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Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions
tsxd − yx 0(1 1 st ) dt Sketch the area represented by tsxd. Then find t9sxd in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating.
tsxd − yx 0s1 1 t 2d dt Sketch the area represented by tsxd. Then find t9sxd in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating.
Let tsxd − yx 0 f std dt, where f is the function whose graph is shown.(a) Evaluate tsxd for x − 0, 1, 2, 3, 4, 5, and 6.(b) Estimate ts7d.(c) Where does t have a maximum value? Where does it have a minimum value?(d) Sketch a rough graph of t. 0 1 4 6
Let tsxd − yx 0 f std dt, where f is the function whose graph is shown.(a) Evaluate ts0d, ts1d, ts2d, ts3d, and ts6d.(b) On what interval is t increasing?(c) Where does t have a maximum value?(d) Sketch a rough graph of t. 1 f 1 5
(a) Show that cossx2d > cos x for 0 < x < 1.(b) Deduce that y y6 0 cossx2d dx > 12.
(a) Show that 1 < s1 1 x3 < 1 1 x3 for x > 0.(b) Show that 1 < y1 0 s1 1 x3 dx < 1.25.
P hotosynthesis Much of the earth’s photosynthesis occurs in the oceans. The rate of primary production (as discussed in Exercise 61) 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
P hotosynthesis The rate of primary production refers to the rate of conversion of inorganic carbon to organic carbon via photosynthesis. It is measured as a mass of carbon fixed per unit biomass, per unit time. The rate of primary production depends on light intensity, measured as the flux of
Incidence and prevalence The incidence istd of an infectious disease at time t is the rate at which new infections are occurring at that time. The prevalence Pstd at time t is the total number of infected individuals at that time.Let’s suppose that Ps0d − 0.(a) Express the total number of new
Medical imaging devices like CT scans work by passing an X-ray beam through part of the body and measuring how much the intensity of the beam attenuates (in other words, is reduced). The amount of attenuation depends on the density and composition of the tissue.(a) If Asxd is the attenuation rate
Niche overlap The extent to which species compete for resources is often measured by the niche overlap. If the horizontal axis represents a continuum of different resource types (for example, seed sizes for certain bird species), then a plot of the degree of preference for these resources is called
Von Bertalanffy growth Many fish grow in a way that is described by the von Bertalanffy growth equation. For a fish that starts life with a length of 1 cm and has a maximum length of 30 cm, this equation predicts that the growth rate is 29e2a cmyyear, where a is the age of the fish. How long will
Water flows into and out of a storage tank. A graph of the rate of change rstd of the volume of water in the tank, in liters per day, is shown. If the amount of water in the tank at time t − 0 is 25,000 L, use the Midpoint Rule to estimate the amount of water in the tank four days later. TA 2000
Water flows from the bottom of a storage tank at a rate of rstd − 200 2 4t liters per minute, where 0 < t < 50. Find the amount of water that flows from the tank during the first 10 minutes.
Suppose that a volcano is erupting and readings of the rate rstd at which solid materials are spewed into the atmosphere are given in the table. The time t is measured in seconds and the units for rstd are tonnes (metric tons) per second.(a) Give upper and lower estimates for the total quantity
If oil leaks from a tank at a rate of rstd gallons per minute at time t, what does y120 0 rstd dt represent?
A honeybee population starts with 100 bees and increases at a rate of n9std bees per week. What does the expression 100 1 y15 0 n9std dt represent?
In a chemical reaction, the rate of reaction is the derivative of the concentration fCgstd of the product of the reaction.What doesrepresent? 12 d[C] S -dt dt
Bacteria growth A bacteria colony increases in size at a rate of 4.0553e1.8t bacteria per hour. If the initial population is 46 bacteria, find the population four hours later.
S ea urchins Integration is sometimes used when censusing a population. For example, suppose the density of sea urchins at different points x along a coastline is given by the function f sxd individuals per meter, where x is the distance (in meters) along the coast from the start of the species’
Age-structured populations Suppose the number of individuals of age a is given by the function Nsad (number of individuals per age a). What does the integral y15 0 Nsad da represent?
Growth rate If w9std is the rate of growth of a child in pounds per year, what does y10 5 w9std dt represent?
If V9std is the rate at which water flows into a reservoir at time t, what does the integral yt2 t1 V9std dt represent?
Measles pathogenesis The function f std − 2tst 2 21dst 1 1d has been used to model the measles virus concentration in an infected individual. The area under the graph of f represents the total amount of infection. We saw in Section 5.1 that at t − 12 days this total amount of infection reaches
y sin 2x sin x dx Find the general indefinite integral.
y sin x 1 2 sin2x dx Find the general indefinite integral.
y sec t ssec t 1 tan td dt Find the general indefinite integral.
y s1 1 tan2 d d Find the general indefinite integral.
y vsv 2 1 2d2 dv Find the general indefinite integral.
y s1 2 tds2 1 t 2d dt Find the general indefinite integral.
y sex 2 2x2d dx Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen.
y scos x 1 12 xd dx Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen.
y x cos x dx − x sin x 1 cos x 1 C Verify by differentiation that the formula is correct.
y cos3 x dx − sin x 2 13 sin3 x 1 C Verify by differentiation that the formula is correct.
y2 2 y2 cos x dx Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.
y2 21 x3 dx Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.
y − sec2x, 0 < x < y3 Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.
y − sin x, 0 < x < Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.
What is wrong with the equation? 30. secx dx = tan x] = 0 0 0
What is wrong with the equation? 29. xp. x -1 x 1- [ 4 3
y2 0|2x 2 1| dx Evaluate the integral.
y1ys3 0t 2 2 1 t 4 2 1 dt Evaluate the integral.
y2 1sx 2 1d3 x2 dx Evaluate the integral.
y y4 01 1 cos2 cos2 d Evaluate the integral.
y y3 0sin 1 sin tan2 sec2 d Evaluate the integral.
y2 1v3 1 3v6 v4 dv Evaluate the integral.
y1 04 t 2 1 1 dt Evaluate the integral.
y1 21 eu11 du Evaluate the integral.
y1 010x dx Evaluate the integral.
y1 0sxe 1 exd dx Evaluate the integral.
y5 0s2ex 1 4 cos xd dx Evaluate the integral.
y9 11 2x dx Evaluate the integral.
y18 1 Î3 zdz Evaluate the integral.
y y4 0sec2t dt Evaluate the integral.
y y4 0sec tan d Evaluate the integral.
y1 0xss3 x 1 s4 x d dx Evaluate the integral.
y1 21 ts1 2 td2 dt Evaluate the integral.
y9 1x 2 1 sx dx Evaluate the integral.
y2 0s y 2 1ds2y 1 1d dy Evaluate the integral.
y2 1s1 1 2yd2 dy Evaluate the integral.
y5 25 e dx Evaluate the integral.
y0 21 s2x 2 exd dx Evaluate the integral.
y8 1s3 x dx Evaluate the integral.
y1 0x4y5 dx Evaluate the integral.
y1 0s1 1 12 u4 2 25 u9d du Evaluate the integral.
y2 0(x4 2 34 x2 1 23 x 2 1) dx Evaluate the integral.
y2 1x22 dx Evaluate the integral.
y3 22 sx2 2 3d dx Evaluate the integral.
Use the properties of integrals to verify that 2 = 1 + x dx = 22 2=
Suppose f has absolute minimum value m and absolute maximum value M. Between what two values must y2 0 f sxd dx lie? Which property of integrals allows you to make your conclusion?
Each of the regions A, B, and C bounded by the graph of f and the x-axis has area 3. Find the value of [f(x) + 2x + 5] dx y. B A -2 0 C x
If Fsxd − yx 2 f std dt, where f is the function whose graph is given, which of the following values is largest?(A) Fs0d (B) Fs1d (C) Fs2d (D) Fs3d (E) Fs4d y y = f(t) + + 0 2 3 4
For the function f whose graph is shown, list the following quantities in increasing order, from smallest to largest, and explain your reasoning. (A) so f(x) dx (D) f(x) dx (B) fo f(x) dx (E) f'(1) (C) f(x) dx y -2 0 5 X
Find y5 0 f sxd dx if f(x) = X for x
If y9 0 f sxd dx − 37 and y9 0 tsxd dx − 16, find y9 0 f2 f sxd 1 3tsxdg dx.
If y5 1 f sxd dx − 12 and y5 4 f sxd dx − 3.6, find y4 1 f sxd dx.
Write as a single integral in the form yb a f sxd dx:y2 22 f sxd dx 1 y5 2f sxd dx 2 y21 22 f sxd dx
Given that y1 03xsx2 1 4 dx − 5s5 2 8, what is y0 13usu2 1 4 du?
Evaluate y sin2x cos4x dx.
y10 0|x 2 5 |dx Evaluate the integral by interpreting it in terms of areas.
y2 21|x | dx Evaluate the integral by interpreting it in terms of areas.
y3 21 s3 2 2xd dx Evaluate the integral by interpreting it in terms of areas.
y0 23 s1 1 s9 2 x2 d dx Evaluate the integral by interpreting it in terms of areas.
y2 22 s4 2 x2 dx Evaluate the integral by interpreting it in terms of areas.
y3 0(12 x 2 1) dx Evaluate the integral by interpreting it in terms of areas.
The graph of t consists of two straight lines and a semi-circle. Use it to evaluate each integral. (a) g(x) dx (b) Log(x) dx (c) g(x) dx 0 2- y= g(x) 4 7
y10 1sx 2 4 ln xd dx Express the integral as a limit of Riemann sums. Do not evaluate the limit.
y6 2x 1 1 x5 dx Express the integral as a limit of Riemann sums. Do not evaluate the limit.
(a) Find an approximation to the integral y4 0 sx2 2 3xd dx using a Riemann sum with right endpoints and n − 8.(b) Draw a diagram like Figure 3 to illustrate the approximation in part (a).(c) Use Theorem 4 to evaluate y4 0 sx2 2 3xd dx.(d) Interpret the integral in part (c) as a difference of
y2 1x3 dx Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.
y5 0s1 1 2x3d dx Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.
y2 0s2 2 x2d dx Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.
y4 1sx2 1 2x 2 5d dx Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.
y5 21 s1 1 3xd dx Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.
Salicylic acid pharmacokinetics In the study cited in Example 5, the metabolite salicylic acid (SA) was rapidly formed and peak SA levels of about 4.2 mgymL were reached after an hour. The concentration of SA was modeled by the function Cstd − 11.4te2t where t is measured in hours and C is
Drug pharmacokinetics During testing of a new drug, researchers measured the plasma drug concentration of each test subject at 10-minute intervals. The average concentrations Cstd are shown in the table, where t is measured in minutes and C is measured in mgymL. Use the Midpoint Rule to estimate
y5 1x2e2x dx, n − 4 Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.
y1 0sinsx2d dx, n − 5 Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.
y y2 0cos4x dx, n − 4 Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.
y10 2sx3 1 1 dx, n − 4 Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.
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