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Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions
Find the limit. lim (x+ ax x + bx) x+x
Find the limit. lim (9x2 x3x) + x-x
Find the limit. lim x x-00 x+1
Find the limit. lim xxx (2x + 1) (x 1)(x + x)
Find the limit. lim 1-1T 1021 3/2 + 31 - 5
Find the limit. lim 7 +12 0021-
Find the limit. lim 5 7% 10' r
Find the limit. lim 0.6' -11
Find the limit. lim 8-18 4x3 + 6x22 2x-4x+5
Find the limit. 1-x-x2 x lim *-* 2x-7
Find the limit. lim 3 1-x2 * x-x+1
Find the limit. lim 3x-2 x-2.x + 1
Find the limit. lim 3x+5 xxx-4
Find the limit. lim 1 x2x+3
(a) Use a graph ofto estimate the value of limxl` f sxd correct to two decimal places.(b) Use a table of values of f sxd to estimate the limit to four decimal places. x f(x)=1- 2 x
Guess the value of the limit lim x l `x2 2x by evaluating the function f sxd − x2y2x for x − 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. Then use a graph of f to support your guess.
(a) Can the graph of y − f sxd intersect a horizontal asymptote? If so, how many times? Illustrate by sketching graphs.(b) How many horizontal asymptotes can the graph of y − f sxd have? Sketch graphs to illustrate the possibilities.
Explain in your own words the meaning of each of the following.(a) lim x l `f sxd − 5 (b) lim x l 2`f sxd − 3
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 perpendicularsin terms of b and . |CD| + |DE| + |EF | + |FG| +
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
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
The Ricker e quation 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
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
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
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.]
Logistic equation: Dependence on initial values Compare plots of the first 20 terms of the logistic equation xt11 − 14 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?
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.
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.
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.
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.
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.
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.
Express the number as a ratio of integers. 7.12345
Express the number as a ratio of integers. 1.5342
Express the number as a ratio of integers. 10.135 − 10.135353535 . . .
Express the number as a ratio of integers. 2.516 − 2.516516516 . . .
Express the number as a ratio of integers. 0.46 − 0.46464646 . . .
Express the number as a ratio of integers. 0.8 − 0.8888 . . .
A sequence is defined recursively by an − s5 2 ndan21, a1 − 1. Find the sum of all the terms of the sequence.
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?
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 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
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
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
A ntibiotic 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
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. a1 − 100, an11 −1 2 San 1 25 an D
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. a1 − 1, an11 − s2 1 an
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. a1 − 3, an11 − 8 2 an
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. a1 − 1, an11 −6 1 1 an
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. a1 − 1, an11 − s5an
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. a1 − 2, an11 − 2an 2 1
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. a1 − 2, an11 − 1 2 13 an
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. a1 − 1, an11 − 12 an 1 1
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. = an In(n + 1) - In n Inn
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an -n en + en e2n - 1
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an 3n+2 5"
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an = In(2n+1)In(n + 1)
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. + an
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an 10" 1 + 9"
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an 10" 1 + 9"
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an sin(n/2)
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an n n + 4n
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. n an 3"
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an =2n+6n
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an = 1 - (0.2)"
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an n 3 3 - 1 n +1
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an 3 + 5n 2 + 7n
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an n 3. - 1 n
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an 2n2n-1 n
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an 5 3"
Determine whether the sequence is convergent or divergent.If it is convergent, find the limit. an 1 3n4
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. an −n sn 1 1
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. an − 3 1 (223)n
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. an − 4 2 2n 13 n2
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. an −n2 2n 1 3n2
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 tl`.(b) Do you think this sequence is convergent or divergent?Explain. 78 18
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 tl`? What would that mean for this sporting event? 10.6 seconds 10.4 + 10.2 10.0 9.8 + 9.6 + + 1920 1940 1960
(a) What is a convergent sequence? Give two examples.(b) What is a divergent sequence? Give two examples.
(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` an − `?
Find the first 40 terms of the sequence defined by an11 − H12 an if an is an even number 3an 1 1 if an is an odd number and a1 − 11. Do the same if a1 − 25. Make a conjecture about this type of sequence.
T wo bacteria strains Suppose the population sizes of two strains of bacteria each grow as described by the recursions at11 − Raat and bt11 − Rbbt, respectively.The frequency of the first strain at time t is defined as pt − atysat 1 btd. Derive a recursion for pt and show that it can be
Methyl groups in DNA DNA sometimes has chemical groups attached, called methyl groups, that affect gene expression. Suppose that, during each hour, first a fraction m of unmethylated locations on the DNA become methylated, and then a fraction u of methylated locations become unmethylated. Find
Salmon and bears Pacific salmon populations have discrete breeding cycles in which they return from the ocean to streams to reproduce and then die. This occurs every one to five years, depending on the species.(a) Suppose that each fish must first survive predation by bears while swimming upstream,
Spherical colonies Suppose the volume of a spherical colony is proportional to the number of individuals in it and growth occurs only at the surface-resource interface of the colony. Find a difference equation that models the population.
Bacteria colonies on agar plates Bacteria are often grown on agar plates and form circular colonies. The area of a colony is proportional to the number of bacteria it contains. The agar (a gelatinous substance obtained from red seaweed) is the resource that bacteria use to reproduce and so only
Drug concentration Suppose Ct is the concentration of a drug in the bloodstream at time t, A is the concentration of the drug that is administered at each time step, and k is the fraction of the drug metabolized in a time step.(a) What is the recursion that models how the drug concentration
Repeat Exercise 32(a) for x0 − 78 and c − 3.42. Compare with Exercise 29.
R icker equation In the logistic difference equation the factor s1 2 xtd decreases linearly from 1 to 0 as xt increases from 0 to 1. If, instead, we introduce the decreasing exponential factor e2xt, we get what is called the Ricker difference equation:xt11 − cxte2xt This model has the advantage
Logistic equation For the logistic difference equation xt11 − cxts1 2 xtd and the given values of x0 andc, calculate xt to four decimal places for t − 1, 2, . . . , 10 and graph xt.Comment on the behavior of the sequence. x0 − 0.5, c − 3.7
Logistic equation For the logistic difference equation xt11 − cxts1 2 xtd and the given values of x0 andc, calculate xt to four decimal places for t − 1, 2, . . . , 10 and graph xt.Comment on the behavior of the sequence. x0 − 78, c − 3.45
Logistic equation For the logistic difference equation xt11 − cxts1 2 xtd and the given values of x0 andc, calculate xt to four decimal places for t − 1, 2, . . . , 10 and graph xt.Comment on the behavior of the sequence. x0 − 78, c − 3.42
Logistic equation For the logistic difference equation xt11 − cxts1 2 xtd and the given values of x0 andc, calculate xt to four decimal places for t − 1, 2, . . . , 10 and graph xt.Comment on the behavior of the sequence. x0 − 0.5, c − 2.5
Logistic equation For the logistic difference equation xt11 − cxts1 2 xtd and the given values of x0 andc, calculate xt to four decimal places for t − 1, 2, . . . , 10 and graph xt.Comment on the behavior of the sequence. x0 − 0.5, c − 1.5
(a) For a difference equation of the form Nt11 − f sNtd, calculate the composition s f 8 f dsNtd. What is the meaning of f 8 f in this context?(b) If Nt11 − f sNtd, where f is one-to-one, what is f 21sNt11d? What is the meaning of the inverse function f 21 in this context?
Consider the difference equation N0 − 1 Nt11 − RNt What can you say about the solution of this equation as t becomes large in the following three cases?(a) R , 1 (b) R − 1 (c) R . 1
H arvesting fish A fish farmer has 5000 catfish in his pond. The number of catfish increases by 8% per month and the farmer harvests 300 catfish per month.(a) Show that the catfish population Pn after n months is given recursively by Pn − 1.08Pn21 2 300 P0 − 5000 (b) How many catfish are in the
Breeding rabbits Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair, which becomes reproductive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth month? Show that the answer
Find the first six terms of the recursive sequence. a1 − 1, a2 − 2, an12 − an11 1 2an
Find the first six terms of the recursive sequence. a1 − 2, a2 − 1, an11 − an 2 an21
Find the first six terms of the recursive sequence. a1 − 3, an11 − s3an
Find the first six terms of the recursive sequence. a1 − 1, an11 − s3an
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