New Semester
Started
Get
50% OFF
Study Help!
--h --m --s
Claim Now
Question Answers
Textbooks
Find textbooks, questions and answers
Oops, something went wrong!
Change your search query and then try again
S
Books
FREE
Study Help
Expert Questions
Accounting
General Management
Mathematics
Finance
Organizational Behaviour
Law
Physics
Operating System
Management Leadership
Sociology
Programming
Marketing
Database
Computer Network
Economics
Textbooks Solutions
Accounting
Managerial Accounting
Management Leadership
Cost Accounting
Statistics
Business Law
Corporate Finance
Finance
Economics
Auditing
Tutors
Online Tutors
Find a Tutor
Hire a Tutor
Become a Tutor
AI Tutor
AI Study Planner
NEW
Sell Books
Search
Search
Sign In
Register
study help
business
statistics econometrics
Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions
(a) Show that every member of the family of functions y − sln x 1 Cdyx is a solution of the differential equation x2y9 1 xy − 1.; (b) Illustrate part (a) by graphing several members of the family of solutions on a common screen.(c) Find a solution of the differential equation that satisfies the
Show that y − e2at cos t is a solution of the differential equation y9 − 2e2atsa cos t 1 sin td. Is this differential equation pure-time, autonomous, or nonautonomous?
Verify that y − 2t cos t 2 t is a solution of the initial-value problem tdy dt− y 1 t 2 sin t ys d − 0 Is this differential equation pure-time, autonomous, or nonautonomous?
Show that y − 23 ex 1 e22x is a solution of the differential equation y9 1 2y − 2ex. Is this differential equation pure-time, autonomous, or nonautonomous?
The height of a monument is 20 m. A horizontal crosssection at a distance x meters from the top is an equilateral triangle with side 14 x meters. Find the volume of the monument.
The base of a solid is a circular disk with radius 3. Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.
Let 5 be the region bounded by the curves y − 1 2 x2 and y − x6 2 x 1 1. Estimate the following quantities.(a) The x-coordinates of the points of intersection of the curves(b) The area of 5(c) The volume generated when 5 is rotated about the x-axis
Find the volumes of the solids obtained by rotating the region bounded by the curves y − x and y − x2 about the following lines.(a) The x-axis (b) The y-axis
Let 5 be the region in the first quadrant bounded by the curves y − x3 and y − 2x 2 x2. Calculate the following quantities.(a) The area of 5(b) The volume obtained by rotating 5 about the x-axis
Let 5 be the region bounded by the curves y − tansx2 d, x − 1, and y − 0. Use the Midpoint Rule with n − 4 to estimate the following quantities.(a) The area of 5(b) The volume obtained by rotating 5 about the x-axis
Find the volume of the solid obtained by rotating about the x-axis the region bounded by the curves y − e22x, y − 1 1 x, and x − 1.
Cardiac output After a 6-mg injection of dye into a heart, the readings of dye concentration, in mgyL, at two-second intervals are as shown in the table. Use the Midpoint Rule to estimate the cardiac output. t C(t) t C(t) 0 0 14 4.7 2 1.9 16 3.3 4 3.3 18 2.1 6 5.1 20 8 7.6 22 10 7.1 24 222 1.1
A nimal survival and renewal The fish population in a lake is currently 3400 and is increasing at a rate of Rstd − 650e0.04t fish per month. If the proportion of fish that remain after t months is Sstd − e20.09t, how many fish will be in the lake in three years?
Survival and renewal Suppose a city’s population is currently 75,000 and the renewal function is Rstd − 3200e0.05t If the survival function is Sstd − e20.1t, predict the population in 10 years.
Salicylic acid pharmacokinetics In a study of the effects of aspirin, salicylic acid was formed and its concentration was modeled by the function Cstd − 11.4te2t where the time t is measured in hours and C is measured in mgymL. What is the average concentration of the salicylic acid during the
A ntibiotic pharmacokinetics When an antibiotic tablet is taken, the concentration of the antibiotic in the bloodstream is modeled by the function Cstd − 8se20.4t 2 e20.6td where the time t is measured in hours and C is measured in mgymL. What is the average concentration of the antibiotic during
Find the average value of the function f sxd − x2s1 1 x3 on the interval f0, 2g.
Find the average value of the function f std − t sinst2d on the interval f0, 10g.
Birth and death rates The birth rate of a population is bstd − 1240e0.0197t people per year and the death rate is dstd − 682e0.008t people per year. Find the area between these curves for 0 < t < 20. What does this area represent?
MRI brain scan Shown is a cross-section of a human brain obtained with an MRI. Use the Midpoint Rule to estimate the area of the cross-section. 0 Allison Herreid/Shutterstock.com 5 15 y (cm) 10 15 x (cm)
x 1 y − 0, x − y2 1 3y Find the area of the region bounded by the given curves.
y − 1 2 2x2, y − |x |Find the area of the region bounded by the given curves.
y − 1yx, y − x2, y − 0, x − e Find the area of the region bounded by the given curves.
y − x2, y − 4x 2 x2 Find the area of the region bounded by the given curves.
(a) Suppose S is a solid with known cross-sectional areas.Explain how to approximate the volume of S by a Riemann sum. Then write an expression for the exact volume.(b) If S is a solid of revolution, how do you find the crosssectional areas?
(a) What is the cardiac output of the heart?(b) Explain how the cardiac output can be measured by the dye dilution method.
If we have survival and renewal functions for a population, how do we predict the size of the population T years from now?
(a) What is the average value of a function f on an interval fa, bg?(b) What does the Mean Value Theorem for Integrals say?What is its geometric interpretation?
Suppose that Sue runs faster than Kathy throughout a 1500-meter race. What is the physical meaning of the area between their velocity curves for the first minute of the race?
Draw two typical curves y − f sxd and y − tsxd, where f sxd > tsxd for a < x
Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles.
Find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere.
The base of S is a circular disk with radius r. Parallel crosssections perpendicular to the base are isosceles triangles with height h and unequal side in the base.(a) Set up an integral for the volume of S.(b) By interpreting the integral as an area, find the volume of S.
The base of S is an elliptical region with boundary curve 9x2 1 4y2 − 36. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.Find the volume of the described solid S.
The base of S is a circular disk with radius r. Parallel crosssections perpendicular to the base are squares.Find the volume of the described solid S.
S is a right circular cone with height h and base radius r Find the volume of the described solid S.
Volume of a bird’s egg(a) A model for the shape of a bird’s egg is obtained by rotating about the x-axis the region under the graph of f sxd − sax3 1 bx2 1 cx 1 dds1 2 x2 Use a CAS to find the volume of such an egg.(b) For a red-throated loon, a − 20.06, b − 0.04, c − 0.1, and d −
(a) If the region shown in the figure is rotated about the x-axis to form a solid, use the Midpoint Rule with n − 4 to estimate the volume of the solid.(b) Estimate the volume if the region is rotated about the y-axis. Again use the Midpoint Rule with n − 4. 74 y 2 0 2 4 6 8 10 x
A log 10 m long is cut at 1-meter intervals and its cross-sectional areas A (at a distance x from the end of the log)are listed in the table. Use the Midpoint Rule with n − 5 to estimate the volume of the log. x (m) A (m) x (m) A (m) 012345 0.68 0.65 7 0.64 819 6 0.53 0.55 8 0.52 0.61 9 0.50 0.58
Volume of a pancreas A CAT scan of a human pancreas shows cross-sections spaced 1 cm apart. The pancreas is 12 cm long and the cross-sectional areas, in square centimeters, are 0, 7.7, 15.2, 18.0, 10.3, 10.8, 9.7, 8.7, 7.7, 5.5, 4.0, 2.7, and 0. Use the Midpoint Rule to estimate the volume of the
y − ln x, y − 1, y − 2, x − 0 Here we rotate about the y-axis instead of the x-axis. Find the volume of the solid obtained by rotating the region bounded by the given curves about the y-axis. Sketch the region, the solid, and a typical disk.
x − 2sy , x − 0, y − 9 Here we rotate about the y-axis instead of the x-axis. Find the volume of the solid obtained by rotating the region bounded by the given curves about the y-axis. Sketch the region, the solid, and a typical disk.
y − 14 x2, y − 5 2 x2 Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region, the solid, and a typical disk or washer.
y − x3, y − x, x > 0 Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region, the solid, and a typical disk or washer.
y − s25 2 x2 , y − 0, x − 2, x − 4 Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region, the solid, and a typical disk or washer.
y − sx 2 1 , y − 0, x − 5 Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region, the solid, and a typical disk or washer.
y − 1 2 x2, y − 0 Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region, the solid, and a typical disk or washer.
y − 2 2 12 x, y − 0, x − 1, x − 2 Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region, the solid, and a typical disk or washer.
Drug administration A patient is continually receiving a drug. If the drug is eliminated from the body over time so that the fraction that remains after t hours is e20.4t, at what constant rate should the drug be administered to maintain a steady level of the drug in the bloodstream?
Cardiac output The graph of the concentration function cstd is shown after a 7-mg injection of dye into a heart. Use the Midpoint Rule to estimate the cardiac output. y (mg/L) 6 4 2 0 2 4 6 8 10 12 14 (seconds)
Cardiac output After an 8-mg injection of dye, the readings of dye concentration, in mgyL, at two-second intervals are as shown in the table. Use the Midpoint Rule to estimate the cardiac output. t c(t) 124 0 12 12 c(t) 3.9 2.4 14 2.3 4 5.1 16 1.6 6 7.8 18 0.7 8 7.6 20 0 10 10 5.4
Cardiac output The dye dilution method is used to measure cardiac output with 6 mg of dye. The dye concentrations, in mgyL, are modeled by cstd − 20te20.6t, 0 < t < 10, where t is measured in seconds. Find the cardiac output.[Hint: Integration by parts is required.]
Blood flow High blood pressure results from constriction of the arteries. To maintain a normal flow rate (flux), the heart has to pump harder, thus increasing the blood pressure.Use Poiseuille’s Law to show that if R0 and P0 are normal values of the radius and pressure in an artery and the
Blood flow Use Poiseuille’s Law to calculate the rate of flow in a small human artery where we can take − 0.027 dyn ? sycm2, R − 0.008 cm, l − 2 cm, and P − 4000 dynycm2.
I nsect survival and renewal Sterile fruit flies are used in an experiment where the proportion that survive at least t days is given by e20.15t. If the experiment begins with 200 fruit flies, and flies are added at the rate of 5 per hour, how many flies are present 14 days after the start of the
Water pollution A contaminant is leaking into a lake at a rate of Rstd − 1600e0.06t gallonsyh Enzymes have been added to the lake that neutralize the contaminant over time so that after t hours the fraction of the contaminant that remains is Sstd − e20.32t. If there are currently 10,000 gallons
Drug concentration A patient receives a drug at a constant rate of 30 mgyh. The drug is eliminated from the bloodstream over time so that the fraction e20.2t remains after t hours. The patient currently has 80 mg of the drug present in the bloodstream. How much will be present in 24 hours?
Drug concentration A drug is administered intravenously to a patient at the rate of 12 mgyh. The patient’s body eliminates the drug over time so that after t hours the proportion that remains is e20.25t. If the patient currently has 50 mg of the drug in her bloodstream, how much of the drug is
A nimal survival and renewal There are currently 3800 birds of a particular species in a national park and their number is increasing at a rate of Rstd − 525e0.05t birdsyyear.If the proportion of birds that survive t years is given by Sstd − e20.1t, what do you predict the bird population will
I nsect survival and renewal A population of insects currently numbers 22,500 and is increasing at a rate of Rstd − 1225e0.14t insectsyweek. If the survival function for the insects is Sstd − e20.2t, where t is measured in weeks, how many insects are there after 12 weeks?
City population A city currently has 36,000 residents and is adding new residents steadily at the rate of 1600 per year.If the proportion of residents that remain after t years is given by Sstd − 1yst 1 1d, what is the population of the city seven years from now?
A nimal survival and renewal An animal population currently has 7400 members and is reproducing at the rate Rstd − 2240 1 60t membersyyear. The proportion of members that survive after t years is given by Sstd − 1yst 1 1d.(a) How many of the original members survive four years?(b) How many new
Prove the Mean Value Theorem for Integrals by applying the Mean Value Theorem for derivatives (see Section 4.2) to the function Fsxd − yx a f std dt.
Find the numbers b such that the average value of f sxd − 2 1 6x 2 3x2 on the interval f0, bg is equal to 3.
If f is continuous and y3 1 f sxd dx − 8, show that f takes on the value 4 at least once on the interval f1, 3g.
Blood flow The velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is vsrd −P 4 l sR2 2 r 2 d where P is the pressure difference between the ends of the vessel and is the viscosity of the blood (see Example 3.3.9). Find the average
Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 s. The maximum rate of air flow into the lungs is about 0.5 Lys. This explains, in part, why the function f std −1 2sin 2 t 5has often been used to model the rate of air flow
Length of a fish For a fish that starts life with a length of 1 cm and has a maximum length of 30 cm, the von Bertalanffy growth model predicts that the growth rate is 29e2a cmyyear. What is the average length of the fish over its first five years?
Measles pathogenesis In Section 5.1 we modeled the infection level of the measles virus in a patient by the function f std − 2t st 2 21dst 1 1d where t is measured in days and f std is measured in the number of infected cells per mL of blood plasma. Over the course of the 21-day infection, what
Blood alcohol concentration In Section 3.1 we modeled the BAC of male adult subjects after rapid consumption of 15 mL of ethanol (corresponding to one alcoholic drink) by the concentration function Cstd − 0.0225te20.0467t where t is measured in minutes after consumption and Cstd is measured in
The population of Indonesia from 1950 to 2000 has been modeled with the function Pstd − 83e0.18t where P is measured in millions and t is measured in years with t − 0 in the year 1950. What was the average population of Indonesia in the second half of the 20th century?
If a cup of coffee has temperature 95°C in a room where the temperature is 20°C, then, according to Newton’s Law of Cooling, the temperature of the coffee after t minutes is Tstd − 20 1 75e2ty50. What is the average temperature of the coffee during the first half hour?
In a certain city the temperature (in °F) t hours after 9 am was modeled by the function Tstd − 50 1 14 sin t 12 Find the average temperature during the period from 9 am to 9 pm.
The velocity graph of an accelerating car is shown.(a) Use the Midpoint rule to estimate the average velocity of the car during the first 12 seconds.(b) At what time was the instantaneous velocity equal to the average velocity? (km/h) 8. 60 40 20 0 4 8 12 t (seconds)
Find the average value of f on f0, 8g. 0 2 4 X
f sxd − 2xys1 1 x2d2, f0, 2g(a) Find the average value of f on the given interval.(b) Find c such that fave − f scd.(c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f.
f sxd − 2 sin x 2 sin 2x, f0, g(a) Find the average value of f on the given interval.(b) Find c such that fave − f scd.(c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f.
f sxd − ln x, f1, 3g(a) Find the average value of f on the given interval.(b) Find c such that fave − f scd.(c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f.
f sxd − sx 2 3d2, f2, 5g(a) Find the average value of f on the given interval.(b) Find c such that fave − f scd.(c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f.
hsud − s3 2 2ud21, f21, 1g Find the average value of the function on the given interval.
hsxd − cos4x sin x, f0, g Find the average value of the function on the given interval.
f s d − sec2s y2d, f0, y2g Find the average value of the function on the given interval.
tsxd − s3 x , f1, 8g Find the average value of the function on the given interval.
f sxd − sin 4x, f2 , g Find the average value of the function on the given interval.
f sxd − 4x 2 x2, f0, 4g Find the average value of the function on the given interval.
Find the area of the region bounded by the parabola y − x2, the tangent line to this parabola at s1, 1d, and the x-axis.
Find the values of c such that the area of the region bounded by the parabolas y − x2 2 c2 and y − c2 2 x2 is 576.
Find the number a such that the line x − a bisects the area under the curve y − 1yx2, 1 < x < 4.
Birth and death rates If the birth rate of a population is bstd − 2200e0.024t people per year and the death rate is dstd − 1460e0.018t people per year, find the area between these curves for 0 < t < 10. What does this area represent?
Two cars, A and B, start side by side and accelerate from rest. The figure shows the graphs of their velocity functions.(a) Which car is ahead after one minute? Explain.(b) What is the meaning of the area of the shaded region?(c) Which car is ahead after two minutes? Explain.(d) Estimate the time
Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first 10 seconds of the race. Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first 10 seconds. t VC
Cerebral blood flow Models for the arterial and venous concentration functions in Figure 8 are given by(a) Find the area between the graphs of A and V for 0 (b) If the volume of N2O absorbed by the brain in the first 10 minutes is 60 mL, determine the cerebral blood flow. A(t) = 0.05t 12+1 V(t) =
C erebral blood flow The table shows measurements of Astd, the concentration of N2O flowing into a patient’s brain, and Vstd, the concentration of N2O flowing out of the brain, where t is measured in minutes and Astd and Vstd are measured in mL of N2O per mL of blood.(a) Use the Midpoint Rule to
The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use the Midpoint Rule to estimate the area of the pool. 5.6 5.0 4.8 6.8 4.8 7.2 6.2
A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 20-centimeter intervals are 5.8, 20.3, 26.7, 29.0, 27.6, 27.3, 23.8, 20.5, 15.1, 8.7, and 2.8. Use the Midpoint Rule to estimate the area of the wing’s cross-section. -200 cm-
A cicada wing is shown. Estimate its area using the Midpoint Rule with six subintervals. y(cm) 0.5 0 1 2 x (cm) Tropper2000 / Shutterstock.com
A laurel leaf is shown. Estimate its area using the Midpoint Rule with six subintervals. 2 Vasilius/Shutterstock.com 0 ya (cm) 12 3 9 S + 4 + x (cm)
Find the area of the region enclosed by the curves y − x and 4x 1 y2 − 12.
Sometimes it’s easier to find an area by regarding x as a function of y instead of y as a function of x. To illustrate this idea, let S be the region enclosed by the line y − x 2 1 and the parabola y2 − 2x 1 6.(a) By sketching S, observe that if you want to integrate with respect to x you
Sketch the curves y − cos x and y − 1 2 cos x, 0 < x < , and observe that the region between them consists of two separate parts. Find the area of this region.
Showing 200 - 300
of 7357
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Last
Step by Step Answers
Get 50% OFF
Claim Now
