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Math Calculus Calculus & Its Applications

Autocatalytic Reaction In an autocatalytic reaction, one substance is converted into a second substance in such a way that the second substance catalyzes its own formation. This is the process by which trypsinogen is converted into the enzyme trypsin. The reaction starts only in the presence of some trypsin, and each molecule of trypsinogen yields 1 molecule of trypsin. The rate of formation of trypsin is proportional to the product of the amounts of the two substances present. Set up the differential equation that is satisfied by y = f ( t ) , the amount (number of molecules) of trypsin present at time t . Sketch the solution. For what value of y is the reaction proceeding the fastest? [ Note: Letting M be the total amount of the two substances, the amount of trypsinogen present at time t is M − f ( t ) .]

Autocatalytic Reaction In an autocatalytic reaction, one substance is converted into a second substance in such a way that the second substance catalyzes its own formation. This is the process by which trypsinogen is converted into the enzyme trypsin. The reaction starts only in the presence of some trypsin, and each molecule of trypsinogen yields 1 molecule of trypsin. The rate of formation of trypsin is proportional to the product of the amounts of the two substances present. Set up the differential equation that is satisfied by y = f ( t ) , the amount (number of molecules) of trypsin present at time t . Sketch the solution. For what value of y is the reaction proceeding the fastest? [ Note: Letting M be the total amount of the two substances, the amount of trypsinogen present at time t is M − f ( t ) .]

Calculus & Its Applications

Calculus & Its Applications

15th Edition

ISBN: 9780137590896

Author: Larry J. Goldstein; David C. Lay; David I. Schneider; Nakhle H. Asmar; William Edward Tavernetti

Publisher: Pearson Education (US)

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Chapter 10.6, Problem 9E

Autocatalytic Reaction In an autocatalytic reaction, one substance is converted into a second substance in such a way that the second substance catalyzes its own formation. This is the process by which trypsinogen is converted into the enzyme trypsin. The reaction starts only in the presence of some trypsin, and each molecule of trypsinogen yields 1 molecule of trypsin. The rate of formation of trypsin is proportional to the product of the amounts of the two substances present. Set up the differential equation that is satisfied by y = f ( t ) , the amount (number of molecules) of trypsin present at time t. Sketch the solution. For what value of y is the reaction proceeding the fastest? [Note: Letting M be the total amount of the two substances, the amount of trypsinogen present at time t is M − f ( t ) .]

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Chapter 10.6, Problem 8E

Chapter 10.6, Problem 10E

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Chapter 10 Solutions

Calculus & Its Applications

Ch. 10.1 - Show that any function of the form y=Aet3/3, where...Ch. 10.1 - If the function f(t) is a solution of the...Ch. 10.1 - Prob. 3CYU Ch. 10.1 - Show that the function f(t)=32et212 is a solution...Ch. 10.1 - Show that the function f(t)=t212 is a solution of...Ch. 10.1 - Show that the function f(t)=5e2t satisfies...Ch. 10.1 - Show that the function f(t)=(et+1)1 satisfies...Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Is the constant function f(t)=3 a solution of the...

Ch. 10.1 - Prob. 8E Ch. 10.1 - Find a constant solution of y=t2y5t2.Ch. 10.1 - Prob. 10E Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - Prob. 15E Ch. 10.1 - Savings Account Let f(t) be the balance in a...Ch. 10.1 - Spread of News A certain piece of news is being...Ch. 10.1 - Paramecium Growth Let f(t) be the size of...Ch. 10.1 - Rate of Net Investment Let f(t) denote the amount...Ch. 10.1 - Newtons Law of Cooling A cool object is placed in...Ch. 10.1 - Carbon Dioxide Diffusion in Lungs during Breath...Ch. 10.1 - Slope Field The slope field in Fig4(a) suggests...Ch. 10.1 - Prob. 23E Ch. 10.1 - On the slope field in Fig5(a), or a copy of it,...Ch. 10.1 - Prob. 25E Ch. 10.1 - On the slope field in Fig4(a), or a copy of it,...Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Technology Exercise Consider the differential...Ch. 10.1 - Technology Exercise The function f(t)=50001+49et...Ch. 10.2 - Solve the initial-value problem y=5y,y(0)=2, by...Ch. 10.2 - Solve y=ty,y(1)=4.Ch. 10.2 - Solve the following differential equations:...Ch. 10.2 - Solve the following differential equations:...Ch. 10.2 - Solve the following differential equations:...Ch. 10.2 - Solve the following differential equations:...Ch. 10.2 - Solve the following differential equations:...Ch. 10.2 - Solve the following differential equations:...Ch. 10.2 - Solve the following differential equations:...Ch. 10.2 - Prob. 8E Ch. 10.2 - Prob. 9E Ch. 10.2 - Solve the following differential equations:...Ch. 10.2 - Solve the following differential equations:...Ch. 10.2 - Prob. 12E Ch. 10.2 - Prob. 13E Ch. 10.2 - Solve the following differential equations:...Ch. 10.2 - Prob. 15E Ch. 10.2 - Prob. 16E Ch. 10.2 - Prob. 17E Ch. 10.2 - Prob. 18E Ch. 10.2 - Solve the following differential equations with...Ch. 10.2 - Solve the following differential equations with...Ch. 10.2 - Solve the following differential equations with...Ch. 10.2 - Solve the following differential equations with...Ch. 10.2 - Prob. 23E Ch. 10.2 - Solve the following differential equations with...Ch. 10.2 - Prob. 25E Ch. 10.2 - Prob. 26E Ch. 10.2 - Solve the following differential equations with...Ch. 10.2 - Solve the following differential equations with...Ch. 10.2 - Prob. 29E Ch. 10.2 - Prob. 30E Ch. 10.2 - Prob. 31E Ch. 10.2 - Prob. 32E Ch. 10.2 - Probability of AccidentsLet t represent the total...Ch. 10.2 - Amount of Information LearnedIn certain learning...Ch. 10.2 - Prob. 35E Ch. 10.2 - Prob. 36E Ch. 10.2 - Prob. 37E Ch. 10.2 - Rate of DecompositionWhen a certain liquid...Ch. 10.2 - Prob. 39E Ch. 10.2 - Prob. 40E Ch. 10.3 - Using an integrating factor, solve y+y=1+et.Ch. 10.3 - Find an integrating factor for the differential...Ch. 10.3 - Find an integrating factor for an equation:...Ch. 10.3 - Find an integrating factor for an equation:...Ch. 10.3 - Find an integrating factor for an equation:...Ch. 10.3 - Find an integrating factor for an equation:...Ch. 10.3 - Find an integrating factor for the equation:...Ch. 10.3 - Find an integrating factor for the equation:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the equation using an integrating factor:...Ch. 10.3 - Solve the initial value problem: y+2y=1,y(0)=1.Ch. 10.3 - Solve the initial value problem:...Ch. 10.3 - Solve the initial value problem:...Ch. 10.3 - Solve the initial value problem: y=2(10y),y(0)=1.Ch. 10.3 - Solve the initial value problem: y+y=e2t,y(0)=1.Ch. 10.3 - Solve the initial value problem: tyy=1,y(1)=1,t0.Ch. 10.3 - Solve the initial value problem:...Ch. 10.3 - Solve the initial value problem:...Ch. 10.3 - Consider the initial value problem...Ch. 10.4 - Solutions can be found following the section...Ch. 10.4 - A Retirement Account refer toExample 1 a. How fast...Ch. 10.4 - Prob. 2E Ch. 10.4 - A Retirement Account A person planning for her...Ch. 10.4 - A Savings Account A person deposits 10,000 in bank...Ch. 10.4 - Prob. 5E Ch. 10.4 - Prob. 6E Ch. 10.4 - Aperson took out a loan of 100,000 from a bank...Ch. 10.4 - Car Prices in 2012 The National Automobile Dealers...Ch. 10.4 - New Home Prices in 2012 The Federal Housing...Ch. 10.4 - Answer parts (a), (b), and (c) of Exercise 9 if...Ch. 10.4 - Prob. 11E Ch. 10.4 - Find the demand function if the elasticity of...Ch. 10.4 - Temperature of a Steel Rod When a red-hot steel...Ch. 10.4 - Prob. 14E Ch. 10.4 - Determining the Time of Death A body was found in...Ch. 10.4 - Prob. 16E Ch. 10.4 - Prob. 17E Ch. 10.4 - Prob. 18E Ch. 10.4 - Prob. 19E Ch. 10.4 - Radioactive Decay Radium 226 is a radioactive...Ch. 10.4 - In Exercises 2125, solving the differential...Ch. 10.4 - Prob. 22E Ch. 10.4 - In Exercises 2125, solving the differential...Ch. 10.4 - Prob. 24E Ch. 10.4 - Prob. 25E Ch. 10.4 - Technology Exercise Therapeutic Level of a Drug A...Ch. 10.5 - Consider the differential equation y=g(y) where...Ch. 10.5 - Prob. 2CYU Ch. 10.5 - Prob. 3CYU Ch. 10.5 - Prob. 4CYU Ch. 10.5 - Exercise 1-6 review concepts that are important in...Ch. 10.5 - Prob. 2E Ch. 10.5 - Prob. 3E Ch. 10.5 - Prob. 4E Ch. 10.5 - Prob. 5E Ch. 10.5 - Prob. 6E Ch. 10.5 - One or more initial conditions are given for each...Ch. 10.5 - One or more initial conditions are given for each...Ch. 10.5 - One or more initial conditions are given for each...Ch. 10.5 - One or more initial conditions are given for each...Ch. 10.5 - Prob. 11E Ch. 10.5 - Prob. 12E Ch. 10.5 - Prob. 13E Ch. 10.5 - Prob. 14E Ch. 10.5 - Prob. 15E Ch. 10.5 - Prob. 16E Ch. 10.5 - One or more initial conditions are given for each...Ch. 10.5 - Prob. 18E Ch. 10.5 - Prob. 19E Ch. 10.5 - Prob. 20E Ch. 10.5 - Ch. 10.5 - Prob. 22E Ch. 10.5 - Prob. 23E Ch. 10.5 - Prob. 24E Ch. 10.5 - Prob. 25E Ch. 10.5 - Ch. 10.5 - Prob. 27E Ch. 10.5 - Prob. 28E Ch. 10.5 - Prob. 29E Ch. 10.5 - Prob. 30E Ch. 10.5 - Prob. 31E Ch. 10.5 - Prob. 32E Ch. 10.5 - Prob. 33E Ch. 10.5 - , where , and Ch. 10.5 - Prob. 35E Ch. 10.5 - Prob. 36E Ch. 10.5 - Growth of a plant Suppose that, once a sunflower...Ch. 10.5 - Prob. 38E Ch. 10.5 - Technology Exercises Draw the graph of, and use...Ch. 10.5 - Technology Exercises Draw the graph of...Ch. 10.6 - Refer to Example 4, involving the flow of...Ch. 10.6 - Prob. 2CYU Ch. 10.6 - In Exercises 1- 4, you are given a logistic...Ch. 10.6 - Prob. 2E Ch. 10.6 - In Exercises 1- 4, you are given a logistic...Ch. 10.6 - Prob. 4E Ch. 10.6 - Answer part (a) in Example 2, if the pond was...Ch. 10.6 - Prob. 6E Ch. 10.6 - Social Diffusion For information being spread by...Ch. 10.6 - Gravity At one point in his study of a falling...Ch. 10.6 - Autocatalytic Reaction In an autocatalytic...Ch. 10.6 - Drying A porous material dries outdoors at a rate...Ch. 10.6 - Movement of Solutes through a Cell Membrane Let c...Ch. 10.6 - Bacteria Growth An experimenter reports that a...Ch. 10.6 - Chemical Reaction Suppose that substance A is...Ch. 10.6 - War Fever L. 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Rate of evaporation The Los Angeles plans to...Ch. 10.7 - Prob. 13E Ch. 10.7 - The differential equation y=0.5(1y)(4y) has five...Ch. 10.7 - Prob. 15E Ch. 10.7 - Prob. 16E Ch. 10 - What is a differential equation?Ch. 10 - Prob. 2FCCE Ch. 10 - Prob. 3FCCE Ch. 10 - Prob. 4FCCE Ch. 10 - Prob. 5FCCE Ch. 10 - Prob. 6FCCE Ch. 10 - Prob. 7FCCE Ch. 10 - Prob. 8FCCE Ch. 10 - Prob. 9FCCE Ch. 10 - Prob. 10FCCE Ch. 10 - Prob. 11FCCE Ch. 10 - Prob. 12FCCE Ch. 10 - Describe Eulers method for approximating the...Ch. 10 - Prob. 1RE Ch. 10 - Prob. 2RE Ch. 10 - Prob. 3RE Ch. 10 - Prob. 4RE Ch. 10 - Prob. 5RE Ch. 10 - Prob. 6RE Ch. 10 - Prob. 7RE Ch. 10 - Solve the differential equation in Exercises 1-10....Ch. 10 - Prob. 9RE Ch. 10 - Prob. 10RE Ch. 10 - Prob. 11RE Ch. 10 - Let P(t) denote the price in dollars of a certain...Ch. 10 - Prob. 13RE Ch. 10 - Prob. 14RE Ch. 10 - Prob. 15RE Ch. 10 - Prob. 16RE Ch. 10 - Prob. 17RE Ch. 10 - Prob. 18RE Ch. 10 - Prob. 19RE Ch. 10 - Sketch the solutions of the differential equations...Ch. 10 - Sketch the solutions of the differential equations...Ch. 10 - Prob. 22RE Ch. 10 - Prob. 23RE Ch. 10 - Prob. 24RE Ch. 10 - Prob. 25RE Ch. 10 - Suppose that in a chemical reaction, each gram of...Ch. 10 - Prob. 27RE Ch. 10 - Prob. 28RE Ch. 10 - Let f(t) be the solution to y=2e2ty,y(0)=0. 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