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mathematics
calculus
Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions
Find all functions of the form g(x) = mx + b that commute with the function f(x) = x + 1. Can you explain your answer in words? Not very many functions commute with each other. The above problem asks you to find all linear functions that commute with the given linear function.
Find all functions of the form g(x) = mx + b that commute with the function f(x) = 2x. Can you explain your answer in words? Not very many functions commute with each other. The above problem asks you to find all linear functions that commute with the given linear function.
y - 5 = - 3(x + 2) - 6, using points with x = 1 and x = 3. For the above line, find the slopes between the two given points by finding the change in output divided by the change in input. What is the ratio of the output to the input at each of the points? Which are proportional relations? Which are
Graph these data. The above data give the elevation of the surface of the Great Salt Lake in Utah. Year, y Elevation, E (ft) 1965 ............................. 4,193 1970 ............................. 4,196 1975 ............................. 4,199 1980 ............................. 4,199 1985
Mass as a function of age. Find the mass on day 1.75.Graph the following relations between measurements of a growing plant, checking that the points lie on a line. Find the equations in both point-slope and slope-intercept form.
Volume as a function of age. Find the volume on day 2.75.Graph the following relations between measurements of a growing plant, checking that the points lie on a line. Find the equations in both point-slope and slope-intercept form.
Glucose production as a function of mass. Estimate glucose production when the mass reaches 20.0 g.Graph the following relations between measurements of a growing plant, checking that the points lie on a line. Find the equations in both point-slope and slope-intercept form.
The line f(x) = 2x + 3 and the point (2, 7). Check that the point indicated lies on the line and find the equation of the line in point-slope form using the given point. Multiply out to check that the point-slope form matches the original equation.
Volume as a function of mass. Estimate the volume when the mass reaches 30.0 g. How will the density at that time compare with the density when a = 0.5?Graph the following relations between measurements of a growing plant, checking that the points lie on a line. Find the equations in both
Graph these data. Which point does not lie on the line? Consider the data in the following table (adapted from Parasitoids by H. C. F. God fray), describing the number of wasps that can develop inside caterpillars of different weights. Weight of Caterpillar (g) Number of Wasps 0.5
The men's Olympic record for the 1500 meters was 3:36.8 in 1972 and 3:35.9 in 1988. Find and graph the line connecting these. (Don't forget to convert everything into seconds.) The world record times for various races are decreasing at roughly linear rates.
The women's Olympic record for the 1500 meters was 4:01.4 in 1972 and 3:53.9 in 1988. Find and graph the line connecting these. The world record times for various races are decreasing at roughly linear rates.
If things continue at this rate, when will women finish the race in exactly no time? What might happen before that date? The world record times for various races are decreasing at roughly linear rates.
If things continue at this rate, when will women be running this race faster than men? The world record times for various races are decreasing at roughly linear rates.
The line g(y) = -2y + 7 and the point (3, 1). Check that the point indicated lies on the line and find the equation of the line in point-slope form using the given point. Multiply out to check that the point-slope form matches the original equation.
The line f(x) = 2(x - 1) + 3. Find equations in slope-intercept form for the above line. Sketch a graph indicating the original point from point-slope form.
The line g(z) = -3(z + l) - 3. Find equations in slope-intercept form for the above line. Sketch a graph indicating the original point from point-slope form.
A line passing through the point (1, 6) with slope -2. Find equations in slope-intercept form for the above line. Sketch a graph indicating the original point from point-slope form.
Write the updating function associated with each of the following discrete-time dynamical systems and evaluate it at the given arguments. Which are linear? pt+1 = pt - 2, evaluate at pt = 5, pt = 10, and pt = 15.
The updating function f(x) = x/1 + x. Put things over a common denominator to simplify the composition. Find the composition of the above mathematically elegant updating functions with itself, and find the inverse function.
The updating function h(x) = x/x - 1. Put things over a common denominator to simplify the composition. Find the composition of the above mathematically elegant updating functions with itself, and find the inverse function.
Find and graph the solutions of the following discrete-time dynamical systems for five steps with the given initial condition. Compare the graph of the solution with the graph of the updating function. vt+1 = l.5vt, with v0 = 1220μm3.
Find and graph the solutions of the following discrete-time dynamical systems for five steps with the given initial condition. Compare the graph of the solution with the graph of the updating function. lt+1 = lt - 1.7, with l0 = 13.1 cm.
Find and graph the solutions of the following discrete-time dynamical systems for five steps with the given initial condition. Compare the graph of the solution with the graph of the updating function. nt+1 = 0.5nt, with n0 = 1200.
Find and graph the solutions of the following discrete-time dynamical systems for five steps with the given initial condition. Compare the graph of the solution with the graph of the updating function. Mt+1 = 0.75Mt + 2.0 with M0 = 16.0.
Find a formula for vt, for the discrete-time dynamical system in Exercise 15, and use it to find the volume at t = 20.
Write the updating function associated with each of the following discrete-time dynamical systems and evaluate it at the given arguments. Which are linear? ψt+1 = ψt/2, evaluate at ψt = 4, ψt = 8, and ψt = 12.
Find a formula for vt, for the discrete-time dynamical system in Exercise 16, and use it to find the length at t = 20.
Find a formula for vt, for the discrete-time dynamical system in Exercise 17, and use it to find the number at t = 20.
Find a formula for vt, for the discrete-time dynamical system in Exercise 18, and use it to find the concentration at t = 20
Consider the updating function f(x) = x/1 + x from Exercise 13. Starting from an initial condition of x0 = 1, compute x1, x2, x3, and x4, and try to spot the pattern.
Use the updating function in Exercise 23 but start from the initial condition x0 = 2.
Consider the updating function h(x) = x/x - 1 Start from initial condition of x0 = 3, and try to spot the pattern. Experiment with a couple of other initial conditions. How would you describe your results in words?
A population doubles in size; 10 individuals are removed from a population. Try starting with 100 individuals, and then try to figure out what happens in general. Consider the above actions. Which of them commute (produce the same answer when done in either order)?
A population doubles in size; population size is divided by 4. Try starting with 100 individuals, and then try to figure out what happens in general. Consider the above actions. Which of them commute (produce the same answer when done in either order)?
Write the updating function associated with each of the following discrete-time dynamical systems and evaluate it at the given arguments. Which are linear? xt+1 = x2t + 2, evaluate at xt = 0, xt = 2, and xt = 4.
Find the pattern in the number of mites on a lizard with x0 = 10 and following the discrete-time dynamical system xt+1 = 2xt + 30.
Find the pattern in the number of mites on a lizard with x0 = 10 and following the discrete-time dynamical system xt+1 = 2xt + 20.
The following table display data from four experiments:a. Cell volume after 10 minutes in a watery bathb. Fish mass after 1 week in a chilly tankc. Gnat population size after 3 days without foodd. Yield of several varieties of soybean before and after fertilizationFor each, graph the new value as a
The following table display data from four experiments:a. Cell volume after 10 minutes in a watery bathb. Fish mass after 1 week in a chilly tankc. Gnat population size after 3 days without foodd. Yield of several varieties of soybean before and after fertilizationFor each, graph the new value as a
The following table display data from four experiments:a. Cell volume after 10 minutes in a watery bathb. Fish mass after 1 week in a chilly tankc. Gnat population size after 3 days without foodd. Yield of several varieties of soybean before and after fertilizationFor each, graph the new value as a
Write the updating function associated with each of the following discrete-time dynamical systems and evaluate it at the given arguments. Which are linear? Qt+1 = 1/Qt + 1, evaluate at Qt = 0, Qt = 1, and Qt = 2.
The following table display data from four experiments:a. Cell volume after 10 minutes in a watery bathb. Fish mass after 1 week in a chilly tankc. Gnat population size after 3 days without foodd. Yield of several varieties of soybean before and after fertilizationFor each, graph the new value as a
Find and graph the discrete-time dynamical system for tail length.These data define several discrete-time dynamical systems. For example, between the first measurement (on day 0.5) and the second (on day 1.0), the length increases by 1.5 cm. Between the second measurement (on day 1.0) and the third
Find and graph the discrete-time dynamical system for mass.These data define several discrete-time dynamical systems. For example, between the first measurement (on day 0.5) and the second (on day 1.0), the length increases by 1.5 cm. Between the second measurement (on day 1.0) and the third (on
Find and graph the discrete-time dynamical system for age.These data define several discrete-time dynamical systems. For example, between the first measurement (on day 0.5) and the second (on day 1.0), the length increases by 1.5 cm. Between the second measurement (on day 1.0) and the third (on day
Sketch a possible graph of the updating function. Suppose students are permitted to take a test again and again until they get a perfect score of 100. We wish to write a discrete-time dynamical system describing these dynamics.
Write a discrete-time dynamical system for the total volume of bacteria (suppose each bacterium takes up 104μm3). Consider the discrete-time dynamical system bt+1 = 2.0bt for a bacterial population.
Volume follows vt+1 = 1.5vt, with v0 = 1220μm3. Compose the updating function associated with each discrete-time dynamical system with itself. Find the two-step discrete-time dynamical system. Check that the result of applying the original discrete-time dynamical system twice to the given initial
Write a discrete-time dynamical system for the total area taken up by the bacteria (suppose the thickness is 20 μm). Consider the discrete-time dynamical system bt+1 = 2.0bt for a bacterial population.
Write a discrete-time dynamical system for the total volume of the cylindrical trees in Section 1.3, Exercise 27. Recall the equation ht+1 = ht + 1.0 for tree height.
Write a discrete-time dynamical system for the total volume of a spherical tree (this is kind of tricky). Recall the equation ht+1 = ht + 1.0 for tree height.
Graph three points on the updating function for the first patient. Find the discrete-time dynamical system for the first patient.Consider the following data describing the level of medication in the blood of two patients over the course of several days.
Graph three points on the updating function for the second patient and find the discrete-time dynamical system.Consider the following data describing the level of medication in the blood of two patients over the course of several days.
Two bacterial populations follow the discrete-time dynamical system bt+1 = 2.0bt, but the first starts with initial condition b0 = 1.0 x 106 and the second starts with initial condition b0 = 3.0 x 105.
Two trees follow the discrete-time dynamical system ht+1 = ht + 1.0, but the first starts with initial condition h0 = 10.0 m and the second starts with initial condition h0 = 2.0 m.
A population of bacteria doubles every hour, but 1.0 x 106 individuals are removed after reproduction to be converted into valuable biological by-products. The population begins with b0 = 3.0 x 106 bacteria.a. Find the population after 1, 2, and 3 hours.b. How many bacteria were harvested?c. Write
Suppose a population of bacteria doubles every hour, but that 1.0 x 106 individuals are removed before reproduction to be converted into valuable biological by-products. Suppose the population begins with b0 = 3.0 x 106 bacteria. a. Find the population after 1, 2, and 3 hours. b. Write the
Suppose the fraction of individuals with some superior gene increases by 10% each generation.a. Write the discrete-time dynamical system for the fraction of organisms with the gene (denote the fraction at time t by ft and figure out the formula for ft+1).b. Write the solution with f0 = 0.0001.c.
Length obeys lt+1 = lt - 1.7, with l0 = 13.1 cm. Compose the updating function associated with each discrete-time dynamical system with itself. Find the two-step discrete-time dynamical system. Check that the result of applying the original discrete-time dynamical system twice to the given initial
The Weber-Fechner law describes how human beings perceive differences. Suppose, for example, that a person first hears a tone with a frequency of 400 hertz (cycles per second). He is then tested with higher tones until he can hear the difference. The ratio between these values describes how well
The number of bacteria doubles each hour, and the mass of each bacterium triples during the same time.The total mass of a population of bacteria will change if either the number of bacteria changes, the mass per bacterium changes, or both. The above problem derives discrete-time dynamical systems
The number of bacteria doubles each hour, and the mass of each bacterium increases by 1.0 x 10-9g. What seems to go wrong with this calculation? Can you explain why? The total mass of a population of bacteria will change if either the number of bacteria changes, the mass per bacterium changes, or
Population size follows nt+1 = 0.5nt, with n0 = 1200. Compose the updating function associated with each discrete-time dynamical system with itself. Find the two-step discrete-time dynamical system. Check that the result of applying the original discrete-time dynamical system twice to the given
Medication concentration obeys Mt+1 = 0.75Mt + 2.0 with M0 = 16.0. Compose the updating function associated with each discrete-time dynamical system with itself. Find the two-step discrete-time dynamical system. Check that the result of applying the original discrete-time dynamical system twice to
The discrete-time dynamical system bt+1 = 2.0bt with b0 = 1.0. The following steps are used to build a cobweb diagram. Follow them for the given discrete time dynamical system based on bacterial populations. a. Graph the updating function. b. Use your graph of the updating function to find the
Graph the updating functions associated with the following discrete-time dynamical systems, and cobweb for five steps starting from the given initial condition. xt+1 = 4 - xt, starting from x0 = 1 For Information: (as in Section 1.5, Exercise 25).
Graph the updating functions associated with the following discrete-time dynamical systems, and cobweb for five steps starting from the given initial condition. xt+1 = xt/1 + xt, starting from x0 = 1. For Information: (as in Section 1.5, Exercise 23)
Graph the updating functions associated with the following discrete-time dynamical systems, and cobweb for five steps starting from the given initial condition. xt+1 = xt/xt - 1, for xt > 1, starting from x0 = 3. For Information: (as in Section 1.5, Exercise 26)
Find the equilibria of the following discrete-time dynamical system from the graphs of their updating functions Label the coordinates of the equilibria.
Find the equilibria of the following discrete-time dynamical system from the graphs of their updating functions Label the coordinates of the equilibria.
Find the equilibria of the following discrete-time dynamical system from the graphs of their updating functions Label the coordinates of the equilibria.
Find the equilibria of the following discrete-time dynamical system from the graphs of their updating functions Label the coordinates of the equilibria.
Sketch graphs of the following updating functions over the given range and mark the equilibria. Find the equilibria algebraically if possible. f(x) = x2 for 0 ≤ x ≤ 2.
Sketch graphs of the following updating functions over the given range and mark the equilibria. Find the equilibria algebraically if possible. g(y) = y2 - 1 for 0 ≤ y ≤ 2.
Graph the following discrete-time dynamical systems. Solve for the equilibria algebraically, and identify equilibria and the regions where the updating function lies above the diagonal on your graph. ct+1 = 0.5ct + 8.0, for 0 ≤ ct ≤ 30.
The discrete-time dynamical system nt+l = 0.5nt with n0 = 1.0. The following steps are used to build a cobweb diagram. Follow them for the given discrete time dynamical system based on bacterial populations. a. Graph the updating function. b. Use your graph of the updating function to find the
Graph the following discrete-time dynamical systems. Solve for the equilibria algebraically, and identify equilibria and the regions where the updating function lies above the diagonal on your graph. bt+1 =3bt for 0 ≤ bt ≤ 10.
Graph the following discrete-time dynamical systems. Solve for the equilibria algebraically, and identify equilibria and the regions where the updating function lies above the diagonal on your graph. bt+1 = 0.36t, for 0 ≤ bt ≤ 10.
Graph the following discrete-time dynamical systems. Solve for the equilibria algebraically, and identify equilibria and the regions where the updating function lies above the diagonal on your graph. bt+1 = 2.0bt - 5.0, for 0 ≤ bt ≤ 10.
Cobweb the following discrete-time dynamical systems for three steps starting from the given initial condition. Compare with the solution found earlier. vt+1 = l.5vt starting from v0 = 1220μm3 For Information: (as in Section 1.5, Exercise 5).
Find the equilibria of the following discrete-time dynamical systems that include parameters. Identify values of the parameter for which there is no equilibrium, for which the equilibrium is negative, and for which there is more than one equilibrium. wt+1 = awt + 3.
Find the equilibria of the following discrete-time dynamical systems that include parameters. Identify values of the parameter for which there is no equilibrium, for which the equilibrium is negative, and for which there is more than one equilibrium. xt+1 = b - xt.
Find the equilibria of the following discrete-time dynamical systems that include parameters. Identify values of the parameter for which there is no equilibrium, for which the equilibrium is negative, and for which there is more than one equilibrium. xt+1 = axt/1 + xt.
Find the equilibria of the following discrete-time dynamical systems that include parameters. Identify values of the parameter for which there is no equilibrium, for which the equilibrium is negative, and for which there is more than one equilibrium. xt+1 = xt/xt - K.
An alternative tree growth discrete-time dynamical system with form ht+1 = ht + 5.0 with initial condition x0 = 10. Cobweb the above discrete-time dynamical systems for five steps starting from the given initial condition.
The lizard-mite system (Example 1.5.3) xt+1 = 2xt + 30 with initial condition x0 = 0. Cobweb the above discrete-time dynamical systems for five steps starting from the given initial condition.
The model defined in Section 1.5, Exercise 37 starting from an initial volume of 1420. Cobweb the above discrete-time dynamical systems for five steps starting from the given initial condition.
The model defined in Section 1.5, Exercise 38 starting from an initial mass of 13.1. Cobweb the above discrete-time dynamical systems for five steps starting from the given initial condition.
The model defined in Section 1.5, Exercise 39 starting from an initial population of 800. Cobweb the above discrete-time dynamical systems for five steps starting from the given initial condition.
Cobweb the following discrete-time dynamical systems for three steps starting from the given initial condition. Compare with the solution found earlier. lt+1 = lt - 1.7, starting from l0 = 13.1 cm For Information: (as in Section 1.5, Exercise 6).
The model defined in Section 1.5, Exercise 40 starting from an initial yield of 20. Cobweb the above discrete-time dynamical systems for five steps starting from the given initial condition.
For the first patient, graph the updating function and cobweb starting from the initial condition on day 0. Find the equilibrium.Reconsider the data describing the levels of a medication in the blood of two patients over the course of several days (measured in mg per liter).For Information: Section
For the second patient, graph the updating function and cobweb starting from the initial condition on day 0. Find the equilibrium.Reconsider the data describing the levels of a medication in the blood of two patients over the course of several days (measured in mg per liter).For Information:
Consider a bacterial population that doubles every hour, but 1.0 x 106 individuals are removed after reproduction. Cobweb starting from b0 = 3.0 x 106 bacteria. Cobweb and find the equilibrium of the above discrete-time dynamical system. For Information: (Section 1.5, Exercise 57)
Consider a bacterial population that doubles every hour, but 1.0 x 106 individuals are removed before reproduction (Section 1.5, Exercise 58). Cobweb starting from b0 = 3.0 x 106 bacteria. Cobweb and find the equilibrium of the above discrete-time dynamical system.
Consider a bacterial population that doubles every hour, but h individuals are removed after reproduction. Find the equilibrium. Does it make sense? Consider the above general models for bacterial populations with harvest.
Consider a bacterial population that increases by a factor of r every hour, but 1.0 x 106 individuals are removed after reproduction. Find the equilibrium. What values of r produce a positive equilibrium? Consider the above general models for bacterial populations with harvest.
Consider the general model Mt+1 = (1 - α)Mt + S for medication (Example 1.6.11). Find the loading dose (Example 1.6.7) in the following cases. α = 0.2, S = 2.
Consider the general model Mt+1 = (1 - α)Mt + S for medication (Example 1.6.11). Find the loading dose (Example 1.6.7) in the following cases. α = 0.8, S = 4.
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