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Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions
lim x l 01 sin x ln x Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l 0 cot 2x sin 6x Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim xl2`x2ex Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l `x sins yxd Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l a1 cos x lnsx 2 ad lnsex 2 ea d Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l 0 cos x 2 1 1 12 x2 x4 Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l 0 ex 2 e2x 2 2x x 2 sin x Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l 1 xa 2 ax 1 a 2 1 sx 2 1d2 Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l 0 xtan21s4xd Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l 1 1 2 x 1 ln x 1 1 cos x Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l 0 cos mx 2 cos nx x2 Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l 0 ex 2 1 2 x x2 Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim ul `euy10 u3 Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim tl0 5t 2 3t tFind the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim xl1 ln x sin x Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l 0 s1 1 2x 2 s1 2 4x xFind the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim xl`sln xd2 xFind the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l 01 ln x xFind the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim l y2 1 2 sin csc Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l `ln x sx Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim tl0 e3t 2 1 tFind the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim tl0 et 2 1 t 3 Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim xl0 sin 4x tan 5x Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim xls y2d1 cos x 1 2 sin x Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim xl1 xa 2 1 xb 2 1 Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
lim x l 1 x2 2 1 x2 2 x Find the limit. Use l’Hospital’s Rule where appropriate.If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
For what values of c does the polynomial Psxd − x4 1 cx3 1 x2 have two inflection points? One inflection point? None? Illustrate by graphing P for several values ofc. How does the graph change as c decreases?
Show that a cubic function (a third-degree polynomial)always has exactly one point of inflection. If its graph has three x-intercepts x1, x2, and x3, show that the x-coordinate of the inflection point is sx1 1 x2 1 x3 dy3.
A ntibiotic concentration Suppose an antibiotic is administered orally. It is first absorbed into the bloodstream, from which it passes into the sinus cavity. It is also metabolized from both sites. The concentrations C1std in the blood and C2std in the sinus cavity are shown. Prove that there is a
Growth rate A 20-year-old university student weighs 138 lb and had a birth weight of 6 lb. Prove that at some point in her life she was growing at a rate of 6.6 pounds per year.
f sxd − tan x 1 5 cos x Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f 9 and f 0 to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.
f sxd − x2 2 4x 1 7 cos x, 24 < x < 4 Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f 9 and f 0 to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.
f sxd − x6 2 15x5 1 75x4 2 125x3 2 x Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f 9 and f 0 to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.
f sxd − 4x4 2 32x3 1 89x2 2 95x 1 29 Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f 9 and f 0 to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.
For what values of the numbers a and b does the function f sxd − axebx2 have the maximum value f s2d − 1?
Find a cubic function f sxd − ax3 1 bx2 1 cx 1 d that has a local maximum value of 3 at x − 22 and a local minimum value of 0 at x − 1.
In the theory of relativity, the energy of a particle is E − sm0 2c4 1 h2c2y2 where m0 is the rest mass of the particle, is its wave length, and h is Planck’s constant. Sketch the graph of E as a function of . What does the graph say about the energy?
In the theory of relativity, the mass of a particle is m −m0 s1 2 v2yc2 where m0 is the rest mass of the particle, m is the mass when the particle moves with speed v relative to the observer, and c is the speed of light. Sketch the graph of m as a function of v.
The family of bell-shaped curves y −1s2 e2sx2d2ys22 d occurs in probability and statistics, where it is called the normal density function. The constant is called the mean and the positive constant is called the standard deviation.For simplicity, let’s scale the function so as to remove
A dose response curve in pharmacology is a plot of the effectiveness R of a drug as a function of the drug concentrationc. Such curves typically increase with an S-shape, a simple mathematical model being Rscd −c2 3 1 c2(a) At what drug concentration does the inflection point occur?(b) Suppose we
Mutation accumulation When a population is subjected to a mutagen, the fraction of the population that contains at least one mutation increases with the duration of the exposure. A commonly used equation describing this fraction is f std − 1 2 e2t, where is the mutation rate and is positive.
A drug-loading curve describes the level of medication in the bloodstream after a drug is administered. A surge function Sstd − At pe2kt is often used to model the loading curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, A − 0.01,
A ntibiotic pharmacokinetics Suppose that antibiotics are injected into a patient to treat a sinus infection. The antibiotics circulate in the blood, slowly diffusing into the sinus cavity while simultaneously being filtered out of the blood by the liver. In Chapter 10 we will derive a model for
Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavity.What is the significance of the inflection point?
Suppose f s3d − 2, f 9s3d − 12, and f 9sxd . 0 and f 0sxd , 0 for all x.(a) Sketch a possible graph for f .(b) How many solutions does the equation f sxd − 0 have?Why?(c) Is it possible that f 9s2d − 13? Why?
Let f std be the temperature at time t where you live and suppose that at time t − 3 you feel uncomfortably hot. How do you feel about the given data in each case?(a) f 9s3d − 2, f 0s3d − 4(b) f 9s3d − 2, f 0s3d − 24(c) f 9s3d − 22, f 0s3d − 4(d) f 9s3d − 22, f 0s3d − 24
Use the methods of this section to sketch the curve y − x3 2 3a2x 1 2a3, where a is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?
Suppose the derivative of a function f is f 9sxd − sx 1 1d2sx 2 3d5sx 2 6d4. On what interval is f increasing?
f sxd − earctan x(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection points.(e) Use the information from parts (a)–(d) to sketch the graph of f.
f sxd − e21ysx11d(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection points.(e) Use the information from parts (a)–(d) to sketch the graph of f.
f sxd −ex 1 1 ex(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection points.(e) Use the information from parts (a)–(d) to sketch the graph of f.
f sxd − lns1 2 ln xd(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection points.(e) Use the information from parts (a)–(d) to sketch the graph of f.
f sxd − x tan x, 2 y2 , x , y2(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection points.(e) Use the information from parts (a)–(d) to sketch the
f sxd − sx2 1 1 2 x(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection points.(e) Use the information from parts (a)–(d) to sketch the graph of f.
f sxd −x2 sx 2 2d2(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection points.(e) Use the information from parts (a)–(d) to sketch the graph of f.
f sxd −x2 x2 2 1(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection points.(e) Use the information from parts (a)–(d) to sketch the graph of f.
f std − t 1 cos t, 22 < t < 2 (a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c) to sketch the graph.Check your work with a graphing device if you
f s d − 2 cos 1 cos2 , 0 < < 2 (a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c) to sketch the graph.Check your work with a graphing device if
f sxd − lnsx4 1 27d(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c) to sketch the graph.Check your work with a graphing device if you have one.
Csxd − x1y3sx 1 4d(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c) to sketch the graph.Check your work with a graphing device if you have one.
Bsxd − 3x2y3 2 x(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c) to sketch the graph.Check your work with a graphing device if you have one.
Asxd − xsx 1 3(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c) to sketch the graph.Check your work with a graphing device if you have one.
hsxd − x5 2 2x3 1 x(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c) to sketch the graph.Check your work with a graphing device if you have one.
hsxd − sx 1 1d5 2 5x 2 2(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c) to sketch the graph.Check your work with a graphing device if you have
tsxd − 200 1 8x3 1 x4(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c) to sketch the graph.Check your work with a graphing device if you have one.
f sxd − 2 1 2x2 2 x4(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c) to sketch the graph.Check your work with a graphing device if you have one.
f sxd − 2 1 3x 2 x3(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c) to sketch the graph.Check your work with a graphing device if you have one.
f sxd − 2x3 2 3x2 2 12x(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c) to sketch the graph.Check your work with a graphing device if you have one.
(a) Find the critical numbers of f sxd − x4sx 2 1d3.(b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers?(c) What does the First Derivative Test tell you?
Suppose f 0 is continuous on s2`, `d.(a) If f 9s2d − 0 and f 0s2d − 25, what can you say about f ?(b) If f 9s6d − 0 and f 0s6d − 0, what can you say about f ?
f sxd −x x2 1 4 Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer?
f sxd − x 1 s1 2 x Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer?
f sxd − sx e2x(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f .(c) Find the intervals of concavity and the inflection points.
f sxd − sln xdysx(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f .(c) Find the intervals of concavity and the inflection points.
f sxd − x2 ln x(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f .(c) Find the intervals of concavity and the inflection points.
f sxd − e2x 1 e2x(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f .(c) Find the intervals of concavity and the inflection points.
f sxd − cos2x 2 2 sin x, 0 < x < 2 (a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f .(c) Find the intervals of concavity and the inflection points.
f sxd − sin x 1 cos x, 0 < x < 2 (a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f .(c) Find the intervals of concavity and the inflection points.
f sxd − x2 x2 1 3(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f .(c) Find the intervals of concavity and the inflection points.
f sxd − x4 2 2x2 1 3(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f .(c) Find the intervals of concavity and the inflection points.
f sxd − 4x3 1 3x2 2 6x 1 1(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f .(c) Find the intervals of concavity and the inflection points.
f sxd − 2x3 1 3x2 2 36x(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f .(c) Find the intervals of concavity and the inflection points.
HI V prevalence The table gives the number of HIV-infected men in San Francisco from 1982 to 1991.(a) If Hstd is the number of infected men at time t, plot the values of Hstd. What does the direction of concavity appear to be? Provide a biological interpretation.(b) Use the table to construct a
In each part state the x-coordinates of the inflection points of f . Give reasons for your answers.(a) The curve is the graph of f .(b) The curve is the graph of f 9.(c) The curve is the graph of f 0. y 0 2 4 6 8 x *
The graph of the first derivative f 9 of a function f is shown.(a) On what intervals is f increasing? Explain.(b) At what values of x does f have a local maximum or minimum? Explain.(c) On what intervals is f concave upward or concave downward?Explain.(d) What are the x-coordinates of the
(a) State the First Derivative Test.(b) State the Second Derivative Test. Under what circumstances is it inconclusive? What do you do if it fails?
Suppose you are given a formula for a function f .(a) How do you determine where f is increasing or decreasing?(b) How do you determine where the graph of f is concave upward or concave downward?(c) How do you locate inflection points?
Use the given graph of f to find the following.(a) The open intervals on which f is increasing.(b) The open intervals on which f is decreasing.(c) The open intervals on which f is concave upward.(d) The open intervals on which f is concave downward.(e) The coordinates of the points of inflection.
Use the given graph of f to find the following.(a) The open intervals on which f is increasing.(b) The open intervals on which f is decreasing.(c) The open intervals on which f is concave upward.(d) The open intervals on which f is concave downward.(e) The coordinates of the points of inflection.
Suppose that 3 < f 9sxd < 5 for all values of x. Show that 18 < f s8d 2 f s2d < 30.
Foraging Many animals forage on resources that are distributed in discrete patches. For example, bumblebees visit many flowers, foraging on nectar from each. The amount of nectar Nstd consumed from any flower increases with the amount of time spent at that flower, but with diminishing returns, as
Use the graph of f to estimate the values of c that satisfy the conclusion of the Mean Value Theorem for the interval f0, 8g. y. 1 0 1 y = f(x) x
A cubic function is a polynomial of degree 3; that is, it has the form f sxd − ax3 1 bx2 1 cx 1d, where a ± 0.(a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities.(b) How many local extreme values can a cubic
Between 0°C and 30°C the volume V (in cubic centimeters)of 1 kg of water at a temperature T is given approximately by the formula V − 999.87 2 0.06426T 1 0.0085043T 2 2 0.0000679T 3 Find the temperature at which water has its maximum density.
On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite.The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters.(a) Use
Coughing When a foreign object lodged in the trachea(windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. For a given amount
The Maynard Smith and Slatkin model for population growth is a discrete-time model of the form nt11 − nt 1 1 nk tFor the constants − 2, − 0.25, and k − 2, the model is nt11 − f sntd, where the updating function is f snd −2n 1 1 0.25n2 Find the largest value of f and interpret it.
Disease virulence The Kermack-McKendrick model for infectious disease transmission (see Exercise 7.6.23) can be used to predict the population size P as a function of the disease’s virulence (that is, the extent to which the disease kills people). The population size P is large when virulence v
A ntibiotic pharmacokinetics After an antibiotic tablet is taken, the concentration of the antibiotic in the bloodstream is modeled by the function Cstd − 8se20.4t 2 e20.6t d where the time t is measured in hours and C is measured in mgymL. What is the maximum concentration of the antibiotic
If a and b are positive numbers, find the maximum value of f sxd − xas1 2 xdb, 0 < x < 1.
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