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Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions
f sxd − 2sx 1 6 cos x Find the most general antiderivative of the function.(Check your answer by differentiation.)
ts d − cos 2 5 sin Find the most general antiderivative of the function.(Check your answer by differentiation.)
hsmd −2 sm Find the most general antiderivative of the function.(Check your answer by differentiation.)
cstd −3 t 2 , t . 0 Find the most general antiderivative of the function.(Check your answer by differentiation.)
f sxd − e2 Find the most general antiderivative of the function.(Check your answer by differentiation.)
f sxd − s2 Find the most general antiderivative of the function.(Check your answer by differentiation.)
f sxd − s4 x3 1 s3 x4 Find the most general antiderivative of the function.(Check your answer by differentiation.)
f sxd − 6sx 2 s6 x Find the most general antiderivative of the function.(Check your answer by differentiation.)
f sxd − 2x 1 3x1.7 Find the most general antiderivative of the function.(Check your answer by differentiation.)
f sxd − 5x1y4 2 7x3y4 Find the most general antiderivative of the function.(Check your answer by differentiation.)
f sxd − xs2 2 xd2 Find the most general antiderivative of the function.(Check your answer by differentiation.)
f sxd − sx 1 1ds2x 2 1d Find the most general antiderivative of the function.(Check your answer by differentiation.)
f sxd − 8x9 2 3x6 1 12x3 Find the most general antiderivative of the function.(Check your answer by differentiation.)
f sxd − 12 1 34 x2 2 45 x3 Find the most general antiderivative of the function.(Check your answer by differentiation.)
Blood cell production A simple model of blood cell production is given by Rt11 − Rts1 2 dd 1 FsRtd where d is the fraction of red blood cells that die from one day to the next and Fsxd is a function specifying the number of new cells produced in a day, given that the current number is x. Find the
Drug resistance in malaria In the project on page 78 we developed the following recursion equation for the spread of a gene for drug resistance in malaria:where WRR, WRS, and WSS are constants representing the probability of survival of the three genotypes. In fact this model applies to the
Species discovery curves A common assumption is that the rate of discovery of new species is proportional to the fraction of currently undiscovered species. If dt is the fraction of species discovered by time t, a recursion equation describing this process is dt11 − dt 1 as1 2 dtd where a is a
H eart excitation A simple model for the time xt it takes for an electrical impulse in the heart to travel through the atrioventricular node of the heart is(a) Find the relevant equilibrium and determine when it is stable.(b) Draw a cobwebbing diagram. 375 X1+1 + 100 xt > 90 x- 90
Sustainable harvesting In Example 4.4.5 we looked at a model of sustainable harvesting, which can be formulated as a discrete-time model:Find the equilibria and determine when each is stable. N N+N+rN 1- - hN K
c − 3.6, x0 − 0.4 Logistic difference equation Illustrate the results of Example 4 for the logistic difference equation by cobwebbing and by graphing the first ten terms of the sequence for the given values of c and x0.
c − 2.7, x0 − 0.1 Logistic difference equation Illustrate the results of Example 4 for the logistic difference equation by cobwebbing and by graphing the first ten terms of the sequence for the given values of c and x0.
c − 1.8, x0 − 0.1 Logistic difference equation Illustrate the results of Example 4 for the logistic difference equation by cobwebbing and by graphing the first ten terms of the sequence for the given values of c and x0.
c − 0.8, x0 − 0.6 Logistic difference equation Illustrate the results of Example 4 for the logistic difference equation by cobwebbing and by graphing the first ten terms of the sequence for the given values of c and x0.
Drug pharmacokinetics A patient is injected with a drug every 8 hours. Immediately before each injection the concentration of the drug has been reduced by 40% and the new dose increases the concentration by 1.2 mgymL.(a) If Qn is the concentration of the drug in the body just after the nth
Drug pharmacokinetics A patient takes 200 mg of a drug at the same time every day. Just before each tablet is taken, 10% of the drug remains in the body.(a) If Qn is the quantity of the drug in the body just after the nth tablet is taken, write a difference equation expressing Qn11 in terms of
xt11 −xt c 1 xt Find the equilibria of the difference equation. Determine the values of c for which each equilibrium is stable.
xt11 −cxt 1 1 xt Find the equilibria of the difference equation. Determine the values of c for which each equilibrium is stable.
xt11 −7xt 2xt 2 1 10, x0 − 1, x0 − 3 Find the equilibria of the difference equation and classify them as stable or unstable. Use cobwebbing to find limtl` xt for the given initial values.
xt11 −4xt 2xt 2 1 3, x0 − 0.5, x0 − 2 Find the equilibria of the difference equation and classify them as stable or unstable. Use cobwebbing to find limtl` xt for the given initial values.
xt11 − xt 3 2 3xt 2 1 3xt Find the equilibria of the difference equation and classify them as stable or unstable.
xt11 − 10xte22xt Find the equilibria of the difference equation and classify them as stable or unstable.
xt11 −3xt 1 1 xt Find the equilibria of the difference equation and classify them as stable or unstable.
xt11 −xt 0.2 1 xt Find the equilibria of the difference equation and classify them as stable or unstable.
xt11 − 1 2 xt 2Find the equilibria of the difference equation and classify them as stable or unstable.
xt11 − 12 xt 2Find the equilibria of the difference equation and classify them as stable or unstable.
The graph of the function f for a recursive sequence xt11 − f sxtd is shown. Estimate the equilibria and classify them as stable or unstable. Confirm your answer by cobwebbing. X+14 f 2+ 0 1 2 x
The graph of the function f for a recursive sequence xt11 − f sxtd is shown. Estimate the equilibria and classify them as stable or unstable. Confirm your answer by cobwebbing. X+1 0 f + 2 + x
The graph of the function f for a recursive sequence xt11 − f sxtd is shown. Estimate the equilibria and classify them as stable or unstable. Confirm your answer by cobwebbing. 0.5 f 0 0.5 x
The graph of the function f for a recursive sequence xt11 − f sxtd is shown. Estimate the equilibria and classify them as stable or unstable. Confirm your answer by cobwebbing. x+11 0.5+ 0 + 0.5
Crows and whelks Crows on the west coast of Canada feed on whelks by carrying them to heights of about 5 m and dropping them onto rocks (several times if necessary)to break open their shells. Two of the questions raised by the author of a study of this phenomenon were “Do crows drop whelks from
Bird flight paths Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than over land because air generally rises over land and falls over water during the day.
Swimming speed of fish For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v 3. It is believed that migrating fish try to minimize the total energy required to swim a fixed distance.If the fish are swimming against a current u su , vd,
Beehives In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end as in the figure.It is believed that bees form their cells in such a way as to minimize the surface area for a given side length and height, thus using the least amount of wax in
The graph shows the fuel consumption c of a car (measured in gallons per hour) as a function of the speed v of the car. At very low speeds the engine runs inefficiently, so initially c decreases as the speed increases. But at high speeds the fuel consumption increases. You can see that csvd is
Let v1 be the velocity of light in air and v2 the velocity of light in water. According to Fermat’s Principle, a ray of light will travel from a point A in the air to a point B in the water by a path ACB that minimizes the time taken. Show that sin 1 sin 2−v1 v2 where 1 (the angle of incidence)
Aquatic birds Suppose that, instead of the specific oxygen function in Example 4, we have an arbitrary function O with Os0d − 0, Ostd > 0, O9std . 0, O0std , 0, and arbitrary travel time T.(a) Interpret the conditions on the function O.(b) Show that the optimal surface time t satisfies the
N ectar foraging by bumblebees Suppose that, instead of the specific nectar function in Example 2, we have an arbitrary function N with Ns0d − 0, Nstd > 0, N9std . 0, N0std , 0, and arbitrary travel time T.(a) Interpret the conditions on the function N.(b) Show that the optimal foraging time t
The von Bertalanffy model for the growth of an individual fish assumes that an individual acquires energy at a rate that is proportional to its length squared, but that it uses energy for metabolism at a rate proportional to its length cubed. The underlying idea is that energy acquisition is
Sustainable harvesting Example 5 was based on the assumption that we want to maximize the total harvest size H, but instead we might want to maximize profit.(a) Suppose the selling price of a unit of harvest is p dollars and the cost per unit harvested is C dollars. (Assume p . C.) Show that the
If Rsxd is the revenue that a company receives when it sells x units of a product, then the marginal revenue function is the derivative R9sxd. The profit function is Psxd − Rsxd 2 Csxd where C is the cost function from Exercise 19.(a) Show that if the profit Psxd is a maximum, then the marginal
If Csxd is the cost of producing x units of a commodity, then the average cost per unit is csxd − Csxdyx. The marginal cost is the rate of change of the cost with respect to the number of items produced, that is, the derivative C9sxd.(a) Show that if the average cost is a minimum, then the
A ge and size at maturity Most organisms grow for a period of time before maturing reproductively. For many species of insects and fish, the later the age a at maturity, the larger the individual will be, and this translates into a greater reproductive output. At the same time, however, the
Find the point on the curve y − sx that is closest to the point s3, 0d.
Find the point on the line y − 2x 1 3 that is closest to the origin.
A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $10 per square meter.Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.
(a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square.(b) Show that of all the rectangles with a given perimeter, the one with greatest area is a square.
A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used.
If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.(a) Draw several diagrams to illustrate the
Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle.What is the largest possible total area of the four pens?(a) Draw several diagrams illustrating the situation,
The measles pathogenesis function f std − 2tst 2 21dst 1 1d is used in Section 5.1 to model the development of the disease, where t is measured in days and f std represents the number of infected cells per milliliter of plasma. What is the peak infection time for the measles virus?
Crop yield A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is where k is a positive constant. What nitrogen level gives the best yield? Y= Y = kN 1+ N
P hotosynthesis The rate sin mg carbonym3yhd at which photosynthesis takes place for a species of phytoplankton is modeled by the functionwhere I is the light intensity (measured in thousands of footcandles).For what light intensity is P a maximum? P= 1001 12+1+4
Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.
The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares?
Find two positive numbers whose product is 100 and whose sum is a minimum.
Find two numbers whose difference is 100 and whose product is a minimum.
Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum.(a) Make a table of values, like the following one, so that the sum of the numbers in the first two columns is always 23.On the basis of the evidence in your table, estimate the answer to the problem.(b)
If f 9 is continuous, f s2d − 0, and f 9s2d − 7, evaluate lim x l 0 f s2 1 3xd 1 f s2 1 5xd x
The first appearance in print of l’Hospital’s Rule was in the book Analyse des Infiniment Petits published by the Marquis de l’Hospital in 1696. This was the first calculus textbook ever published and the example that the Marquis used in that book to illustrate his rule was to find the limit
Prove that lim x l `ln x x p − 0 for any number p . 0. This shows that the logarithmic function approaches infinity more slowly than any power of x.
Prove that lim x l `ex xn − `for any positive integer n. This shows that the exponential function approaches infinity faster than any power of x.
Drug pharmacokinetics The level of medication in the bloodstream after a drug is administered is often modeled by a function of the form Sstd − At pe2kt where A, p, and k are positive constants. This is called a surge function because its graph shows an initial surge in the drug level followed by
Drug pharmacokinetics So-called two-compartment models are often used to describe drug pharmacokinetics, with the blood being one compartment and the internal organs being the other (see Section 10.3). If the rate of flow from the blood to the organs is and the rate of metabolism from the organs
A ntibiotic concentration for large patients Suppose an antibiotic is administered at a constant rate through intravenous supply to a patient and is metabolized. It can be shown using the type of mixing models discussed in Exercises 7.4.45–48 that the concentration of antibiotic after one unit of
Stiles-Crawford effect Light enters the eye through the pupil and strikes the retina, where photoreceptor cells sense light and color. W. Stanley Stiles and B. H. Crawford studied the phenomenon in which measured brightness decreases as light enters farther from the center of the pupil.(See the
Foraging In Exercise 4.2.2 we let Nstd be the amount of nectar foraged from a flower by a bumblebee in t seconds.(a) What is the average rate of nectar consumption over a period of t seconds?(b) Find the average rate of nectar consumption for very short foraging visits by using l’Hospital’s
Models of population growth have the general form dNydt − f sNd, where f sNd is a function such that f s0d − 0 and f sNd is positive for some positive values of N. The per capita growth rate is defined to be the population growth rate divided by the population size.(a) What is the per capita
lim x l s y2d2 sec x tan x What happens if you try to use l’Hospital’s Rule to find the limit? Evaluate the limit using another method.
lim xl `x sx2 1 1 What happens if you try to use l’Hospital’s Rule to find the limit? Evaluate the limit using another method.
y − e2x, y − xe2x, y − e2x 2, y − xe2x 2 Rank the functions in order of how quickly they approach 0 as x l `.
y −1 x, y −1 x2 , y − e2x, y − x21y2 Rank the functions in order of how quickly they approach 0 as x l `.
lim xl`x2 2 x 1 ln x x 1 2x Guess the value of the limit by considering the dominant terms in the numerator and denominator. Then use l’Hospital’s Rule to confirm your guess.
lim xl`e22x 1 x 1 e0.1x x3 2 x2 Guess the value of the limit by considering the dominant terms in the numerator and denominator. Then use l’Hospital’s Rule to confirm your guess.
y − x 1 e2x, y − 10 ln x, y − 5sx , y − xsx Rank the functions in order of how quickly they grow as x l `.
y − sln xd2, y − sln xd3, y − sx , y − s3 x Rank the functions in order of how quickly they grow as x l `.
y − 2x, y − 3x, y − exy2, y − exy3 Rank the functions in order of how quickly they grow as x l `.
y − x5, y − lnsx10d, y − e2x, y − e3x Rank the functions in order of how quickly they grow as x l `.
f sxd − xe2x2 Use l’Hospital’s Rule to help find the asymptotes of f .Then use them, together with information from f 9 and f 0, to sketch the graph of f . Check your work with a graphing device.
f sxd − sln xdyx Use l’Hospital’s Rule to help find the asymptotes of f .Then use them, together with information from f 9 and f 0, to sketch the graph of f . Check your work with a graphing device.
f sxd − exyx Use l’Hospital’s Rule to help find the asymptotes of f .Then use them, together with information from f 9 and f 0, to sketch the graph of f . Check your work with a graphing device.
f sxd − xe2x Use l’Hospital’s Rule to help find the asymptotes of f .Then use them, together with information from f 9 and f 0, to sketch the graph of f . Check your work with a graphing device.
lim xl`sxe1yx 2 xd 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 `sx 2 ln xd 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 01 Scot x 2 1x 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 xl`(sx2 1 x 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.
lim x l 0 scsc x 2 cot xd 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 S x x 2 1 21 ln xD 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 tans1yxd 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 `x3e2x2 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.
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