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mathematics
calculus
Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions
A population of bacteria has per capita production r < 1, and 1.0 x 106 bacteria are added each generation. What happens to the equilibrium if r = 0? What happens if r is close to 1? Do these results make biological sense? Find the equilibrium population of bacteria in the above case with
There is no evaporation, and 3.0 x 106m3 of water flows out each year. Lakes receive water from streams each year and lose water to out flowing streams and evaporation. The following values are based on the Great Salt Lake in Utah. The lake receives 3.0 x 106m3 of water per year with salinity of 1
1.5 x 106m3 of water flows out each year, and 1.5 x 106m3 evaporates. No salt is lost through evaporation. Lakes receive water from streams each year and lose water to out flowing streams and evaporation. The following values are based on the Great Salt Lake in Utah. The lake receives 3.0 x 106m3
A total of 3.0 x 106m3 of water evaporates, and there is no outflow. Lakes receive water from streams each year and lose water to out flowing streams and evaporation. The following values are based on the Great Salt Lake in Utah. The lake receives 3.0 x 106m3 of water per year with salinity of 1
Assume instead that 2.0 x 106m3 of water evaporates and there is no outflow. The volume of this lake is increasing. Lakes receive water from streams each year and lose water to out flowing streams and evaporation. The following values are based on the Great Salt Lake in Utah. The lake receives 3.0
The situation described in Exercise 41.Find the equilibrium concentration of salt in a lake in the above case. Describe the result in words by comparing the equilibrium salt level with the salt level of the water flowing in.Exercise 41There is no evaporation, and 3.0 x 106m3 of water flows out each
The situation described in Exercise 42.Find the equilibrium concentration of salt in a lake in the above case. Describe the result in words by comparing the equilibrium salt level with the salt level of the water flowing in.Exercise 421.5 x 106m3 of water flows out each year, and 1.5 x 106m3
The situation described in Exercise 43. Find the equilibrium concentration of salt in a lake in the above case. Describe the result in words by comparing the equilibrium salt level with the salt level of the water flowing in. Exercise 43 A total of 3.0 x 106m3 of water evaporates, and there is no
The situation described in Exercise 44.Find the equilibrium concentration of salt in a lake in the above case. Describe the result in words by comparing the equilibrium salt level with the salt level of the water flowing in.Exercise 44Assume instead that 2.0 x 106m3 of water evaporates and there is
Suppose that r = 1.5 and h = l.0 x 106 bacteria. Sketch the updating function, and find the equilibrium both algebraically and graphically. A lab is growing and harvesting a culture of valuable bacteria described by the discrete-time dynamical system bt+1 = rbt - h. The bacteria have per capita
1.0 L of water at temperature T1 is mixed with 2.0 L of water at temperature T2. What is the temperature of the resulting mixture? Set T1 = 30 and T2 = 100 and compare with the result of Exercise 1. Express the above weighted averages in terms of the given variables.
Without setting r and h to particular values, find the equilibrium algebraically. Does the equilibrium get larger when h gets larger? Does it get larger when r gets larger? If the answers seem odd (as they should), look at a cobweb diagram to try to figure out why. A lab is growing and harvesting a
V1 liters of water at 30°C is mixed with V2 liters of water at 100°C. What is the temperature of the resulting mixture? Set V1 = 1.0 and V2 = 2.0 and compare with the result of Exercise 1. Express the above weighted averages in terms of the given variables.
V1 liters of water at temperature T1 is mixed with V2 liters of water at temperature T2. What is the temperature of the resulting mixture? Express the above weighted averages in terms of the given variables.
V1 liters of water at temperature T1 is mixed with V2 liters of water at temperature T2 and V3 liters of water at temperature T3. What is the temperature of the resulting mixture? Express the above weighted averages in terms of the given variables.
1.0 L of water at 30°C is to be mixed with 2.0 L of water at 100°C, as in Exercise 1. Before mixing, however, the temperature of each moves half-way to 0 °C (so the 30°C water cools to 15°C). What is the temperature of the resulting mixture? Is this half the temperature of the result in
A culture of bacteria has mass 3.0 × 10−3 grams and consists of spherical cells of mass 2.0 × 10−10 grams and density 1.5 grams/cm3. a. How many bacteria are in the culture? b. What is the radius of each bacterium? c. If the bacteria were mashed into mush, how much volume would they take up?
A person develops a small liver tumor. It grows according toS(t) = S(0)eαtwhere S(0) = 1.0 gram and α = 0.1/day. At time t = 30 days, the tumor is detected and treatment begins. The size of the tumor then decreases linearly with slope of −0.4 grams/day.a. Write the equation for tumor size at t
Two similar objects are left to cool for one hour. One starts at 80oC and cools to 70oC and the other starts at 60oC and cools to 55oC. Suppose the discrete-time dynamical system for cooling objects is linear.a. Find the discrete-time dynamical system. Find the temperature of the first object after
A culture of bacteria increases in area by 10% each hour. Suppose the area is 2.0 cm2 at 2:00 P.M. a. What will the area be at 5:00 P.M.? b. Write the relevant discrete-time dynamical system and cobweb starting from 2.0. c. What was the area at 1:00 P.M.? d. If all bacteria are the same size and
Candidates Dewey and Howe are competing for fickle voters. A total of 100,000 people are registered to vote in the election, and each will vote for one of these two candidates. Voters often switch their allegiance. 20% of Dewey's supporters switch to Howe. Howe's supporters are more likely to
A certain bacterial population has the following odd behavior. If the population is less than 1.5 × 108 in a given generation, each bacterium produces two offspring. If the population is greater than or equal to 1.5 × 108 in a given generation, it will be exactly 1.0 × 108 in the next.a. Cobweb
An organism is breathing a chemical that modifies the depth of its breaths. In particular, suppose that the fraction q of air exchanged is given bywhere γ is the ambient concentration and ct is the concentration in the lung. After a breath, a fraction q of the air came from outside, and a
Lint is building up in a dryer. With each use, the old amount of lint xt is divided by 1 + xt and 0.5 lintons (the units of lint) are added.a. Find the discrete-time dynamical system and graph the updating function.b. Cobweb starting from x0 = 0. Graph the associated solution.c. Find the
Suppose people in a bank are waiting in two separate lines. Each minute several things happen: some people are served, some people join the lines, and some people switch lines. In particular, suppose that 1/10 of the people in the first line are served, and 3/10 of the people in the second line are
A gambler faces off against a small casino. She begins with $1000, and the casino with $11,000. In each round, the gambler loses 10% of her current funds to the casino, and the casino loses 2% of its current funds to the gambler. a. Find the amount of money each has after one round. b. Find a
Suppose the number of bacteria in a culture is a linear function of time.a. If there are 2.0 × 108 bacteria in your lab at 5 P.M. on Tuesday, and 5.0 × 108 bacteria the next morning at 9 A.M., find the equation of the line describing the number of bacteria in your culture as a function of time.b.
Let V represent the volume of a lung and c the concentration of some chemical inside. Suppose the internal surface area is proportional to volume and that a lung with volume 400 cm3 has a surface area of 100 cm2. The lung absorbs the chemical at a rate per unit surface area ofTime is measured in
Suppose a person's head diameter D and height H grow according to D(t) = 10.0e0.03t H(t) = 50.0e0.09t during the first 15 years of life. a. Find D and H at t =0, t =7.5, and t =15. b. Sketch graphs of these two measurements as functions of time. c. Sketch semi log graphs of these two measurements
On another planet, people have three hands and like to compute tripling times instead of doubling times.a. Suppose a population follows the equation b(t) = 3.0 × 103e0.333t where t is measured in hours. Find the tripling time.b. Suppose a population has a tripling time of 33 hours. Find the
A Texas millionaire (with $1,000,001 in assets in 2010) got rich by clever investments. She managed to earn 10% interest per year for the last 20 years and plans to do the same in the future.a. How much did she have in 1990?b. When will she have $5,000,001?c. Write the discrete-time dynamical
A major university hires a famous Texas millionaire to manage its endowment. The millionaire decides to follow this plan each year: • Spend 25% of all funds above $100 million on university operations. • Invest the remainder at 10% interest. • Collect $50 million in donations from wealthy
Another major university hires a different famous Texas millionaire to manage its endowment. This millionaire starts with $340 million, brings back $355 million the next year, and claims to be able to guarantee a linear increase in funds thereafter.a. How much money will this university have after
A heart receives a signal to beat every second. If the voltage when the signal arrives is below 50 micro volts, the heart beats and increases its voltage by 30 micro volts. If the voltage when the signal arrives is greater than 50 micro volts, the heart does not beat and the voltage does not
Suppose vehicles are moving at 72 kph (kilometers per hour). Each car carries an average of 1.5 people, and all are carefully keeping a 2- second following distance (getting no closer than the distance a car travels in 2 seconds) on a three-lane highway.a. How far between vehicles?b. How many
Suppose traffic volume on a particular road has been as follows:Year Vehicles1970 ...... 40,0001980 ...... 60,0001990 ...... 90,0002000 ......135,000a. Sketch a graph of traffic over time.b. Find the discrete-time dynamical system that describes this traffic.c. What was the traffic volume in
In order to improve both the economy and quality of life, policies are designed to encourage growth and decrease traffic flow. In particular, the number of vehicles is encouraged to increase by a factor of 1.6 over each 10-year period, but the commuters from 10,000 vehicles are to choose to ride
Consider the functions f (x) = e−2x and g(x) = x3 + 1. a. Find the inverses of f and g, and use these to find when f (x) = 2 and when g(x) = 2. b. Find f ο g and g ο f and evaluate each at x = 2. c. Find the inverse of g ο f. What is the domain of this function?
A lab has a culture of a new kind of bacteria where each individual takes 2 hours to split into three bacteria. Suppose that these bacteria never die and that all offspring are OK. a. Write an updating function describing this system. b. Suppose there are 2.0 × 107 bacteria at 9 A.M. How many will
The number of bacteria (in millions) in a lab are as followsTime, t (h)Number, bt0.0 ........................1.51.0 ........................3.02.0 ........................4.53.0 ........................5.04.0 ........................7.55.0 ........................9.0a. Graph these points.b. Find
The number of bacteria in another lab follows the discrete-time dynamical systemwhere t is measured in hours and bt in millions of bacteria.a. Graph the updating function. For what values of bt does it make sense?b. Find the equilibrium.c. Cobweb starting from b0 = 0.4 million bacteria. What do you
Convert the following angles from degrees to radians and find the sine and cosine of each. Plot the related point both on a circle and on a graph of the sine or cosine.a. θ = 60ο.b. θ = −60ο.c. θ = 110ο.d. θ = −190ο.e. θ = 1160ο.
Suppose the temperature H of a bird follows the equationWhere t is measured in days and H is measured in degrees C.a. Sketch a graph of the temperature of this bird.b. Write the equation if the period changes to 1.1 days. Sketch a graph.c. Write the equation if the amplitude increases to 3.5
The butterflies on a particular island are not doing too well. Each autumn, every butterfly produces on average 1.2 eggs and then dies. Half of these eggs survive the winter and produce new butterflies by late summer. At this time, 1000 butterflies arrive from the mainland to escape overcrowding.a.
A continuous function that is -1 for x ≤ -0.1, 1 for x ≥ 0.1, and is linear for -0.1 < x < 0.1. We can build different continuous approximations of signum (the function giving the sign of a number) as follows. For each, case a. Graph the continuous function. b. Find the formula. c. How close
A continuous function that is -1 for x ≤ -0.01, 1 for x ≥ 0.01, and is linear for -0.01 < x < 0.01.We can build different continuous approximations of signum (the function giving the sign of a number) as follows. For each, casea. Graph the continuous function.b. Find the formula.c. How
Suppose the mass of an object as a function of volume is given by M = pV. If p = 2.0 g/cm3, how close must V be to 2.5 cm3 for M to be within 0.2 g of 5.0 g? Find the accuracy of input necessary to achieve the desired output accuracy.
Describe how the following functions are built out of the basic continuous functions. Identify points where they might not be continuous. f(x) = ex / x + 1.
The area of a disk as a function of radius is given by A = πr2. How close must r be to 2.0 cm to guarantee an area within 0.5 cm2 of 4π?Find the accuracy of input necessary to achieve the desired output accuracy.
The flow rate F through a vessel is proportional to the fourth power of the radius, or F(r) = ar4. Suppose a = 1.0/cm s. How close must r be to 1.0 cm to guarantee a flow within 5% of 1 mL/s? Find the accuracy of input necessary to achieve the desired output accuracy.
Consider an organism growing according to S(t) = S(0)eαt. Suppose α = 0.001/s, and S(0) = 1.0 mm. At time 1000 s, S(t) = 2.71828 mm. How close must t be to 1000 s to guarantee a size within 0.1 mm of 2.71828 mm? Find the accuracy of input necessary to achieve the desired output accuracy.
What values of b9 produce a result within the desired tolerance? What is the input tolerance?Suppose a population of bacteria follows the discrete-time dynamical systembt+1 = 2.0btand we wish to have a population within 1.0 x 108 of 1.0 x 109 at t = 10.
What values of b5 produce a result within the desired tolerance? What is the input tolerance? Why is it harder to hit the target from here?Suppose a population of bacteria follows the discrete-time dynamical systembt+1 = 2.0btand we wish to have a population within 1.0 x 108 of 1.0 x 109 at t = 10.
What values of b0 produce a result within the desired tolerance? What is the input tolerance?Suppose a population of bacteria follows the discrete-time dynamical systembt+1 = 2.0btand we wish to have a population within 1.0 x 108 of 1.0 x 109 at t = 10.
How would your answers differ if the discrete-time dynamical system were bt+1 = 5bt? Would the tolerances be larger or smaller? Why? Suppose a population of bacteria follows the discrete-time dynamical system bt+1 = 2.0bt and we wish to have a population within 1.0 x 108 of 1.0 x 109 at t = 10.
What values of T9 produce a result within the desired tolerance? What is the input tolerance? Suppose the amount of toxin in a culture declines according to Tt+1 = 0.5Tt and we wish to have a concentration within 0.02 of 0.5 g/L at t = 10.
What values of T5 produce a result within the desired tolerance? What is the input tolerance? Suppose the amount of toxin in a culture declines according to Tt+1 = 0.5Tt and we wish to have a concentration within 0.02 of 0.5 g/L at t = 10.
Describe how the following functions are built out of the basic continuous functions. Identify points where they might not be continuous. h(y) = y2ln(y - l) for y > l.
How would your answers differ if the discrete-time dynamical system were Tt+1 = 0.1Tt? Would the tolerances be larger or smaller? Why? Suppose the amount of toxin in a culture declines according to Tt+1 = 0.5Tt and we wish to have a concentration within 0.02 of 0.5 g/L at t = 10.
Suppose that k = 2.0, V0 = 50, and V* = 80. Write and graph the function giving output in terms of input as a function defined in pieces. Suppose a neuron has the following response to inputs. If it receives a voltage input V greater than or equal to a threshold of V0, it outputs a voltage of kV
Suppose that k = 1.5, V0 = 60, and V* = 100. Write and graph the function giving output in terms of input as a function defined in pieces. Suppose a neuron has the following response to inputs. If it receives a voltage input V greater than or equal to a threshold of V0, it outputs a voltage of kV
If k = 2.0 and V0 = 50, what would V* have to be for the function to be continuous? Graph the resulting function. Suppose a neuron has the following response to inputs. If it receives a voltage input V greater than or equal to a threshold of V0, it outputs a voltage of kV for some constant k. If it
If V0 = 50 and V* = 80, what would k have to be to make the function continuous? Graph the resulting function. Suppose a neuron has the following response to inputs. If it receives a voltage input V greater than or equal to a threshold of V0, it outputs a voltage of kV for some constant k. If it
A child outside is swinging on a swing that makes a horrible screeching noise. Starting from when the swing is furthest back, the pitch of the screeching noise increases as it swings forward and then decreases as it swings back. a. Draw a graph of the pitch as a function of position without
Little Billy walks due east to school, but must cross from the south side to the north side of the street. Because he is a very careful child, he crosses quickly at the first possible opportunity. a. Graph little Billy's latitude as a function of distance from home on the way to school. b. Graph
Describe how the following functions are built out of the basic continuous functions. Identify points where they might not be continuous. g(z) = ln (z - 1) / z2 for z > 1.
Describe how the following functions are built out of the basic continuous functions. Identify points where they might not be continuous. a(t) = t2 if t > 0 and 0 if t = 0.
For each of the following quadratic functions, find the slope of the secant line connecting x = 1 and x = 1 + Δx, and the slope of the tangent line at x = 1 by taking the limit. f(x) = 4 - x2.
For each of the following quadratic functions, find the slope of the secant line connecting x = 1 and x = 1 + Δx, and the slope of the tangent line at x = 1 by taking the limit. g(x) = x + 2x2.
For each of the following quadratic functions, find the slope of the secant line connecting x and x + Δx, and the slope of the tangent line as a function of x. Write your result in both differential and prime notation. f(x) = 4 - x2 (based on Exercise 11).
For each of the following quadratic functions, find the slope of the secant line connecting x and x + Δx, and the slope of the tangent line as a function of x. Write your result in both differential and prime notation. g(x) = x + 2x2 (based on Exercise 12).
For each of the following quadratic functions, graph the function and the derivative. Identify critical points, points where the function is increasing, and points where the function is decreasing. f(x) = 4 - x2 (based on Exercise 13).
For each of the following quadratic functions, graph the function and the derivative. Identify critical points, points where the function is increasing, and points where the function is decreasing. g(x) = x + 2x2. For Information: (Exercise 14)
On the figures, label the following points and sketch the derivative.a. One point where the derivative is positive.b. One point where the derivative is negative.c. The point with maximum derivative.d. The point with minimum (most negative) derivative.e. Points with derivative of zero (critical
On the figures, label the following points and sketch the derivative.a. One point where the derivative is positive.b. One point where the derivative is negative.c. The point with maximum derivative.d. The point with minimum (most negative) derivative.e. Points with derivative of zero (critical
On the figures, identify which of the curves is a graph of the derivative of the other.
On the figures, identify which of the curves is a graph of the derivative of the other.
On the figures, identify which of the curves is a graph of the derivative of the other.
On the figures, identify which of the curves is a graph of the derivative of the other.
The absolute value function f(x) = |x|. The following functions all fail to be differentiable at x = 0. In each case, graph the function, see what happens if you try to compute the derivative as the limit of the slopes of secant lines, and say something about the tangent line.
The square root function f(x) = √x. (Because this function is only denned for x ≥ 0, you can only use Δx > 0.) The following functions all fail to be differentiable at x = 0. In each case, graph the function, see what happens if you try to compute the derivative as the limit of the slopes of
The Heaviside function (Section 2.3, Exercise 25), defined byThe following functions all fail to be differentiable at x = 0. In each case, graph the function, see what happens if you try to compute the derivative as the limit of the slopes of secant lines, and say something about the tangent line.
The signum function defined byThe following functions all fail to be differentiable at x = 0. In each case, graph the function, see what happens if you try to compute the derivative as the limit of the slopes of secant lines, and say something about the tangent line.
The following graphs show the temperature of different solutions with chemical reactions as functions of time. Graph the rate of change of temperature in each case, and indicate when the solution is warming up and when it is cooling down.
The following graphs show the temperature of different solutions with chemical reactions as functions of time. Graph the rate of change of temperature in each case, and indicate when the solution is warming up and when it is cooling down.
The following graphs show the temperature of different solutions with chemical reactions as functions of time. Graph the rate of change of temperature in each case, and indicate when the solution is warming up and when it is cooling down.
The following graphs show the temperature of different solutions with chemical reactions as functions of time. Graph the rate of change of temperature in each case, and indicate when the solution is warming up and when it is cooling down.
Both move at constant speed, but the bear is faster and eventually catches the hiker. A bear sets off in pursuit of a hiker. Graph the position of a bear and hiker as functions of time from the above descriptions.
Both increase speed until the bear catches the hiker. A bear sets off in pursuit of a hiker. Graph the position of a bear and hiker as functions of time from the above descriptions.
The bear increases speed and the hiker steadily slows down until the bear catches the hiker. A bear sets off in pursuit of a hiker. Graph the position of a bear and hiker as functions of time from the above descriptions.
The bear runs at constant speed, the hiker steadily runs faster until the bear gives up and stops. The hiker slows down and stops soon after that. A bear sets off in pursuit of a hiker. Graph the position of a bear and hiker as functions of time from the above descriptions.
On Earth, where a = 9.78m/sec2. An object dropped from a height of 100 m has distance above the ground of M(t) = 100 - a/2t2 where a is the acceleration of gravity. For each of the above planet with the given acceleration, find the time when the object hits the ground, and the speed of the object
On the moon, where a = 1.62m/sec2. An object dropped from a height of 100 m has distance above the ground of M(t) = 100 - a/2t2 where a is the acceleration of gravity. For each of the above planet with the given acceleration, find the time when the object hits the ground, and the speed of the
On Jupiter, where a = 22.88m/sec2. An object dropped from a height of 100 m has distance above the ground of M(t) = 100 - a/2t2 where a is the acceleration of gravity. For each of the above planet with the given acceleration, find the time when the object hits the ground, and the speed of the
On Mars' moon Deimos, where a = 2.15 × 10-3m/sec2. An object dropped from a height of 100 m has distance above the ground of M(t) = 100 - a/2t2 where a is the acceleration of gravity. For each of the above planet with the given acceleration, find the time when the object hits the ground, and the
On the following graphs, identify points wherea. The function is not continuous,b. The function is not differentiable (and say why),c. The derivative is zero (critical points).
On the following graphs, identify points wherea. The function is not continuous,b. The function is not differentiable (and say why),c. The derivative is zero (critical points).
Find the derivatives of the following polynomial functions. Say where you used the sum, constant product, and power rules. s(x) = 1 - x + x2 - x3 + x4.
Find the derivatives of the following polynomial functions. Say where you used the sum, constant product, and power rules. g(z) = 3z3 + 2z2.
Find the derivatives of the following polynomial functions. Say where you used the sum, constant product, and power rules. p(x) = 1 + x + x2/2 + x3/6 + x4/24.
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