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mathematics
calculus
Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions
Estimate b(1.0) if b obeys the differential equation db/dt = 3b with initial condition b(0) = 10. Use Euler's method with Δt = 0.5 for two steps. Compare with the exact answer in Exercise 16.
Estimate G(1.0) if G obeys the differential equation dG/dt = G - 1 with initial condition G(0) = 2. Use Euler's method with Δt = 0.2 for five steps. Compare with the exact answer in Exercise 17.
Estimate z(4.0) if z obeys the differential equation dz/dt = 1/z - 1 with initial condition z(0) = 2. Use Euler's method with Δt = 1.0 for four steps. Compare with the exact answer in Exercise 18.
The derivation of the differential equation for p in the text requires combining two differential equations for a and b. Often, one can find a differential equation for a new variable derived from a single equation. In the above cases, use the chain rule to derive a new differential
The derivation of the differential equation for p in the text requires combining two differential equations for a and b. Often, one can find a differential equation for a new variable derived from a single equation. In the above cases, use the chain rule to derive a new differential
The derivation of the differential equation for p in the text requires combining two differential equations for a and b. Often, one can find a differential equation for a new variable derived from a single equation. In the above cases, use the chain rule to derive a new differential
The derivation of the differential equation for p in the text requires combining two differential equations for a and b. Often, one can find a differential equation for a new variable derived from a single equation. In the above cases, use the chain rule to derive a new differential
One widely used nonlinear model of competition is the logistic model, where per capita production is a linearly decreasing function of population size. Suppose that per capita production has a maximum at λ(0) = 1 and that it decreases with a slope of -0.002. Find λ(b) and the differential
Suppose that per capita production decreases linearly from a maximum of λ(0) = 4 with slope -0.001. Find λ(b) and the differential equation for b. Is b(t) increasing when b = 1000? Is b(t) increasing when b = 5000? The simple model of bacterial growth assumes that per capita production does not
In some circumstances, individuals reproduce better when the population size is large, and fail to reproduce when the population size is small (the Allee effect introduced in Exercise 46). Suppose that per capita production is an increasing linear function with λ(0) = -2 and a slope of 0.01. Find
Suppose that per capita production increases linearly with λ(0) = -5 and a slope of 0.001. Find λ(b) and the differential equation for b. Is b(t) increasing when b = 1000? Is b(t) increasing when b = 3000? The simple model of bacterial growth assumes that per capita production does not depend on
Suppose that no chemical re-enters the cell. This should look like the differential equation for a population. What would be happening to the concentration? The derivation of the movement of chemical assumed that chemical moved as easily into the cell as out of it. If the membrane can act as a
Suppose that no chemical leaves the cell. What would happen to the concentration? The derivation of the movement of chemical assumed that chemical moved as easily into the cell as out of it. If the membrane can act as a filter, the rates at which chemical enters and leaves might differ, or might
Suppose that the constant of proportionality governing the rate at which chemical enters the cell is three times as large as the constant governing the rate at which it leaves. Would the concentration inside the cell be increasing or decreasing if C = Г? What would this mean for the cell? The
Suppose that the constant of proportionality governing the rate at which chemical enters the cell is half as large as the constant governing the rate at which it leaves. Would the concentration inside the cell be increasing or decreasing if C = Г? What would this mean for the cell? The derivation
Suppose that the constant of proportionality governing the rate at which chemical enters the cell is proportional to 1 + C (because the chemical helps to open special channels). Would the concentration inside the cell be increasing or decreasing if C = Г? What would this mean for the cell?The
Suppose that the constant of proportionality governing the rate at which chemical enters the cell is proportional to 1 - C (because the chemical helps to close special channels). Would the concentration inside the cell be increasing or decreasing if C = Г? What would this mean for the cell?The
The per capita production of each type is reduced by a factor of 1 - p by a factor of 1 - p, so that the per capita production of type a is 2(1 - p). This is a case where a large proportion of type a reduces the production of both types. Will type a take over? The model of selection includes no
The per capita production of type a is reduced by a factor of 1 - p and the per capita production of type b is reduced by a factor of p. This is a case where a large proportion of type a reduces the production of type a, and a large proportion of type b reduces the production of type b. Do you
We will find later (with separation of variables) that the solution for Newton's law of cooling with initial condition H(0) isH(t) = A + [H(0) - A]e-αt.For each set of given parameter values,a. Write and check the solution.b. Find the temperature at t = 1 and t = 2.c. Sketch of graph of your
We will find later (with separation of variables) that the solution for Newton's law of cooling with initial condition H(0) isH(t) = A + [H(0) - A]e-αt.For each set of given parameter values,a. Write and check the solution.b. Find the temperature at t = 1 and t = 2.c. Sketch of graph of your
Use Euler's method to estimate the temperature for the following case of Newton's law of cooling. Compare with the exact answer. α = 0.2/min and A = 10°C and H(0) =40. Estimate H(l) and H(2) using Δt = 1. Compare with Exercise 39.
Use Euler's method to estimate the temperature for the following case of Newton's law of cooling. Compare with the exact answer. α = 0.02/min and A = 30°C and H(0) = 40. Estimate H(1) and H(2) using Δt = 1. Compare with Exercise 40. Why is the result so close?
Use the solution for Newton's law of cooling (Exercises 39 and 40) to find the solution expressing the concentration of chemical inside a cell as a function of time in the following examples. Find the concentration after 10 seconds, 20 seconds, and 60 seconds. Sketch your solutions for the first
Use the solution for Newton's law of cooling (Exercises 39 and 40) to find the solution expressing the concentration of chemical inside a cell as a function of time in the following examples. Find the concentration after 10 seconds, 20 seconds, and 60 seconds. Sketch your solutions for the first
Use Euler's method to estimate the value of pit) from the selection differential equation (Equation 5.1.4) for the given parameter values. Compare with the exact answer using the equation for the solution. Graph the solution, including the estimates from Euler's method. Suppose μ = 2.0/h, λ =
Use Euler's method to estimate the value of pit) from the selection differential equation (Equation 5.1.4) for the given parameter values. Compare with the exact answer using the equation for the solution. Graph the solution, including the estimates from Euler's method. Suppose μ = 2.5/h, λ =
Find the equilibrium temperature as a function of r. For the same values of c1 and c2, which animals stay warmest?Suppose that an endothermic (warm-blooded) animal generates heat at a rate proportional to its metabolic rate with constant c1, and loses heat at a rate proportional to its surface area
Suppose that c2 = 1.0 and that A = -20.0°C. Find the value of c1 required to maintain an equilibrium temperature of 40.0°C when r = 1.0 and when r = 2.0. Which organism needs to generate less heat to maintain its temperature?Suppose that an endothermic (warm-blooded) animal generates heat at a
Use integration to solve the pure-time differential equation starting from the initial condition p(0) = 1, find p(1), and sketch the solution. The above exercise compare the behavior of two similar-looking differential equations, the pure-time differential equation dp/dt = t and the autonomous
Find the equilibria of the following autonomous differential equations.
From the following graphs of the rate of change as a function of the state variable, draw the phase-line diagram.
From the following phase-line diagrams, sketch a solution starting from the specified initial condition.
From the following phase-line diagrams, sketch a solution starting from the specified initial condition.
The phase line in Exercise 11.From the given phase-line diagram, sketch a possible graph of the rate of change of x as a function of x.
Suppose we wish to calculate the proportion of days that the temperature rises above 20°C. Evaluate the following sampling schemes.1. Sample 100 consecutive days beginning on January 1.2. Sample 100 consecutive days beginning on June 1.3. Sample the temperature on March 15 for 100 years.4. What
Let H represent the number of heads in four flips of a fair coin. Compare with Exercise 9. Find the probability distribution associated with the above random variables, and identify which part corresponds to the likelihood found in the earlier problem.
The number N of sixes rolled in four rolls of a fair die. Compare with Exercise 10. Find the probability distribution associated with the above random variables, and identify which part corresponds to the likelihood found in the earlier problem.
Team A wins five out of six games in a series against team B. Find the maximum likelihood estimator of the probability that team A wins a game against team B. If you were willing to gamble, would it make sense to enter a bet about the next game in the series where you win $1 if team A wins, but
One out of 150 people you know wins $500 in a raffle that costs $5 to enter. Find the maximum likelihood estimator of the probability of winning the raffle. What is your best guess of the average payoff? Find the likelihood as a function of the binomial proportion p in each of the above case, and
Twenty events occur in 1 min. Compare the likelihood with the maximum likelihood estimator of Λ with the likelihood if Λ = 10.0. Find the likelihood as a function of the Poisson parameter Λ, find the maximum likelihood estimator, and evaluate the likelihood at the maximum and at the other given
Ten high energy cosmic rays hit detector over the course of 1 yr. Compare the likelihood with the maximum likelihood estimator of Λ with the likelihood if Λ = 8.0. Find the likelihood as a function of the Poisson parameter Λ, find the maximum likelihood estimator, and evaluate the likelihood at
The number of events that occur are counted for 3 min. Twenty events occur the first minute, 16 events occur the second minute, and 21 events occur the third minute. Compare the likelihood with the maximum likelihood estimator of Λ with the likelihood if Λ = 20.0. Find the likelihood as a
Ten high energy cosmic rays hit detector in its first year, 7 in the second year, 11 in the third year, and 8 in the fourth and final year. Compare the likelihood with the maximum likelihood estimator of Λ with the likelihood if Λ = 10.0. Find the likelihood as a function of the Poisson parameter
Flies are tested for the ability to learn to fly toward the smell of potato, and the first to succeed is the 13th. Compare with the likelihood of q = 0.1. Find the likelihood as a function of the parameter q of a geometric distribution, find the maximum likelihood estimator, and evaluate the
Random compounds are tested for the ability to suppress a particular type of tumor, and the first to succeed is the 94th. Compare with the likelihood of q = 0.005. Find the likelihood as a function of the parameter q of a geometric distribution, find the maximum likelihood estimator, and evaluate
The experiment testing flies for the ability to learn to fly toward the smell of potato is repeated three times. In the first experiment the first fly to succeed is the 13th, in the second experiment it is the 8th, and in the third experiment it is the 12th. Compare with the likelihood of q =
The experiment testing compounds for the ability to suppress tumors is repeated twice. In the first experiment the first compound to succeed is the 94th, and in the second experiment it is the 406th. Compare with the likelihood of q = 0.005. Find the likelihood as a function of the parameter q of a
The estimator of q in Exercise 21. Write down the equations that would express the fact that the above estimator is unbiased.
The estimator of A in Exercise 17. If you think about the definition of the expectation, you might be able to demonstrate that this estimator is unbiased. Write down the equations that would express the fact that the above estimator is unbiased.
Two individuals are tested for a particular allele and one has it. In the above case where a very small number of individuals is tested for an allele, find and graph the likelihood function for the proportion p of individuals in the whole population with this allele, find the maximum likelihood
Three individuals are tested for a particular allele, and all three have it. In the above case where a very small number of individuals is tested for an allele, find and graph the likelihood function for the proportion p of individuals in the whole population with this allele, find the maximum
The sample consists of 20 out of 50 infected women and 10 out of 50 infected men. Thirty out of 100 individuals are found to be infected with a disease. Estimate the proportion of infected women and infected men in the above circumstances. Assuming that the whole population is composed of 50%
The sample consists of 20 out of 40 infected women and 10 out of 60 infected men. Thirty out of 100 individuals are found to be infected with a disease. Estimate the proportion of infected women and infected men in the above circumstances. Assuming that the whole population is composed of 50%
Two couples are trying to have more girl babies. For each, find the likelihood function for the fraction q of female sperm and the maximum likelihood estimator, and compare with the likelihood of q = 0.5.1. The first couple has seven boys before having a girl. Use the geometric distribution to
For waiting time 1. Use the method of maximum likelihood to estimate the rate λ from the accompanying table of data drawn from an exponential distribution.In each case, find the likelihood function, find the maximum likelihood, and say whether if seems probable that the true rate is
For waiting time 2. Use the method of maximum likelihood to estimate the rate λ from the accompanying table of data drawn from an exponential distribution.In each case, find the likelihood function, find the maximum likelihood, and say whether if seems probable that the true rate is
Write the likelihood function for the expected number of mutations per million bases in the first piece and find the maximum likelihood estimator. Mutations are counted in four pieces of DNA that are 1 million base pairs long. There are 14 mutations in the first piece, 17 in the second piece, 8 in
Write the likelihood function for the expected number of mutations per million bases in the second piece and find the maximum likelihood estimator. Mutations are counted in four pieces of DNA that are 1 million base pairs long. There are 14 mutations in the first piece, 17 in the second piece, 8 in
Write the likelihood function for the expected number of mutations per million bases in the first two pieces and find the maximum likelihood estimator. Compare this with the estimated expected number for each of the two pieces separately. Mutations are counted in four pieces of DNA that are 1
Write the likelihood function for the expected number of mutations per million bases in the first four pieces and find the maximum likelihood estimator. Compare this with the estimated expected number for each of the four pieces separately. Mutations are counted in four pieces of DNA that are 1
Estimate λs, the mutation rate for synonymous sites, from the given data. Mutation rates differ depending on whether changing the nucleotide base changes the amino acid (non synonymous sites) or not (synonymous sites). A piece of DNA has 200 non synonymous sites with 12 mutations and 100
Estimate λn, the mutation rate for non synonymous sites, from the given data. Mutation rates differ depending on whether changing the nucleotide base changes the amino acid (non synonymous sites) or not (synonymous sites). A piece of DNA has 200 non synonymous sites with 12 mutations and 100
Suppose we assume that the synonymous and non synonymous rates are both equal to the same value λ. Estimate λ, and compare with the values of λs and λn found in Exercises 41 and 42. Mutation rates differ depending on whether changing the nucleotide base changes the amino acid (non synonymous
Suppose instead that the synonymous rate is three times that of the non synonymous rate. Formally, λn = λ and λs = 3λ. Estimate λ, and compare with the values of λ found in Exercises 41 and 42. Mutation rates differ depending on whether changing the nucleotide base changes the amino acid (non
Estimate p using just the males. Color blindness is due to a recessive allele that appears on the X chromosome. If the color blindness allele has frequency p, a fraction p of males will show the phenotype, and a fraction p2 of females will show the phenotype (because they require two copies). We
Estimate p using just the females. Color blindness is due to a recessive allele that appears on the X chromosome. If the color blindness allele has frequency p, a fraction p of males will show the phenotype, and a fraction p2 of females will show the phenotype (because they require two copies). We
Write the likelihood function for males and females together. Color blindness is due to a recessive allele that appears on the X chromosome. If the color blindness allele has frequency p, a fraction p of males will show the phenotype, and a fraction p2 of females will show the phenotype (because
Evaluate the likelihood function in Exercise 47 at p = 0.09, p = 0.114, and p = 0.1. Where do you think the maximum might be? If you are very determined, it is possible to solve for the maximum. Color blindness is due to a recessive allele that appears on the X chromosome. If the color blindness
What distribution describes the result of sampling 20 people with cellular phones? Use the rule of thumb that 95% of the distribution lies within two standard deviations of the mean to give a probable range. Compare this with the true average of the population. Consider again the situation in
What would be the results of sampling 20 people without cellular phones? Use the rule of thumb that 95% of the distribution lies within two standard deviations of the mean to give a probable range. Compare this with the true average of the population. Consider again the situation in Exercises 5 and
Find the likelihood as a function of the binomial proportion p for each of the following.1. Flipping 2 out of 4 heads with a fair coin. Evaluate at p = 0.5, the value for a fair coin.2. Rolling 2 out of 4 sixes with a fair die. Evaluate at p = 1/6, the value for a fair die.3. Flipping 2 out of 12
A coin is flipped five times and comes up heads every time. Does the value p = 0.5 for the probability of heads lie within the 95% confidence limits? In the above situation, find the probability of a result as extreme as or more extreme than the actual result for the given value of the parameter.
Find the 98% confidence limits if a one cosmic ray hits a detector in 1 yr (Exercise 6). Find the exact confidence limits and check if the value for the earlier problem lies within them.
A coin is flipped five times and comes up heads every time (Exercise 7). Find the approximate 95% confidence limits using the method of support and compare with earlier exercises.
A coin is flipped seven times and comes up heads six out of seven times (Exercise 8). Find the approximate 95% confidence limits using the method of support and compare with earlier exercises.
A person wins the lottery the second time he plays (Exercise 9, but recall that the earlier exercise found 99% confidence limits). Find the approximate 95% confidence limits using the method of support and compare with earlier exercises.
One cosmic ray hits a detector in 1 yr (Exercise 6, but recall that the earlier exercise found 99% confidence limits). Find the approximate 95% confidence limits using the method of support and compare with earlier exercises.
A coin is flipped five times and comes up heads every time, and we wish to find the upper and lower 95% confidence limits. Explain how you would use the Monte Carlo method to estimate confidence limits in the above case.
A coin is flipped seven times and comes up heads six out of seven times, and we wish to find upper and lower 95% confidence limits. Explain how you would use the Monte Carlo method to estimate confidence limits in the above case.
A person wins the lottery the second time he plays, and we wish to find upper and lower 99% confidence limits. Explain how you would use the Monte Carlo method to estimate confidence limits in the above case.
One cosmic ray hits a detector in 1 yr, and we wish to find upper and lower 98% confidence limits. Explain how you would use the Monte Carlo method to estimate confidence limits in the above case.
Flipping 2 out of 4 heads with a fair coin. Check whether the expected value of p (p = 1/2 for a fair coin and p = 1 /6 for a fair die) lies within the approximate 95% confidence limits given by the method of support.
A coin is flipped seven times and comes up heads six out of seven times. Does the value p = 0.5 for the probability of heads lie within the 95% confidence limits? In the above situation, find the probability of a result as extreme as or more extreme than the actual result for the given value of the
Rolling 2 out of 4 sixes with a fair die. Check whether the expected value of p (p = 1/2 for a fair coin and p = 1 /6 for a fair die) lies within the approximate 95% confidence limits given by the method of support.
Flipping 2 out of 12 heads with a fair coin. Check whether the expected value of p (p = 1/2 for a fair coin and p = 1 /6 for a fair die) lies within the approximate 95% confidence limits given by the method of support.
Twenty events occur in 1 min with a given value of A = 10.0. Check whether the given value of A lies within the approximate 95% confidence limits given by the method of support.
Ten high energy cosmic rays hit detector over the course of 1 yr, with a given value of A = 8.0. Check whether the given value of A lies within the approximate 95% confidence limits given by the method of support.
The confidence limits if 20 out of 100 individuals are measured with a particular allele (Example 8.2.13). Use experimentation and Newton's method to solve the equations for the approximate confidence limits with the method of support.
The confidence limits if 23 seeds are found in 1 m2 (Example 8.2.14).Use experimentation and Newton's method to solve the equations for the approximate confidence limits with the method of support.
Using the likelihood function in Section 8.1, Exercise 9. Prove that the support takes on its maximum where the likelihood does.
Using the likelihood function in Section 8.1, Exercise 17. Prove that the support takes on its maximum where the likelihood does.
Using the likelihood function in Section 8.1, Exercise 19. Prove that the support takes on its maximum where the likelihood does.
A person wins the lottery the second time he plays. Does the value q = 0.001 for the probability of success lie within the 99% confidence limits? In the above situation, find the probability of a result as extreme as or more extreme than the actual result for the given value of the parameter.
For a general likelihood function. Prove that the support takes on its maximum where the likelihood does.
Find the exact 95% confidence limits. Consider a tiny data set where one out of two people is found with an allele.
Find the exact 99% confidence limits. Consider a tiny data set where one out of two people is found with an allele.
How would you use the Monte Carlo method to estimate the 99% confidence limits? Consider a tiny data set where one out of two people is found with an allele.
Use the method of support to estimate 95% confidence limits and compare your results with Exercise 31. Consider a tiny data set where one out of two people is found with an allele.
Find the exact 95% confidence limits. Consider a data set where three out of three people are found with an allele.
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