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Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions
y − arctansarcsin sx d Calculate y9.
y − sinstan s1 1 x3 d Calculate y9.
y − ln Z x2 2 4 2x 1 5 Z Calculate y9.
y − tan2ssin d Calculate y9.
y − 10tan Calculate y9.
y − ln |sec 5x 1 tan 5x |Calculate y9.
y − ecos x 1 cossex d Calculate y9.
y − x tan21s4xd Calculate y9.
y −sx2 1 1d4 s2x 1 1d3s3x 2 1d5 Calculate y9.
y − ln sin x 2 12 sin2x Calculate y9.
xey − y 2 1 Calculate y9.
y − 3x ln x Calculate y9.
y − st lnst 4d Calculate y9.
sinsxyd − x2 2 y Calculate y9.
y − sln xd cos x Calculate y9.
y − log5s1 1 2xd Calculate y9.
y − lnsx2exd Calculate y9.
y − ecxsc sin x 2 cos xd Calculate y9.
x2 cos y 1 sin 2y − xy Calculate y9.
y −sec 2 1 1 tan 2 Calculate y9.
y − lnscsc 5xd Calculate y9.
xy4 1 x2y − x 1 3y Calculate y9.
y − S u 2 1 u2 1 u 1 1D4 Calculate y9.
y −Calculate y9.e1yx x2
y − emx cos nx Calculate y9.
y −t 1 2 t 2 Calculate y9.
y − e2tst 2 2 2t 1 2d Calculate y9.
y − esin 2 Calculate y9.
y −ex 1 1 x2 Calculate y9.
y − 2xsx2 1 1 Calculate y9.
y −3x 2 2 s2x 1 1 Calculate y9.
y − sx 1 1s3 x4 Calculate y9.
y − cosstan xd Calculate y9.
y − sx4 2 3x2 1 5d3 Calculate y9.
The total fertility rate at time t, denoted by Fstd, is an estimate of the average number of children born to each woman (assuming that current birth rates remain constant).The graph of the total fertility rate in the United States shows the fluctuations from 1940 to 2010.(a) Estimate the values of
The graph of f is shown. State, with reasons, the numbers at which f is not differentiable. Fr y. 0 -1 2 4 6 x
The figure shows the graphs of f , f 9, and f 0. Identify each curve, and explain your choices. y a 0 b C x
(a) If f sxd − s3 2 5x , use the definition of a derivative to find f 9sxd.(b) Find the domains of f and f 9.; (c) Graph f and f 9 on a common screen. Compare the graphs to see whether your answer to part (a) is reasonable.
(a) Find the asymptotes of the graph of f sxd −4 2 x 3 1 x and use them to sketch the graph.(b) Use your graph from part (a) to sketch the graph of f 9.(c) Use the definition of a derivative to find f 9sxd.; (d) Use a calculator to graph f 9 and compare with your sketch in part (b).
A ntihypertension medication The figure shows the heart rate Hstd after a patient has taken nifedipine tablets.(a) What is the meaning of the derivative H9std?(b) Sketch the graph of H9std. HA 70- Beats per minute 65- 65 60 60 1 3 5 t (hours)
Bacteria count Shown is a typical graph of the number N of bacteria grown in a bacteria culture as a function of time t.(a) What is the meaning of the derivative N9std?(b) Sketch the graph of N9std.
Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath. 7.
Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath. 6. 0 x
Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath. 5. YA 0 x
(a) Use the definition of a derivative to find f 9s2d, where f sxd − x3 2 2x.(b) Find an equation of the tangent line to the curve y − x3 2 2x at the point (2, 4).;(c) Illustrate part (b) by graphing the curve and the tangent line on the same screen.
The total cost of repaying a student loan at an interest rate of r% per year is C − f srd.(a) What is the meaning of the derivative f 9srd? What are its units?(b) What does the statement f 9s10d − 1200 mean?(c) Is f 9srd always positive or does it change sign?
Life expectancy The table shows how the life expectancy Lstd in Bangladesh has changed from 1990 to 2010.(a) Calculate the average rate of change of the life expectancy Lstd with respect to time over the following time intervals.(i) f1990, 2000g (ii) f1995, 2000g (iii) f2000, 2010g (iv) f2000,
For the function f whose graph is shown, arrange the following numbers in increasing order:0 1 f 9s2d f 9s3d f 9s5d f 0s5d 1 0 1 X
An equation of the tangent line to the parabola y − x2 at s22, 4d is y 2 4 − 2xsx 1 2d.
If tsxd − x5, then lim x l 2 tsxd 2 ts2d x 2 2− 80
If y − e2, then y9 − 2e.==9.d dx s10x d − x10x21==10.d dx sln 10d −1 10 ==11.d dx stan2xd −d dx ssec2xd==12.d2y dx2 − Sdy dxD2
If f is differentiable, then ddx f ssx d −f 9sxd 2sx .
If f is differentiable, then ddx sf sxd −f 9sxd 2sf sxd .
If f and t are differentiable, then ddx f f stsxddg − f 9stsxdd t9sxd
If f and t are differentiable, then ddx f f sxd 1 tsxdg − f 9sxd 1 t9sxd 4. If f and t are differentiable, then ddx f f sxd tsxdg − f 9sxd t9sxd
If f 9srd exists, then limx l r f sxd − f srd.
If f is continuous ata, then f is differentiable at a.
Write an expression for the nth-degree Taylor polynomial of f centered at a.
Write an expression for the linearization of f at a.
(a) Explain how implicit differentiation works. When should you use it?(b) Explain how logarithmic differentiation works. When should you use it?
(a) How is the number e defined?(b) Express e as a limit.(c) Why is the natural exponential function y − ex used more often in calculus than the other exponential functions y − bx?(d) Why is the natural logarithmic function y − ln x used more often in calculus than the other logarithmic
State the derivative of each function.(a) y − xn (b) y − ex (c) y − bx(d) y − ln x (e) y − logb x (f) y − sin x(g) y − cos x (h) y − tan x (i) y − csc x( j) y − sec x (k) y − cot x (l) y − tan21x
State each differentiation rule both in symbols and in words.(a) The Power Rule (b) The Constant Multiple Rule(c) The Sum Rule (d) The Difference Rule(e) The Product Rule (f) The Quotient Rule(g) The Chain Rule
Describe several ways in which a function can fail to be differentiable. Illustrate with sketches.
(a) What does it mean for f to be differentiable at a?(b) What is the relation between the differentiability and continuity of a function?(c) Sketch the graph of a function that is continuous but not differentiable at a − 2.
Define the second derivative of f . If f std is the position function of a particle, how can you interpret the second derivative?
If y − f sxd and x changes from x1 to x2, write expressions for the following.(a) The average rate of change of y with respect to x over the interval fx1, x2 g(b) The instantaneous rate of change of y with respect to x at x − x1
Define the derivative f 9sad. Discuss two ways of interpreting this number.
Write an expression for the slope of the tangent line to the curve y − f sxd at the point sa, f sadd.
Show that if a polynomial Tn of the form given in Equation 5 has the same value at a and the same derivatives at x − a as a function f , then its coefficients are given by the formula ck −f skdsad k!
H abitat fragmentation and species conservation The size of a class-structured population is modeled in Section 8.8. In certain situations the long-term per capita growth rate of the population is given by r − 12(1 1 s1 1 8s )where s is the annual survival probability of juveniles.(a) Approximate
Find the 8th-degree Taylor polynomial centered at a − 0 for the function f sxd − cos x. Graph f together with the Taylor polynomials T2, T4, T6, T8 in the viewing rectangle f25, 5g by f21.4, 1.4g and comment on how well they approximate f .
Find the first five Taylor polynomials for f sxd − sin x centered at 0. Graph them on the interval f24, 4g and comment on how well they approximate f .
Determine the values of x for which the quadratic approximation f sxd < T2sxd in Example 7 is accurate to within 0.1.[Hint: Graph y − T2sxd, y − cos x 2 0.1, and y − cos x 1 0.1 on a common screen.]
Find the quadratic approximation to f sxd − sx 1 3 near a − 1. Graph f , the quadratic approximation, and the linear approximation from Example 1 on a common screen. What do you conclude?
f sxd − sx , n − 2, a − 4 Find the Taylor polynomial of degree n centered at the number a.
f sxd − 1yx, n − 4, a − 1 Find the Taylor polynomial of degree n centered at the number a.
f sxd − sin x, n − 3, a − 0 Find the Taylor polynomial of degree n centered at the number a.
f sxd − ex, n − 3, a − 0 Find the Taylor polynomial of degree n centered at the number a.
(a) Use Newton’s method with x1 − 1 to find the root of the equation x3 2 x − 1 correct to six decimal places.(b) Solve the equation in part (a) using x1 − 0.6 as the initial approximation.(c) Solve the equation in part (a) using x1 − 0.57. (You definitely need a programmable calculator
Explain why Newton’s method doesn’t work for finding the root of the equation x3 2 3x 1 6 − 0 if the initial approximation is chosen to be x1 − 1.
I nfectious disease outbreak size If 99% of a population is initially uninfected and each initial infected person generates, on average, two new infections, then, according to the model we considered in Example 3.5.13, 0.99e22A − 1 2 A where A is the fraction of the population infected at the end
4e2x 2 sin x − x2 2 x 1 1 Use Newton’s method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.
3 sinsx2d − 2x Use Newton’s method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.
x2s2 2 x 2 x2 − 1 Use Newton’s method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.
x2s4 2 x2d −4 x2 1 1 Use Newton’s method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.
x6 2 x5 2 6x4 2 x2 1 x 1 10 − 0 Use Newton’s method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.
sx 2 2d2 − ln x Use Newton’s method to find all roots of the equation correct to six decimal places.==26.1 x − 1 1 x3 Use Newton’s method to find all roots of the equation correct to six decimal places.
ex − 3 2 2x Use Newton’s method to find all roots of the equation correct to six decimal places.
x4 − 1 1 x Use Newton’s method to find all roots of the equation correct to six decimal places.
13 x3 1 12 x2 1 3 − 0, x1 − 23 Use Newton’s method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Give your answer to four decimal places.)
x3 1 2x 2 4 − 0, x1 − 1 Use Newton’s method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Give your answer to four decimal places.)
Follow the instructions for Exercise 19(a) but use x1 − 9 as the starting approximation for finding the root s.
The figure shows the graph of a function f . Suppose that Newton’s method is used to approximate the root r of the equation f sxd − 0 with initial approximation x1 − 1.(a) Draw the tangent lines that are used to find x2 and x3, and estimate the numerical values of x2 and x3.(b) Would x1 − 5
Volume and surface area of a tumor The diameter of a tumor was measured to be 19 mm. If the diameter increases by 1 mm, use linear approximations to estimate the relative changes in the volume (V − 43 r 3) and surface area sS − 4 r 2d.
R elative change in blood flow Another law of Poiseuille says that when blood flows along a blood vessel, the flux F(the volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius R of the blood vessel:F − kR4(We will show why this is true in
R elative change in blood velocity Suppose y − f sxd and x and y change by amounts Dx and Dy. A way of expressing a linear approximation is to write Dy < f 9sxd Dx.The relative change in y is Dyyy.A special case of Poiseuille’s law of laminar flow (see Example 3.3.9) is that at the central axis
I nsecticide resistance If the frequency of a gene for insecticide resistance is p (a constant), then its frequency in the next generation is given by the expressionwhere s is the reproductive advantage this gene has over the wild type in the presence of the insecticide. Often the selective
s1.01d6 < 1.06 Explain, in terms of linear approximations, why the approximation is reasonable.
ln 1.05 < 0.05 Explain, in terms of linear approximations, why the approximation is reasonable.
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