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Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions
(a) Show that y`2` x dx is divergent.(b) Show thatThis shows that we can’t define lim 100 Sx dx = 0
If y`2` f sxd dx is convergent and a and b are real numbers, show that S_ f(x) dx + f* f(x) dx = _ f(x) dx + f(x) dx -00
A manufacturer of lightbulbs wants to produce bulbs that last about 700 hours but, of course, some bulbs burn out faster than others. Let Fstd be the fraction of the company’s bulbs that burn out before t hours, so Fstd always lies between 0 and 1.(a) Make a rough sketch of what you think the
Dialysis treatment removes urea and other waste products from a patient’s blood by diverting some of the bloodflow externally through a machine called a dialyzer. The rate at which urea is removed from the blood (in mgymin) is often well described by the equationwhere K is the rate of flow of
P hotosynthesis Much of the earth’s photosynthesis occurs in the oceans. The rate of primary production depends on light intensity, measured as the flux of photons(that is, number of photons per unit area per unit time). For monochromatic light, intensity decreases with water depth according to
Spread of drug use In a study of the spread of illicit drug use from an enthusiastic user to a population of N users, the authors model the number of expected new users by the equationwherec, k, and are positive constants. Evaluate this integral to express in terms ofc, N, k, and . =So CN(1-ek)
Drug pharmacokinetics The plasma drug concentration of a new drug was modeled by the function Cstd − 23te22t, where t is measured in hours and C in mgymL.(a) What is the maximum drug concentration and when did it occur?(b) Calculate y`0 Cstd dt and explain its significance.
S − hsx, yd | x > 22, 0 < y < e2xy2j Sketch the region and find its area (if the area is finite).
S − hsx, yd | x < 1, 0 < y < exj Sketch the region and find its area (if the area is finite).
y`0 ex e2x 1 3 dx Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`e 1xsln xd3 dx Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`1 ln x x3 dx Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`2`x2 9 1 x6 dx Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`2`x3e2x4 dx Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`1 ln x xdx Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y6 2`rery3 dr Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`0 se25s ds Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`2`cos t dt Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`1 x 1 1 x2 1 2x dx Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`1 e2sx sx dx Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`2`xe2x 2 dx Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`2`s y3 2 3y2d dy Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`2 sin d Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y21 2`e22t dt Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`4 e2yy2 dy Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`0 xsx2 1 2d2 dx Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y21 2`1 s2 2 w dw Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`0 1s4 1 1 x dx Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
y`3 1sx 2 2d3y2 dx Determine whether each integral is convergent or divergent.Evaluate those that are convergent.
(a) Graph the functions f sxd − 1yx1.1 and tsxd − 1yx0.9 in the viewing rectangles f0, 10g by f0, 1g and f0, 100g by f0, 1g.(b) Find the areas under the graphs of f and t from x − 1 to x − b and evaluate for b − 10, 100, 104, 106, 1010, and 1020.(c) Find the total area under each curve
Find the area under the curve y − 1yx3 from x − 1 to x − b and evaluate it for b − 10, 100, and 1000. Then find the total area under this curve for x > 1.
Computer algebra systems sometimes need a helping hand from human beings. Try to evaluate y s1 1 ln xd s1 1 sx ln xd2 dx with a computer algebra system. If it doesn’t return an answer, make a substitution that changes the integral into one that the CAS can evaluate.
y 1s1 1 s3 x dx Use a computer algebra system to evaluate the integral.Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.
y sin4x dx Use a computer algebra system to evaluate the integral.Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.
y tan5x dx Use a computer algebra system to evaluate the integral.Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.
y dx exs3ex 1 2d Use a computer algebra system to evaluate the integral.Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.
y xs1 1 2x dx Use a computer algebra system to evaluate the integral.Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.
y x2s1 1 x3d4 dx Use a computer algebra system to evaluate the integral.Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.
y sec4x dx Use a computer algebra system to evaluate the integral.Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.
Verify Formula 31 (a) by differentiation and (b) by substituting u − a sin .
Verify Formula 53 in the Table of Integrals (a) by differentiation and (b) by using the substitution t − a 1 bu.
y et sins t 2 3d dt Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y s4 1 sln xd2 xdx Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y sec2 tan2 s9 2 tan2 d Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y x4 dx sx10 2 2 Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y1 0x4e2x dx Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y ex 3 2 e2x dx Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y sin 2 s5 2 sin d Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y sin2x cos x lnssin xd dx Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y x sinsx2d coss3x2d dx Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y tan3s1yzd z 2 dz Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y dx 2x3 2 3x2 Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y 0 x3 sin x dx Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y s2y2 2 3 y2 dy Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y e2x arctansexd dx Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y3 21 x2s4x2 2 7 dx Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y dx x2s4x2 1 9 Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y e2 sin 3 d Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
y tan3 s xd dx Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
S terile insect technique One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no offspring.(The photo shows a screw-worm fly, the first pest effectively eliminated
Suppose that F, G, and Q are polynomials andfor all x except when Qsxd − 0. Prove that Fsxd − Gsxd for all x. [Hint: Use continuity.] F(x) G(x) Q(x) Q(x)
Use the method of Exercise 21 to evaluate 3x- 2x + 3 (x-1)(x + 1) dx
If a factor of the denominator is an irreducible quadratic, such as x2 1 1, the corresponding partial fraction has a linear numerator. For instance,Determine the values of A, B, and C and use them to evaluate y f sxd dx. f(x) = 2x+x+1 x(x + 1) A + x Bx + C x + 1
Use the method of Exercise 19 to evaluate r 5x + 16 (2x + 1)(x-2) dx
If a linear factor in the denominator of a rational function is repeated, there will be two corresponding partial fractions. For instance,Determine the values of A, B, and C and use them to evaluate y f sxd dx. f(x)= = 5x2+3x-2 x(x+2) A B C + x x x + 2
y cos x sin2 x 1 sin x dx Make a substitution to express the integrand as a rational function and then evaluate the integral.
y e2x e2x 1 3ex 1 2 dx Make a substitution to express the integrand as a rational function and then evaluate the integral.
y dx 2sx 1 3 1 x Make a substitution to express the integrand as a rational function and then evaluate the integral.
y16 9sx x 2 4 dx Make a substitution to express the integrand as a rational function and then evaluate the integral.
y x2 1 2x 2 1 x3 2 x dx Evaluate the integral.
y2 14y2 2 7y 2 12 ys y 1 2ds y 2 3d dy Evaluate the integral.
y1 0x3 2 4x 2 10 x2 2 x 2 6 dx Evaluate the integral.
y1 02 2x2 1 3x 1 1 dx Evaluate the integral.
y 1sx 1 adsx 1 bd dx Evaluate the integral.
y ax x2 2 bx dx Evaluate the integral.
y1 0x 2 1 x2 1 3x 1 2 dx Evaluate the integral.
y3 21 x2 2 1 dx Evaluate the integral.
y 1st 1 4dst 2 1d dt Evaluate the integral.
y x 2 9 sx 1 5dsx 2 2d dx Evaluate the integral.
y r 2 r 1 4 dr Evaluate the integral.
y xx 2 6 dx Evaluate the integral.
(a)x x2 1 x 2 2(b)2 2 x x2 2 2x 2 8 Write the function as a sum of partial fractions. Do not determine the numerical values of the coefficients.
(a)1 x2 2 1(b)2 x2 1 x Write the function as a sum of partial fractions. Do not determine the numerical values of the coefficients.
Suppose that f s1d − 2, f s4d − 7, f 9s1d − 5, f 9s4d − 3, and f 0 is continuous. Find the value of y4 1 xf 0sxd dx.38. If f s0d − ts0d − 0 and f 0 and t 0 are continuous, show that ya 0f sxd t0sxd dx − f sad t9sad 2 f 9sad tsad 1 ya 0f 0sxd tsxd dx
I nsect metamorphosis The rate of development of many insects increases gradually with temperature up to a maximum and then rapidly falls to zero. This can be approximated by the function T 2 T k, where k is a positive integer and T is a standardized measure of temperature such that 0 < T < 1 and T
Gene regulation In Section 10.3 a model of gene regulation is analyzed and it is shown that the concentration of protein in a cell as a function of time is given by the equation pstd − 12 2 12 e2t ssin t 1 cos td The bioavailability of this protein is defined as the integral of this concentration
Use Exercise 30 to find y x4ex dx.33. Salicylic acid pharmacokinetics In the article cited in Example 5 the authors also studied the formation and concentration of salicylic acid in the bloodstream of 10 volunteers. A model for the concentration iswhere t is measured in hours and C in mgymL.
Use Exercise 29 to find y sln xd3 dx.
y xnex dx − xnex 2 n y xn21ex dx Use integration by parts to prove the reduction formula.
y sln xdn dx − xsln xdn 2 n y sln xdn21 dx Use integration by parts to prove the reduction formula.
(a) If n > 2 is an integer, show thaty sinnx dx − 2 1n cos x sinn21x 1 n 2 1 n y sinn22x dx This is called a reduction formula because the exponent n has been reduced to n 2 1 and n 2 2.(b) Use the reduction formula in part (a) to show that(c) Use parts (a) and (b) to evaluate y sin4x dx.28.
y sinsln xd dx First make a substitution and then use integration by parts to evaluate the integral.
y x lns1 1 xd dx First make a substitution and then use integration by parts to evaluate the integral.
y 0 ecos t sin 2t dt First make a substitution and then use integration by parts to evaluate the integral.
ys s y2 3 coss 2 d d First make a substitution and then use integration by parts to evaluate the integral.
y t 3e2t 2 dt First make a substitution and then use integration by parts to evaluate the integral.
y cos sx dx First make a substitution and then use integration by parts to evaluate the integral.
y1 0r 3 s4 1 r 2 dr Evaluate the integral.
y2 1sln xd2 dx Evaluate the integral.
ys3 1arctans1yxd dx Evaluate the integral.
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