Biocalculus Calculus for the Life Sciences

James Stewart

Chapter 2

Limits - all with Video Answers

Educators

Section 1

Limits of Sequences

Problem 1

(a) What is a sequence?
(b) What does it mean to say that $\lim _{n \rightarrow \infty} a_{n}=8 ?$
(c) What does it mean to say that $\lim _{n \rightarrow \infty} a_{n}=\infty ?$

Lucas G.

Lucas G.

Numerade Educator

Problem 2

(a) What is a convergent sequence? Give two examples.
(b) What is a divergent sequence? Give two examples.

Paula S.

Numerade Educator

Problem 3

World record sprint times The graph plots the sequence of the world record times for the men's 100 -meter sprint every five years $t$ . Do you think that this sequence has a nonzero limit as $t \rightarrow \infty ?$ What would that mean for this sporting event?

Lucas G.

Numerade Educator

Problem 4

World record hammer throws The graph plots the sequence of the world record distances for the women's hammer throw by year $t$ .
(a) Explain what it would mean for this sporting event if the sequence does not have a limit as $t \rightarrow \infty$ .
(b) Do you think this sequence is convergent or divergent? Explain.

Paula S.

Numerade Educator

Problem 5

$5-8$ 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.

$a_{n}=\frac{n^{2}}{2 n+3 n^{2}}$

Lucas G.

Numerade Educator

Problem 6

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.
$a_{n}=4-\frac{2}{n}+\frac{3}{n^{2}}$

Paula S.

Numerade Educator

Problem 7

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.
$a_{n}=3+\left(-\frac{2}{3}\right)^{n}$

Lucas G.

Numerade Educator

Problem 8

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.
$a_{n}=\frac{n}{\sqrt{n}+1}$

Paula S.

Numerade Educator

Problem 9

9 - 26 Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\frac{1}{3 n^{4}}$

Lucas G.

Numerade Educator

Problem 10

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\frac{5}{3^{n}}$

Kevin S.

Numerade Educator

Problem 11

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\frac{2 n^{2}+n-1}{n^{2}}$

Lucas G.

Numerade Educator

Problem 12

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\frac{n^{3}-1}{n}$

Paula S.

Numerade Educator

Problem 13

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\frac{3+5 n}{2+7 n}$

Lucas G.

Numerade Educator

Problem 14

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\frac{n^{3}-1}{n^{3}+1}$

Paula S.

Numerade Educator

Problem 15

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=1-(0.2)^{n}$

Lucas G.

Numerade Educator

Problem 16

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=2^{-n}+6^{-n}$

Paula S.

Numerade Educator

Problem 17

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\frac{n^{2}}{\sqrt{n^{3}+4 n}}$

Lucas G.

Numerade Educator

Problem 18

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\sin (n \pi / 2)$

Paula S.

Numerade Educator

Problem 19

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\cos (n \pi / 2)$

Lucas G.

Numerade Educator

Problem 20

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\frac{\pi^{n}}{3^{n}}$

Paula S.

Numerade Educator

Problem 21

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\frac{10^{n}}{1+9^{n}}$

Lucas G.

Numerade Educator

Problem 22

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\frac{\sqrt[3]{n}}{\sqrt{n}+\sqrt[4]{n}}$

Paula S.

Numerade Educator

Problem 23

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\ln \left(2 n^{2}+1\right)-\ln \left(n^{2}+1\right)$

Lucas G.

Numerade Educator

Problem 24

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\frac{3^{n+2}}{5^{n}}$

Paula S.

Numerade Educator

Problem 25

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\frac{e^{n}+e^{-n}}{e^{2 n}-1}$

Lucas G.

Numerade Educator

Problem 26

Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.
$a_{n}=\ln (n+1)-\ln n$

Paula S.

Numerade Educator

Problem 27

$27-34$ 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.
$a_{1}=1, a_{n+1}=\frac{1}{2} a_{n}+1$

Lucas G.

Numerade Educator

Problem 28

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.
$a_{1}=2, a_{n+1}=1-\frac{1}{3} a_{n}$

Paula S.

Numerade Educator

Problem 29

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.
$a_{1}=2, a_{n+1}=2 a_{n}-1$

Lucas G.

Numerade Educator

Problem 30

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.
$a_{1}=1, a_{n+1}=\sqrt{5 a_{n}}$

Paula S.

Numerade Educator

Problem 31

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.
$a_{1}=1, a_{n+1}=\frac{6}{1+a_{n}}$

Lucas G.

Numerade Educator

Problem 32

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.
$a_{1}=3, a_{n+1}=8-a_{n}$

Paula S.

Numerade Educator

Problem 33

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.
$a_{1}=1, a_{n+1}=\sqrt{2+a_{n}}$

Lucas G.

Numerade Educator

Problem 34

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.
$a_{1}=100, a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{25}{a}\right)$

Paula S.

Numerade Educator

Problem 35

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 sec-
ond tablet is taken? After the third tablet?
(b) If $Q_{n}$ is the quantity of the antibiotic in the body just after the $n$ th tablet is taken, write a difference equation that expresses $Q_{n+1}$ in terms of $Q_{n}$
(c) Find a formula for $Q_{n}$ as a function of $n .$
(d) What quantity of the antibiotic remains in the body in the long run?

Lucas G.

Numerade Educator

Problem 36

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 mg/mL.
(a) What is the concentration after three doses?
(b) If $C_{n}$ is the concentration after the $n$ th dose, write a difference equation that expresses $C_{n+1}$ in terms of $C_{n}$ .
(c) Find a formula for $C_{n}$ as a function of $n$
(d) What is the limiting value of the concentration?

Paula S.

Numerade Educator

Problem 37

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 $n$ th tablet?
(b) What quantity of the drug remains in the body in the long run?

Lucas G.

Numerade Educator

Problem 38

Insulin injection After injection of a dose $D$ of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as $D e^{-a t},$ 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 of the residual concentrations just before the $(n+1)$ st injection.
(b) Determine the limiting pre-injection concentration.
(c) If the concentration of insulin must always remain at or above a critical value $C,$ determine a minimal dosage $D$ in terms of $C, a,$ and $T .$

Paula S.

Numerade Educator

Problem 39

Let $x=0.99999 \ldots$
(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?

Lucas G.

Numerade Educator

Problem 40

A sequence is defined recursively by $a_{n}=(5-n) a_{n-1}$ $a_{1}=1 .$ Find the sum of all the terms of the sequence.

Paula S.

Numerade Educator

Problem 41

$41-46$ Express the number as a ratio of integers.
$0.8=0.8888 \dots$

Lucas G.

Numerade Educator

Problem 42

Express the number as a ratio of integers.
$0.46=0.46464646 \ldots$

Paula S.

Numerade Educator

Problem 43

Express the number as a ratio of integers.
$2 . \overline{516}=2.516516516 \ldots$

Lucas G.

Numerade Educator

Problem 44

Express the number as a ratio of integers.
$10.1 \overline{35}=10.135353535 \ldots$

Paula S.

Numerade Educator

Problem 45

Express the number as a ratio of integers.
1.53$\overline{42}$

Lucas G.

Numerade Educator

Problem 46

Express the number as a ratio of integers.
$46.7 . \overline{12345}$

Paula S.

Numerade Educator

Problem 47

$47-52$ Logistic equation Plot enough terms of the logistic
difference equation $x_{i+1}=c x_{t}\left(1-x_{1}\right)$ 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.
$x_{0}=0.1, c=2$

Lucas G.

Numerade Educator

Problem 48

Logistic equation Plot enough terms of the logistic
difference equation $x_{i+1}=c x_{t}\left(1-x_{1}\right)$ 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.
$x_{0}=0.8, \quad c=2.6$

Paula S.

Numerade Educator

Problem 49

Logistic equation Plot enough terms of the logistic
difference equation $x_{i+1}=c x_{t}\left(1-x_{1}\right)$ 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.
$x_{0}=0.2, c=3.2$

Lucas G.

Numerade Educator

Problem 50

Logistic equation Plot enough terms of the logistic
difference equation $x_{i+1}=c x_{t}\left(1-x_{1}\right)$ 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.
$x_{0}=0.4, c=3.5$

Paula S.

Numerade Educator

Problem 51

Logistic equation Plot enough terms of the logistic
difference equation $x_{i+1}=c x_{t}\left(1-x_{1}\right)$ 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.
$x_{0}=0.1, c=3.8$

Lucas G.

Numerade Educator

Problem 52

Logistic equation Plot enough terms of the logistic
difference equation $x_{i+1}=c x_{t}\left(1-x_{1}\right)$ 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.
$x_{0}=0.6, c=3.9$

Paula S.

Numerade Educator

Problem 53

Logistic equation: Dependence on initial values Compare plots of the first 20 terms of the logistic equation $x_{t+1}=\frac{1}{4} x_{t}\left(1-x_{t}\right)$ for the initial values $x_{0}=0.2$ and
$x_{0}=0.2001 .$ When the initial value changes slightly, how does the solution change?

Lucas G.

Numerade Educator

Problem 54

Logistic equation: Dependence on initial values Repeat Exercise 53 for the equation $x_{t+1}=4 x_{t}\left(1-x_{t}\right)$ and compare with the results of Exercise $53 .$ . IThis behavior is another part of what it means to be chaotic.

Paula S.

Numerade Educator

Problem 55

$55-58$ The Ricker equation $x_{t+1}=c x_{t} e^{-x_{r}}$ 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 behavior of the terms.

$x_{0}=0.2, \quad c=2$

Lucas G.

Numerade Educator

Problem 56

$x_{0}=0.4, \quad c=3$

Paula S.

Numerade Educator

Problem 57

$x_{0}=0.8, \quad c=10$

Lucas G.

Numerade Educator

Problem 58

$x_{0}=0.9, \quad c=20$

Paula S.

Numerade Educator

Problem 59

The Sierpinski carpet is constructed by removing the enter 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 implies that the Sierpinski carpet has area $0 .$

Lucas G.

Numerade Educator

Problem 60

A right triangle $A B C$ is given with $\angle A=\theta$ and $|A C|=b$ . $C D$ is drawn perpendicular to $A B, D E$ is drawn perpen- dicular to $B C, E F \perp A B,$ and this process is continued Indehinitely, as shown in the hgure. Find the total length of all the perpendiculars $|C D|+|D E|+|E F|+|F G|+\cdots$ in terms of $b$ and $\theta.$

Lucas G.

Numerade Educator