, calculate its exact value. If not, describe the behavior of the terms.

Biocalculus: Calculus for Life Sciences 1
Logistic equation: Dependence on initial values Compare plots of the first 20 terms of the logistic equation xt11 1 4 xts1 2 xtd for the initial values x0 0.2 and x0 0.2001. When the initial value changes slightly, how does the solution change?

Biocalculus: Calculus for Life Sciences 1
Logistic equation: Dependence on initial values Repeat Exercise 53 for the equation xt11 4xts1 2 xtd and compare with the results of Exercise 53. [This behavior is another part of what it means to be chaotic.]

Biocalculus: Calculus for Life Sciences 1
5558 The Ricker equation xt11 cxte2xt was introduced in Exercise 1.6.32. Plot enough terms of the Ricker equation to see how the terms behave. Does the sequence appear to be convergent? If so, estimate the limit and then, assuming the limit exists, calculate its exact value. If not, describe the beh

Biocalculus: Calculus for Life Sciences 1
5558 The Ricker equation xt11 cxte2xt was introduced in Exercise 1.6.32. Plot enough terms of the Ricker equation to see how the terms behave. Does the sequence appear to be convergent? If so, estimate the limit and then, assuming the limit exists, calculate its exact value. If not, describe the beh

Biocalculus: Calculus for Life Sciences 1
5558 The Ricker equation xt11 cxte2xt was introduced in Exercise 1.6.32. Plot enough terms of the Ricker equation to see how the terms behave. Does the sequence appear to be convergent? If so, estimate the limit and then, assuming the limit exists, calculate its exact value. If not, describe the beh

Biocalculus: Calculus for Life Sciences 1
5558 The Ricker equation xt11 cxte2xt was introduced in Exercise 1.6.32. Plot enough terms of the Ricker equation to see how the terms behave. Does the sequence appear to be convergent? If so, estimate the limit and then, assuming the limit exists, calculate its exact value. If not, describe the beh

Biocalculus: Calculus for Life Sciences 1
The Sierpinski carpet is constructed by removing the center one-ninth of a square of side 1, then removing the centers of the eight smaller remaining squares, and so on. (The figure shows the first three steps of the construction.) Show that the sum of the areas of the removed squares is 1. This impl

Biocalculus: Calculus for Life Sciences 1
A right triangle ABC is given with /A  and | AC | b. CD is drawn perpendicular to AB, DE is drawn perpendicular to BC, EF  AB, and this process is continued indefinitely, as shown in the figure. Find the total length of all the perpendiculars | CD | 1 | DE | 1 | EF | 1 | FG | 1 in terms of b a