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Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions
I nfectious disease outbreak size In Example 13 we used the equation e2qA − 1 2 A to determine the rate of increase of the outbreak size A with respect to the transmissibility q. Use this same equation to find the rate of change of A with respect to , the fraction of the population that is
x2y3 1 y2y3 − 4, s23s3, 1d (astroid) 8 x
x2 1 y2 − s2x2 1 2y2 2 xd2, s0, 12 d (cardioid) x
x2 1 2xy 2 y2 1 x − 2, s1, 2d (hyperbola)
x2 1 xy 1 y2 − 3, s1, 1d (ellipse)
sin x 1 cos y − sin x cos y Find dyydx by implicit differentiation.
ey cos x − 1 1 sinsxyd Find dyydx by implicit differentiation.
tansx 2 yd −y 1 1 x2 Find dyydx by implicit differentiation.
exyy − x 2 y Find dyydx by implicit differentiation.
1 1 x − sinsxy2d Find dyydx by implicit differentiation.
4 cos x sin y − 1 Find dyydx by implicit differentiation.
y5 1 x2y3 − 1 1 yex2 Find dyydx by implicit differentiation.
x4sx 1 yd − y2s3x 2 yd Find dyydx by implicit differentiation.
2x3 1 x2y 2 xy3 − 2 Find dyydx by implicit differentiation.
x2 1 xy 2 y2 − 4 Find dyydx by implicit differentiation.
2sx 1 sy − 3 Find dyydx by implicit differentiation.
x3 1 y3 − 1 Find dyydx by implicit differentiation.
cos x 1 sy − 5(a) Find y9 by implicit differentiation.(b) Solve the equation explicitly for y and differentiate to get y9 in terms of x.(c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a).
xy 1 2x 1 3x2 − 4(a) Find y9 by implicit differentiation.(b) Solve the equation explicitly for y and differentiate to get y9 in terms of x.(c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a).
In Example 1.3.4 we arrived at a model for the length of daylight (in hours) in Philadelphia on the tth day of the year:Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 and May 21. 2T == L(t) 12+2.8 sin (t - 80) 365
Under certain circumstances a rumor spreads according to the equation pstd −1 1 1 ae2k t where pstd is the proportion of the population that has heard the rumor at time t and a and k are positive constants.(a) Find limt l ` pstd.(b) Find the rate of spread of the rumor.; (c) Graph p for the case
Logistic growth in Japan The midyear population in Japan from 1960 to 2010 has been modeled by the function Pstd − 94,000 1 32,658.5 1 1 12.75e20.1706t where t is measured in years since 1960 and Pstd is measured in thousands. According to this model, how quickly was the Japanese population
Blood alcohol concentration In Section 3.1 we discussed an experiment in which the average BAC of eight male subjects was measured after consumption of 15 mL of ethanol (corresponding to one alcoholic drink). The resulting data were modeled by the concentration function Cstd − 0.0225te20.0467t
World population growth In Example 1.4.3 we modeled the world population from 1900 to 2010 with the exponential function Pstd − s1436.53d ? s1.01395dt where t − 0 corresponds to the year 1900 and Pstd is measured in millions. According to this model, what was the rate of increase of world
Gene regulation Genes produce molecules called mRNA that go on to produce proteins. High concentrations of protein inhibit the production of mRNA, leading to stable gene regulation. This process has been modeled (see Section 10.3) to show that the concentration of mRNA over time is given by the
O ral antibiotics In Example 1.3.7 we studied a model for antibiotic use in sinus infections. If x is the amount of antibiotic taken orally (in mg), then the function hsxd gives the amount entering the bloodstream through the stomach.If x mg reaches the bloodstream, then tsxd gives the amount that
The displacement of a particle on a vibrating string is given by the equation sstd − 10 1 14 sins10 td where s is measured in centimeters and t in seconds. Find the velocity and acceleration of the particle after t seconds.
Find the 1000th derivative of f sxd − xe2x.
Find the 50th derivative of y − cos 2x.
If t is a twice differentiable function and f sxd − xtsx2 d, find f 99 in terms of t, t9, and t99.
Let rsxd − f s tshsxddd, where hs1d − 2, ts2d − 3, h9s1d − 4, t9s2d − 5, and f 9s3d − 6. Find r9s1d.
Suppose f is differentiable on R and is a real number.Let Fsxd − f sx d and Gsxd − f f sxdg. Find expressions for (a) F9sxd and (b) G9sxd.
Suppose f is differentiable on R. Let Fsxd − f sex d and Gsxd − e f sxd. Find expressions for (a) F9sxd and (b) G9sxd.
Let f and t be the functions in Exercise 47.(a) If Fsxd − f s f sxdd, find F9s2d.(b) If Gsxd − tstsxdd, find G9s3d.
A table of values for f , t, f 9, and t9 is given.(a) If hsxd − f stsxdd, find h9s1d.(b) If Hsxd − ts f sxdd, find H9s1d. x f(x) g(x) f'(x) g'(x) 1 3 23 2 1 7 282 2 457 6 7 9
If hsxd − s4 1 3f sxd , where f s1d − 7 and f 9s1d − 4, find h9s1d.
If Fsxd − f stsxdd, where f s22d − 8, f 9s22d − 4, f 9s5d − 3, ts5d − 22, and t9s5d − 6, find F9s5d.
y − sin x 1 sin2x, s0, 0d Find an equation of the tangent line to the curve at the given point.
y − sinssin xd, s , 0d Find an equation of the tangent line to the curve at the given point.
y − s1 1 x3 , s2, 3d Find an equation of the tangent line to the curve at the given point.
y − s1 1 2xd10, s0, 1d Find an equation of the tangent line to the curve at the given point.
y − ee x Find y9 and y99.
y − ex sin x Find y9 and y99.
y − cos2x Find y9 and y99.
y − cossx2d Find y9 and y99.
y − 23x Find the derivative of the function.
y − cosssinstan xd Find the derivative of the function.
y − sx 1 sx 1 sx Find the derivative of the function.
y − cot2ssin d Find the derivative of the function.
y − sinssinssin xdd Find the derivative of the function.
y − 2sin x Find the derivative of the function.
f std −Î t t 2 1 4 Find the derivative of the function.
y − sinstan 2xd Find the derivative of the function.
y − ek tan sx Find the derivative of the function.
y −r sr 2 1 1 Find the derivative of the function.
y −eu 2 e2u eu 1 e2u Find the derivative of the function.
y − sec2x 1 tan2x Find the derivative of the function.
Gs yd − S y2 y 1 1D5 Find the derivative of the function.
y − Sx2 1 1 x2 2 1D3 Find the derivative of the function.
y − 1012x 2 Find the derivative of the function.
y − ex cos x Find the derivative of the function.
hstd − st 4 2 1d3st 3 1 1d4 Find the derivative of the function.
y − s2x 2 5d4s8x2 2 5d23 Find the derivative of the function.
y − e22t cos 4t Find the derivative of the function.
y − xe2kx Find the derivative of the function.
y − 3 cotsn d Find the derivative of the function.
hstd − t 3 2 3t Find the derivative of the function.
y − a3 1 cos3x Find the derivative of the function.
y − cossa3 1 x3d Find the derivative of the function.
f std − s3 1 1 tan t Find the derivative of the function.
f szd −1 z 2 1 1 Find the derivative of the function.
f sxd − s1 1 x4d2y3 Find the derivative of the function.
Fsxd − s1 2 2x Find the derivative of the function.
Fsxd − s4x 2 x2d100 Find the derivative of the function.
Fsxd − sx4 1 3x2 2 2d5 Find the derivative of the function.
y − s2 2 ex Write the composite function in the form f s tsxdd.[Identify the inner function u − tsxd and the outer function y − f sud.] Then find the derivative dyydx.
y − esx Write the composite function in the form f s tsxdd.[Identify the inner function u − tsxd and the outer function y − f sud.] Then find the derivative dyydx.
y − sinscot xd Write the composite function in the form f s tsxdd.[Identify the inner function u − tsxd and the outer function y − f sud.] Then find the derivative dyydx.
y − tan x Write the composite function in the form f s tsxdd.[Identify the inner function u − tsxd and the outer function y − f sud.] Then find the derivative dyydx.
y − s2x3 1 5d4 Write the composite function in the form f s tsxdd.[Identify the inner function u − tsxd and the outer function y − f sud.] Then find the derivative dyydx.
y − s3 1 1 4x Write the composite function in the form f s tsxdd.[Identify the inner function u − tsxd and the outer function y − f sud.] Then find the derivative dyydx.
(a) If t is differentiable, the Reciprocal Rule says that ddx F 1 tsxdG− 2 t9sxd f tsxdg2 Use the Quotient Rule to prove the Reciprocal Rule.(b) Use the Reciprocal Rule to differentiate the function y − 1ysx4 1 x2 1 1d.(c) Use the Reciprocal Rule to verify that the Power Rule is valid for
(a) If Fsxd − f sxd tsxd, where f and t have derivatives of all orders, show that F99 − f 99t 1 2f 9t9 1 f t99.(b) Find similar formulas for F999 and F s4d.(c) Guess a formula for Fsnd.
(a) Use the Product Rule twice to prove that if f , t, and h are differentiable, then s fthd9 − f 9th 1 ft9h 1 fth9.(b) Taking f − t − h in part (a), show that ddx f f sxdg3 − 3f f sxdg2 f 9sxd(c) Use part (b) to differentiate y − e3x.
Find equations of the tangent lines to the curve y −x 2 1 x 1 1 that are parallel to the line x 2 2y − 2.
How many tangent lines to the curve y − xysx 1 1) pass through the point s1, 2d? At which points do these tangent lines touch the curve?
Sensitivity of the eye to brightness If R denotes the reaction of the body to some stimulus of strength x, the sensitivity S is defined to be the rate of change of the reaction with respect to x. A particular example is that when the brightness x of a light source is increased, the eye reacts by
The gas law for an ideal gas at absolute temperature T (in kelvins), pressure P(in atmospheres), and volume V (in liters) is PV − nRT, where n is the number of moles of the gas and R − 0.0821 is the gas constant. Suppose that, at a certain instant, P − 8.0 atm and is increasing at a rate of
The biomass Bstd of a fish population is the total mass of the members of the population at time t. It is the product of the number of individuals Nstd in the population and the average mass Mstd of a fish at time t. In the case of guppies, breeding occurs continually. Suppose that at time t − 4
The Michaelis-Menten equation for the enzyme chymotrypsin is v −0.14fSg 0.015 1 fSg where v is the rate of an enzymatic reaction and [S] is the concentration of a substrate S. Calculate dvydfSg and interpret it.
I nsecticide resistance If the frequency of a gene for insecticide resistance is p, then its frequency in the next generation is given by the expression ps1 1 sd 1 1 sp where s is the reproductive advantage that this gene has over the wild type in the presence of the insecticide.Determine the rate
If t is a differentiable function, find an expression for the derivative of each of the following functions.(a) y − xtsxd (b) y −x tsxd(c) y − tsxd x
Let Psxd − FsxdGsxd and Qsxd − FsxdyGsxd, where F and G are the functions whose graphs are shown.(a) Find P9s2d. (b) Find Q9s7d. y F 0 1 G
If f and t are the functions whose graphs are shown, let usxd − f sxdtsxd and vsxd − f sxdytsxd.(a) Find u9s1d. (b) Find v9s5d. r 6 'f 0
Suppose that f s2d − 23, ts2d − 4, f 9s2d − 22, and t9s2d − 7. Find h9s2d.(a) hsxd − 5f sxd 2 4tsxd (b) hsxd − f sxd tsxd(c) hsxd −f sxd tsxd(d) hsxd − tsxd 1 1 f sxd
Suppose that f s5d − 1, f 9s5d − 6, ts5d − 23, and t9s5d − 2. Find the following values.(a) s ftd9s5d (b) s fytd9s5d (c) s tyf d9s5d
Suppose that f s y3d − 4 and f 9s y3d − 22, and let tsxd − f sxd sin x and hsxd − scos xdyf sxd. Find(a) t9s y3d (b) h9s y3d
Prove that ddx scot xd − 2csc2x.
Prove that ddx ssec xd − sec x tan x.
Prove that ddx scsc xd − 2csc x cot x.
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