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1. (a) Verify that the function y = x³ 2xy + 2y = 8x³ (b) Find a value for c that satisfies the initial condition y(2) = 6. 2. Use separation of variables to solve the following problems dy =-ty, y(0)=1/√ dt y² + 5 y (a) (b) dy dt -, y(0) = -2 dy (c) = 3y(y - 5), y(0) = 8 dt where 3. Consider the following very simple model of blood cholesterol levels based on the fact that cholesterol is manufactured by the body for use in the construction of cell walls and is absorbed from foods containing cholesterol: Let C(t) be the amount (in milligrams per deciliter) of cholesterol in the blood of a particular person at time t (in days). Then is a solution to the differential equation dC dt = k₁(NC) + k₂E N = person's natural cholesterol level k₁ production parameter E k₂ daily rate at which cholesterol is eaten absorption parameter (a) Suppose N = 200, k₁= 0.1, k₂ = 0.1, E = 400, and C(0) = 150. What will the person's cholesterol level be after 2 days on this diet? (Hint: solve the ODE, then plug in t = 2) (b) With the initial conditions as above, what will the person's cholesterol level be after 5 days on this diet? (c) What will the person's cholesterol level be after a long time on this diet? (d) High levels of cholesterol in the blood are known to be a risk factor for heart disease. Suppose that, after a long time on the high cholesterol diet described above, the person goes on a very low cholesterol diet, so E changes to E = 100. (The initial cholesterol level at the starting time of this diet is the result of part (c).) What will the person's cholesterol level be after 1 day on the new diet, after 5 days on the new diet, and after a very long time on the new diet? 1

1. (a) Verify that the function y = x³ 2xy + 2y = 8x³ (b) Find a value for c that satisfies the initial condition y(2) = 6. 2. Use separation of variables to solve the following problems dy =-ty, y(0)=1/√ dt y² + 5 y (a) (b) dy dt -, y(0) = -2 dy (c) = 3y(y - 5), y(0) = 8 dt where 3. Consider the following very simple model of blood cholesterol levels based on the fact that cholesterol is manufactured by the body for use in the construction of cell walls and is absorbed from foods containing cholesterol: Let C(t) be the amount (in milligrams per deciliter) of cholesterol in the blood of a particular person at time t (in days). Then is a solution to the differential equation dC dt = k₁(NC) + k₂E N = person's natural cholesterol level k₁ production parameter E k₂ daily rate at which cholesterol is eaten absorption parameter (a) Suppose N = 200, k₁= 0.1, k₂ = 0.1, E = 400, and C(0) = 150. What will the person's cholesterol level be after 2 days on this diet? (Hint: solve the ODE, then plug in t = 2) (b) With the initial conditions as above, what will the person's cholesterol level be after 5 days on this diet? (c) What will the person's cholesterol level be after a long time on this diet? (d) High levels of cholesterol in the blood are known to be a risk factor for heart disease. Suppose that, after a long time on the high cholesterol diet described above, the person goes on a very low cholesterol diet, so E changes to E = 100. (The initial cholesterol level at the starting time of this diet is the result of part (c).) What will the person's cholesterol level be after 1 day on the new diet, after 5 days on the new diet, and after a very long time on the new diet? 1

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 1CR
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1. (a) Verify that the function y = x³ is a solution to the differential equation 2xy + 2y = 8x³ (b) Find a value for c that satisfies the initial condition y(2) = 6. 2. Use separation of variables to solve the following problems dy =1/√√√T dt y(0) = -2 (a) (b) (c) dy y² +5 dt Y 3y(y-5), y(0) = 8 dy dt = -ty, y(0) = where = 3. Consider the following very simple model of blood cholesterol levels based on the fact that cholesterol is manufactured by the body for use in the construction of cell walls and is absorbed from foods containing cholesterol: Let C(t) be the amount (in milligrams per deciliter) of cholesterol in the blood of a particular person at time t (in days). Then dC dt = k₁(N − C) + k₂E N = person's natural cholesterol level. k₁= production parameter E = daily rate at which cholesterol is eaten k₂ absorption parameter (a) Suppose N = 200, k₁ = 0.1, k₂ = 0.1, E = 400, and C(0) = 150. What will the person's cholesterol level be after 2 days on this diet? (Hint: solve the ODE, then plug in t = 2) (b) With the initial conditions as above, what will the person's cholesterol level be after 5 days on this diet? (c) What will the person's cholesterol level be after a long time on this diet? (d) High levels of cholesterol in the blood are known to be a risk factor for heart disease. Suppose that, after a long time on the high cholesterol diet described above, the person goes on a very low cholesterol diet, so E changes to E = 100. (The initial cholesterol level at the starting time of this diet is the result of part (c).) What will the person's cholesterol level be after 1 day on the new diet, after 5 days on the new diet, and after a very long time on the new diet? 1
Transcribed Image Text:1. (a) Verify that the function y = x³ is a solution to the differential equation 2xy + 2y = 8x³ (b) Find a value for c that satisfies the initial condition y(2) = 6. 2. Use separation of variables to solve the following problems dy =1/√√√T dt y(0) = -2 (a) (b) (c) dy y² +5 dt Y 3y(y-5), y(0) = 8 dy dt = -ty, y(0) = where = 3. Consider the following very simple model of blood cholesterol levels based on the fact that cholesterol is manufactured by the body for use in the construction of cell walls and is absorbed from foods containing cholesterol: Let C(t) be the amount (in milligrams per deciliter) of cholesterol in the blood of a particular person at time t (in days). Then dC dt = k₁(N − C) + k₂E N = person's natural cholesterol level. k₁= production parameter E = daily rate at which cholesterol is eaten k₂ absorption parameter (a) Suppose N = 200, k₁ = 0.1, k₂ = 0.1, E = 400, and C(0) = 150. What will the person's cholesterol level be after 2 days on this diet? (Hint: solve the ODE, then plug in t = 2) (b) With the initial conditions as above, what will the person's cholesterol level be after 5 days on this diet? (c) What will the person's cholesterol level be after a long time on this diet? (d) High levels of cholesterol in the blood are known to be a risk factor for heart disease. Suppose that, after a long time on the high cholesterol diet described above, the person goes on a very low cholesterol diet, so E changes to E = 100. (The initial cholesterol level at the starting time of this diet is the result of part (c).) What will the person's cholesterol level be after 1 day on the new diet, after 5 days on the new diet, and after a very long time on the new diet? 1
(e) Suppose the person stays on the high cholesterol diet but takes drugs that block some of the uptake of cholesterol from food, so k2 changes to k₂ = 0.075. With the cholesterol level from part (c), what will the person's cholesterol level be after 1 day, after 5 days, and after a very long time? 4. Mobile telephone service has increased rapidly in America since the mid 1990s. Today almost all residents have cellular service. The follow table shows the percentage of Americans with cellular service for years 1995 and 2012. Let t represent time in years starting with t = 0 for the year 1995. Let P represent the corresponding percentage of residents with cellular service. Year Americans with cellular service (%) 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 12.69 16.35 20.29 25.08 30.81 38.75 45.00 49.16 55.15 62.85 68.63 76.64 82.47 85.68 89.14 91.86 95.28 98.17 (a) Use exponential growth to fit a model to this data. Show your work for finding the growth rate k. (b) Use logistic growth to fit a model to this data. Show your work and justify how you estimate the carrying capacity and growth rate. Adjust the logistic model starting at your initial estimates for k and N to get a better fit to the data. (c) Use Excel to generate a graph with four lines for the data, exponential growth model, logistic growth model, and adjusted logistic model.
Transcribed Image Text:(e) Suppose the person stays on the high cholesterol diet but takes drugs that block some of the uptake of cholesterol from food, so k2 changes to k₂ = 0.075. With the cholesterol level from part (c), what will the person's cholesterol level be after 1 day, after 5 days, and after a very long time? 4. Mobile telephone service has increased rapidly in America since the mid 1990s. Today almost all residents have cellular service. The follow table shows the percentage of Americans with cellular service for years 1995 and 2012. Let t represent time in years starting with t = 0 for the year 1995. Let P represent the corresponding percentage of residents with cellular service. Year Americans with cellular service (%) 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 12.69 16.35 20.29 25.08 30.81 38.75 45.00 49.16 55.15 62.85 68.63 76.64 82.47 85.68 89.14 91.86 95.28 98.17 (a) Use exponential growth to fit a model to this data. Show your work for finding the growth rate k. (b) Use logistic growth to fit a model to this data. Show your work and justify how you estimate the carrying capacity and growth rate. Adjust the logistic model starting at your initial estimates for k and N to get a better fit to the data. (c) Use Excel to generate a graph with four lines for the data, exponential growth model, logistic growth model, and adjusted logistic model.
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