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Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions
e20.015 Use a linear approximation to estimate the given number.
s2.001d5 Use a linear approximation to estimate the given number.
ex < 1 1 x Verify the given linear approximation at a − 0. Then determine the values of x for which the linear approximation is accurate to within 0.1.
1ys1 1 2xd 4 < 1 2 8x Verify the given linear approximation at a − 0. Then determine the values of x for which the linear approximation is accurate to within 0.1.
tan x < x Verify the given linear approximation at a − 0. Then determine the values of x for which the linear approximation is accurate to within 0.1.
s3 1 2 x < 1 2 13 x Verify the given linear approximation at a − 0. Then determine the values of x for which the linear approximation is accurate to within 0.1.
Find the linear approximation of the function tsxd − s3 1 1 x at a − 0 and use it to approximate the numbers s3 0.95 and s3 1.1 . Illustrate by graphing t and the tangent line.
Find the linear approximation of the function f sxd − s1 2 x at a − 0 and use it to approximate the numbers s0.9 and s0.99 . Illustrate by graphing f and the tangent line.
f sxd − x3y4, a − 16 Find the linearization Lsxd of the function at a.
f sxd − cos x, a − y2 Find the linearization Lsxd of the function at a.
f sxd − ln x, a − 1 Find the linearization Lsxd of the function at a.
f sxd − x4 1 3x2, a − 21 Find the linearization Lsxd of the function at a.
Show that lim nl`S1 1 xnDn− ex for any x . 0.
Use the definition of derivative to prove that lim x l 0 lns1 1 xd x− 1
Find d 9 dx9 sx8 ln xd.
Find a formula for f sndsxd if f sxd − lnsx 2 1d.
Find y9 if xy − yx.
Find y9 if y − lnsx2 1 y2 d.
lim x l 01 tan21sln xd Find the limit.
lim xl`arctansex d Find the limit.
y − arctanÎ1 2 x 1 1 x Find the derivative of the function. Simplify where possible.
y − tan21sx 2s1 1 x2 d Find the derivative of the function. Simplify where possible.
f sxd − x lnsarctan xd Find the derivative of the function. Simplify where possible.
y − arctanscos d Find the derivative of the function. Simplify where possible.
y − tan21sx2d Find the derivative of the function. Simplify where possible.
y − stan21xd2 Find the derivative of the function. Simplify where possible.
P redator-prey dynamics In Chapter 7 we study a model for the population sizes of a predator and its prey species. If ustd and vstd denote the prey and predator population sizes at time t, an equation relating the two is ve2vu e2 u − c where c and are positive constants. Use logarithmic
y − stan xd1yx Use logarithmic differentiation to find the derivative of the function.
y − sx x Use logarithmic differentiation to find the derivative of the function.
y − scos xdx Use logarithmic differentiation to find the derivative of the function.
y − xcos x Use logarithmic differentiation to find the derivative of the function.
y − x x Use logarithmic differentiation to find the derivative of the function.
y −Îx2 1 1 x2 2 1 Use logarithmic differentiation to find the derivative of the function.
y −sin2x tan4x sx2 1 1d2 Use logarithmic differentiation to find the derivative of the function.
y − sx ex 2sx2 1 1d10 Use logarithmic differentiation to find the derivative of the function.
y − s2x 1 1d5sx4 2 3d6 Use logarithmic differentiation to find the derivative of the function.
Let f sxd − logbs3x2 2 2d. For what value of b is f 9s1d − 3?
Carbon dating If N is the measured amount of 14C in a fossil organism and N0 is the amount in living organisms, then the estimated age of the fossil is given by the equationCalculate daydN and interpret it. 5370 No a In In 2 N
Genetic drift A population of fruit flies contains two genetically determined kinds of individuals: white-eyed flies and red-eyed flies. Suppose that a scientist maintains the population at constant size N by randomly choosing N juvenile flies after reproduction to form the next
Dialysis The project on page 458 models the removal of urea from the bloodstream via dialysis. Given that the initial urea concentration, measured in mgymL, is c (where c . 1), the duration of dialysis required for certain conditions is given by the equationCalculate the derivative of t with
Find equations of the tangent lines to the curve y − sln xdyx at the points s1, 0d and se, 1yed. Illustrate by graphing the curve and its tangent lines.
If f sxd −ln x x2 , find f 9s1d.
y − x2 ln x, s1, 0d Find an equation of the tangent line to the curve at the given point.
y − lnsx2 2 3x 1 1d , s3, 0d Find an equation of the tangent line to the curve at the given point.
f sxd − ln ln ln x Differentiate f and find the domain of f .
f sxd −x 1 2 lnsx 2 1d Differentiate f and find the domain of f .
y −ln x x2 Find y9 and y99.
y − x2 lns2xd Find y9 and y99.
y − log2se2x cos xd Differentiate the function.
y − 2x log10 sx Differentiate the function.
y − flns1 1 exdg 2 Differentiate the function.
y − lnse2x 1 xe2x d Differentiate the function.
Hszd − lnÎa2 2 z 2 a2 1 z 2 Differentiate the function.
y − ln|2 2 x 2 5x2 |Differentiate the function.
Fs yd − y lns1 1 eyd Differentiate the function.
tsxd − lnsxsx2 2 1d Differentiate the function.
hsxd − lnsx 1 sx2 2 1d Differentiate the function.Differentiate the function.
Fstd − ln s2t 1 1d3 s3t 2 1d4 Differentiate the function.
f std −1 1 ln t 1 2 ln t Differentiate the function.
f sxd − sin x lns5xd Differentiate the function.
f sxd − ln s5 x Differentiate the function.
f sxd − s5 ln x Differentiate the function.
f sxd − log5sxexd Differentiate the function.
f sxd − log2s1 2 3xd Differentiate the function.
f sxd − lnssin2xd Differentiate the function.
f sxd − sinsln xd Differentiate the function.
f sxd − x ln x 2 x Differentiate the function.
Explain why the natural logarithmic function y − ln x is used much more frequently in calculus than the other logarithmic functions y − logb x.
The rate of change of atmospheric pressure P with respect to altitude h is proportional to P, provided that the temperature is constant. At 15°C the pressure is 101.3 kPa at sea level and 87.14 kPa at h − 1000 m.(a) What is the pressure at an altitude of 3000 m?(b) What is the pressure at the
A freshly brewed cup of coffee has temperature 95°C in a 20°C room. When its temperature is 70°C, it is cooling at a rate of 1°C per minute. When does this occur?
When a cold drink is taken from a refrigerator, its temperature is 5°C. After 25 minutes in a 20°C room its temperature has increased to 10°C.(a) What is the temperature of the drink after 50 minutes?(b) When will its temperature be 15°C?
In a murder investigation, the temperature of the corpse was 32.5°C at 1:30 pm and 30.3°C an hour later. Normal body temperature is 37.0°C and the temperature of the surroundings was 20.0°C. When did the murder take place?
A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 75°F.(a) If the temperature of the turkey is 150°F after half an hour, what is the temperature after 45 minutes?(b) When will the turkey have cooled to 1008F?
Dating dinosaurs with potassium Dinosaur fossils are often dated by using an element other than carbon, such as potassium-40, that has a longer half-life (in this case, approximately 1.25 billion years). Suppose the minimum detectable amount is 0.1% and a dinosaur is dated with 40K to be 68 million
Dating dinosaurs Dinosaur fossils are too old to be reliably dated using carbon-14, which has a half-life of about 5730 years. (See Exercise 10.) Suppose we had a 68-million-year-old dinosaur fossil. What fraction of the living dinosaur’s 14C would be remaining today? Suppose the minimum
Radiometric dating Scientists can determine the age of ancient objects by the method of radiometric dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, 14C, with a half-life of about 5730 years. Vegetation absorbs carbon dioxide
A sample of tritium-3 decayed to 94.5% of its original amount after a year.(a) What is the half-life of tritium-3?(b) How long would it take the sample to decay to 20% of its original amount?
Strontium-90 has a half-life of 28 days.(a) A sample has a mass of 50 mg initially. Find a formula for the mass remaining after t days.(b) Find the mass remaining after 40 days.(c) What is the rate of decay after 40 days?(d) How long does it take the sample to decay to a mass of 2 mg?(e) Sketch the
The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample.(a) Find the mass that remains after t years.(b) How much of the sample remains after 100 years?(c) What is the rate of decay after 100 years?(d) After how long will only 1 mg remain?
I ndonesian population The table gives the population of Indonesia, in millions, for the second half of the 20th century.(a) Assuming the population grows at a rate proportional to its size, use the census figures for 1950 and 1960 to predict the population in 1980. Compare with the actual
World population The table gives estimates of the world population, in millions, from 1750 to 2000.(a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950. Compare with the actual figures.(b) Use the exponential model and the
Bacteria population A bacteria culture grows with constant relative growth rate. The bacteria count was 400 after 2 hours and 25,600 after 6 hours.(a) What is the relative growth rate? Express your answer as a percentage.(b) What was the intitial size of the culture?(c) Find an expression for the
Bacteria population A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420.(a) Find an expression for the number of bacteria after t hours.(b) Find the number of bacteria after 3 hours.(c) Find the rate of
E. coli population A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 60 cells.(a) Find the relative growth rate.(b) Find an expression for the
P rotozoan population A population of protozoa develops with a constant relative growth rate of 0.7944 per member per day. On day zero the population consists of two members.Find the population size after six days.
If y − f sud and u − tsxd, where f and t are twice differentiable functions, show that 2 dy dy_dy d'y dx du du dx + dy d'u du dx
Use the Chain Rule to show that if is measured in degrees, then dd ssin d − 180 cos (This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: The differentiation formulas would not be as simple if we used degree
Brain size in fish Brain weight B as a function of body weight W in fish has been modeled by the power function B − 0.007W2y3, where B and W are measured in grams. A model for body weight as a function of body length L(measured in centimeters) is W − 0.12L2.53. If, over 10 million years, the
Angiotensin-converting enzyme (ACE) inhibitors are a type of blood pressure medication that reduces blood pressure by dilating blood vessels. Suppose that the radius R of a blood vessel is related to the dosage x of the medication by the function Rsxd . One version of Poiseuille’s Law gives the
Blood flow In Example 3.3.9 we discussed Poiseuille’s law of laminar flow:v −P 4 l sR2 2 r 2d where v is the blood velocity at a distance r from the center of a blood vessel (a vein or artery) in the shape of a tube with radius R and length l, P is the pressure difference between the ends of
Habitat fragmentation and species conservation The size of a class-structured population is modeled in Section 8.8.In certain situations the long-term growth rate of a population is given by r − 12(1 1 s1 1 8s ), where s is the annual survival probability of juveniles. Suppose this survival
The von Bertalanffy growth function Lstd − L` 2 sL` 2 L0de2kt where k is a positive constant, models the length L of a fish as a function of t, the age of the fish. Here L0 is the length at birth and L` is the final length. Suppose that the mass of a fish of length L is given by M − aL3, where
Bone mass In Example 1.1.6 we found an expression for the mass m of a human femur of length L in terms of the outer radius r, the inner radius rin, and their ratio k − rinyr. More generally, if the bone density is , measured in gycm3, then bone mass is given by the equation m − r 2Lf 2 s 2
When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV1.4 − C, where C is a constant. Suppose that at a certain instant the volume is 400 cm3 and the pressure is 80 kPa and is decreasing at a rate of 10 kPaymin. At what rate is
Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV − C, where C is a constant. Suppose that at a certain instant the volume is 600 cm3, the pressure is 150 kPa, and the pressure is increasing at a rate of 20
The length of a rectangle is increasing at a rate of 8 cmys and its width is increasing at a rate of 3 cmys. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?
Each side of a square is increasing at a rate of 6 cmys. At what rate is the area of the square increasing when the area of the square is 16 cm2?
(a) If A is the area of a circle with radius r and the circle expands as time passes, find dAydt in terms of drydt.(b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 mys, how fast is the area of the spill
If V is the volume of a cube with edge length x and the cube expands as time passes, find dVydt in terms of dxydt.
The logistic difference equation with migration is of the form Nt11 − Nt 1 Nts1 2 Ntd 1 m where Nt is the population at time t and m is the migration rate. Suppose that as t l ` the population size approaches a limiting value N.(a) What equation does N satisfy?(b) Use implicit differentiation to
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