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All Textbook Solutions for Calculus Volume 2

Find the general solution to the differential equation. 131. tdydt=1y2 Find the general solution to the differential equation. 132. y=exey Find the solution to the initial-value problem. 133. y=eyx,y(0)=0 Find the solution to the initial-value problem. 134. y=y2(x+1),y(0)=2 Find the solution to the initial-value problem. 135. dydx=y3xex2,y(0)=1 Find the solution to the initial-value problem. 136. dydt=y2exsin(3x),y(0)=1 Find the solution to the initial-value problem. 137. y=xsech2y,y(0)=0 Find the solution to the initial-value problem. 138. y=2xy(1+2y),x(0)=0 Find the solution to the initial-value problem. 139. dxdt=In(t)1x2,x(0)=0 Find the solution to the initial-value problem. 140. y=3x2(y2+4),y(0)=0 Find the solution to the initial-value problem. 141. y=ey5x,y(0)=In(In(5))Find the solution to the initial-value problem. 142. y'=-2xtan(y),y(0)=2 For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution? 143. [T] y’=1-2y For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution? 144. [T] y' = y2x3 For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution? 145. [T] y’=y3ex For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution? 146. [T] y' = ey For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution? 147. [T] y' = y In(x)Most drugs in the bloodstream decay according to the equation y' = cy, where y is the concentration of the drug in the bloodstream. If the half-life of a drug is 2 hours, what fraction of the initial dose remains after 6 hours?A drug is administered intravenously to a patient at a rate r mg/h and is cleared from the body at a rate proportional to the amount of drug still present in the body, d Set up and solve the differential equation, assuming there is no drug initially present in the body.[T] How often should a drug be taken if its dose is 3 mg, it is cleared at a rate c = 0.1 mg/h, and 1 mg is required to be in the bloodstream at all times?A tank contains 1 kilogram of salt dissolved in 100 liters of water. A salt solution of 0.1 kg salt/L is pumped into the tank at a rate of 2 L/min and is drained at the same rate. Solve for the salt concentration at time t. Assume the tank is well mixed.A tank containing 10 kilograms of salt dissolved in 1000 liters of water has two salt solutions pumped in. The first solution of 0.2 kg salt/L is pumped in at a rate of 20 L/min and the second solution of 0.05 kg salt/L is pumped in at a rate of 5 L/min. The tank drains at 25 L/min. Assume the tank is well mixed. Solve for the salt concentration at time t.[T] For the preceding problem, find flow much salt is in the tank 1 hour after the process begins.Torricelli’s law states that for a water tank with a hole in the bottom that has a cross-section of A and with a height of water ii above the bottom of the tank, the rate of change of volume of water flowing from the tank is proportional to the square root of the height of water, according to dVdt=A2gh, where g is the acceleration due to gravity. Note that dVdt=Adhdt. Solve the resulting initial-value problem for the height of the water, assuming a tank with a hole of radius 2 ft. The initial height of water is 100 ft.For the preceding problem, determine how long it takes the tank to drain.For the following problems, use Newton’s law of cooling. 156. The liquid base of an ice cream has an initial temperature of 200° F before it is placed in a freezer with a constant temperature of 0°F. After 1 hour, the temperature of the ice-cream base has decreased to 140°F. Formulate and solve the initial-value problem to determine the temperature of the ice cream.For the following problems, use Newton’s law of cooling. 157. [T] The liquid base of an ice cream has an initial temperature of 210°F before it is placed in a freezer with a constant temperature of 20° F. After 2 hours, the temperature of the ice-cream base has decreased to 170°F. At what time will the ice cream be ready to eat? (Assume 30°F is the optimal eating temperature.)For the following problems, use Newton’s law of cooling. 158. [T] You are organizing an ice cream social. The outside temperature is 80°F and the ice cream is at 10°F. After 10 minutes, the ice cream temperature has risen by 10°F. How much longer can you wait before the ice cream melts at 40°F?For the following problems, use Newton’s law of cooling. 159. You have a cup of coffee at temperature 70°C and the ambient temperature in the room is 20°C. Assuming a cooling rate k of 0. 125, write and solve the differential equation to describe the temperature of the coffee with respect to time.For the following problems, use Newton’s law of cooling. 160. [T] You have a cup of coffee at temperature 70°C that you put outside, where the ambient temperature is 0°C. After 5 minutes, how much colder is the coffee?For the following problems, use Newton’s law of cooling. 161. You have a cup of coffee at temperature 70°C and you immediately pour in 1 part milk to 5 parts coffee. The milk is initially at temperature 1°C. Write and solve the differential equation that governs the temperature of this coffee.For the following problems, use Newton’s law of cooling. 162. You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?For the following problems, use Newton’s law of cooling. 163. Solve the generic problem y ' = ay + b with initial condition y(0) = C.For the following problems, use Newton’s law of cooling. 164. Prove the basic continual compounded interest equation. Assuming an initial deposit of P0and an interest rate of r, set up and solve an equation for continually compounded interest.For the following problems, use Newton’s law of cooling. 165. Assume an initial nutrient amount of I kilograms in a tank with L liters. Assume a concentration of c kg/ L being pumped in at a rate of r L/min. The tank is well mixed and is drained at a rate of r L/min. Find the equation describing the amount of nutrient in the tank.For the following problems, use Newton’s law of cooling. 166. Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?For the following problems, use Newton’s law of cooling. 167. Leaves accumulate on the forest floor at a rate of 4 g/cm2/yr. These leaves decompose at a rate of 10% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor. Does this amount approach a steady value? What is that value?Student Project: Logistic Equation with a Threshold Population An improvement to the logistic model includes a threshold population. The threshold population is defined to be the minimum population that is necessary for the species to survive. We use the variable T to represent the threshold population. A differential equation that incoqoraes both the threshold population T and carrying capacity K is dPdt=rP(1PK)(1PT)(4.12) where r represents the growth rate. as before. 1. The threshold population is useful to biologists and can be utilized to determine whether a given species should be placed on the endangered list. A group of Australian researchers say they have determined the threshold population for any species to survive: 5000 adults. (Catherine Gabby. “A Magic Number,” Americon Scientist 98(1): 24. doi:l0.1511/2010.82.24. accessed April 9. 2015. http//www.anwricansoentist.org/iswes/pub/amagic-number). Therefore we use T = 5000 as the threshold population in this project. Suppose that the environmental carrying capacity In Montana for elk Is 25.000. Set up Equation 4.12 using the carrying capacity of 25,000 and threshold population of 5000. Assume an annual net growth rate of 18%.Student Project: Logistic Equation with a Threshold Population An improvement to the logistic model includes a threshold population. The threshold population is defined to be the minimum population that is necessary for the species to survive. We use the variable T to represent the threshold population. A differential equation that incoqoraes both the threshold population T and carrying capacity K is dPdt=rP(1PK)(1PT)(4.12) where r represents the growth rate. as before. 2. Draw the direction field for the differential equation from step 1, along with several solutions for different initial populations. What arc the constant solutions of the differential equation? What do these solutions correspond to In the original population model (I.e., In a biological context)?Student Project: Logistic Equation with a Threshold Population An improvement to the logistic model includes a threshold population. The threshold population is defined to be the minimum population that is necessary for the species to survive. We use the variable T to represent the threshold population. A differential equation that incoqoraes both the threshold population T and carrying capacity K is dPdt=rP(1PK)(1PT)(4.12) where r represents the growth rate. as before. 3. What is the limiting population for each initial population you chose in step 2? (Hint: use the slope field to see what happens for various Initial populations. i.e., look for the horizontal asymptotes of your solutions.)Student Project: Logistic Equation with a Threshold Population An improvement to the logistic model includes a threshold population. The threshold population is defined to be the minimum population that is necessary for the species to survive. We use the variable T to represent the threshold population. A differential equation that incoqoraes both the threshold population T and carrying capacity K is dPdt=rP(1PK)(1PT)(4.12) where r represents the growth rate. as before. 4. This equation can be solved using the method of separation of variables. However, it is very difficult to get the solution as an explicit function oft. Using an Initial population of 18.000 elk, solve the initial-value problem and express the solution as an implicit function of t, or solve the general initial-value problem, finding a solution in terms of r. K, T. and P0.For the following problems, consider the logistic equation in the form P ' = CP P2. Draw the directional field and find the stability of the equilibria. 168. C = 3 For the following problems, consider the logistic equation in the form P ' = CP P2. Draw the directional field and find the stability of the equilibria. 169. C = 0 For the following problems, consider the logistic equation in the form P ' = CP P2. Draw the directional field and find the stability of the equilibria. 170. C = 3 For the following problems, consider the logistic equation in the form P ' = CP P2. Draw the directional field and find the stability of the equilibria. 171. Solve the logistic equation for C = 10 and an initial condition of P(0) = 2.For the following problems, consider the logistic equation in the form P ' = CP P2. Draw the directional field and find the stability of the equilibria. 172. Solve the logistic equation for C = 10 and an initial condition of P(0) = 2.A population of deer inside a park has a carrying capacity of 200 and a growth rate of 2%. If the initial population is 50 deer, what is the population of deer at any given time?A population of frogs in a pond has a growth rate of 5%. If the initial population is 1000 frogs and the carrying capacity is 6000, what is the population of frogs at any given time?[T] Bacteria grow at a rate of 2O per hour in a petri dish. If there is initially one bacterium and a carrying capacity of I million cells, how long does it take to reach 500,000 cells?[T] Rabbits in a park have an initial population of 10 and grow at a rate of 4% per year. If the carrying capacity is 500, at what time does the population reach 100 rabbits?[T] Two monkeys are placed on an island. After 5 years, there are 8 monkeys, and the estimated carrying capacity is 25 monkeys. When does the population of monkeys reach 16 monkeys?[T] A butterfly sanctuary is built that can hold 2000 butterflies, and 400 butterflies are initially moved in. If after 2 months there are now 800 butterflies, when does the population get to 1500 butterflies?The following problems consider the logistic equation with an added term for depletion, either through death or emigration. 179. [T] The population of trout in a pond is given by P=0.4(1P10000)400 , where 400 trout are caught per year. Use your calculator or computer software to draw a directional field and draw a few sample solutions. What do you expect for the behavior?The following problems consider the logistic equation with an added term for depletion, either through death or emigration. 180. In the preceding problem, what are the stabilities of the equilibria 0 < P1< P2?The following problems consider the logistic equation with an added term for depletion, either through death or emigration. 181. [T] For the preceding problem, use software to generate a directional field for the value f = 600. What are the stabilities of the equilibria?The following problems consider the logistic equation with an added term for depletion, either through death or emigration. 182. [T] For the preceding problems, use software to generate a directional field for the value f = 200. What are the stabilities of the equilibria?The following problems consider the logistic equation with an added term for depletion, either through death or emigration. 183. ITI For the preceding problems, consider the case where a certain number of fish are added to the pond, or f = 200. What are the nonnegative equilibria and their stabilities?It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant k, as P=0.4P(1P10000)kP 184. [T] For the previous fishing problem, draw a directional field assuming k = 0.1. Draw some solutions that exhibit this behavior. What are the equilibria and what are their stabilities?It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant k, as P=0.4P(1P10000)kP 185. [T] Use software or a calculator to draw directional fields for k = 0.4. What are the nonnegative equilibria and their stabilities?It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant k, as P=0.4P(1P10000)kP 186. [T] Use software or a calculator to draw directional fields for k = 0.6. What are the equilibria and their stabilities?It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant k, as P=0.4P(1P10000)kP 187. Solve this equation, assuming a value of k = 0.05 and an initial condition of 2000 fish.It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant k, as P=0.4P(1P10000)kP 188. Solve this equation, assuming a value of k = 0.05 and an initial condition of 5000 fish.The following problems add in a minimal threshold value for the species to survive, T, which changes the differential equation to P(t)=rP(1PK)(1TP) 189. Draw the directional field of the threshold logistic equation, assuming K = 10. r = 0.1. T = 2. When does the population survive? When does it go extinct?The following problems add in a minimal threshold value for the species to survive, T, which changes the differential equation to P(t)=rP(1PK)(1TP) 190. For the preceding problem, solve the logistic threshold equation. assuming the initial condition P(0) = P0.The following problems add in a minimal threshold value for the species to survive, T, which changes the differential equation to P(t)=rP(1PK)(1TP) 191. Bengal tigers in a conservation park have a carrying capacity of 100 and need a minimum of 10 to survive. If they grow in population at a rate of 1% per year, with an initial population of 15 tigers, solve for the number of tigers present.The following problems add in a minimal threshold value for the species to survive, T, which changes the differential equation to P(t)=rP(1PK)(1TP) 192. A forest containing ring-tailed lemurs in Madagascar has the potential to support 5000 individuals, and the lemur population grows at a rate of 5% per year. A minimum of 500 individuals is needed for the lemurs to survive. Given an initial population of 6(X) lemurs, solve for the population of lemurs.The following problems add in a minimal threshold value for the species to survive, T, which changes the differential equation to P(t)=rP(1PK)(1TP) 193. The population of mountain lions in Northern Arizona has an estimated carrying capacity of 250 and grows at a rate of 0.25% per year and there must be 25 for the population to survive. With an initial population of 30 mountain lions, how many years will it take to get the mountain lions off the endangered species list (at least 100)?The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. 194. The Gompertz equation is given by P(t)=In(KP(t))P(t). Draw the directional fields for this equation assuming all parameters are positive, and given that K=1.The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. 195. Assume that for a population, K = 1000 and = 0.05. Draw the directional field associated with this differential equation and draw a few solutions. What is the behavior of the population?The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. 196. Solve the Gompertz equation for generic and K and P(0) = P0.The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. 197. [T] The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day 1 and assuming = 0.1 and a carrying capacity of 10 million cells, how long does it take to reach “detection” stage at 5 million cells?The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. 198. [T] It is estimated that the world human population reached 3 billion people in 1959 and 6 billion in 1999. Assuming a carrying capacity of 16 billion humans, write and solve the differential equation for logistic growth. and determine what year the population reached 7 billion.The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. 199. [T] It is estimated that the world human population reached 3 billion people in 1959 and 6 billion in 1999. Assuming a carrying capacity of 16 billion humans, write and solve the differential equation for Gompertz growth, and determine what year the population reached 7 billion. Was logistic growth or Gompertz growth more accurate, considering world population reached 7 billion on October 31, 2011?The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. 200. Show that the population grows fastest when it reaches half the carrying capacity for the logistic equation P=rP(1PK).The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. 201. When does population increase the fastest in the threshold logistic equation P(t)=rP(1PK)(1TP)?The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. 202. When does population increase the fastest for the Gompertz equation P(t)=In(KP(t))P(t)?Below is a table of the populations of whooping cranes in the wild from 1940 to 2000. The population rebounded from near extinction after conservation efforts began. The following problems consider applying population models to fit the data. Assume a carrying capacity of 10,000 cranes. Fit the data assuming years since 1940 (so your initial population at time 0 would be 22 cranes). Year (years since conservation began) Whooping Crane Population 1940(0) 22 1950(10) 31 1960(20) 36 1970(30) 57 I90(40) 91 1990(50) 159 2000(60) 256 Source: hflps:IIwww.savingcranes,o-g/imagesI soriesIsite_imagesIconservationIwtooping_crane/ pdtsThastoric__numbers.pdt 203. Find the equation and parameter r that best fit the data for the logistic equation.Below is a table of the populations of whooping cranes in the wild from 1940 to 2000. The population rebounded from near extinction after conservation efforts began. The following problems consider applying population models to fit the data. Assume a carrying capacity of 10,000 cranes. Fit the data assuming years since 1940 (so your initial population at time 0 would be 22 cranes). Year (years since conservation began) Whooping Crane Population 1940(0) 22 1950(10) 31 1960(20) 36 1970(30) 57 I90(40) 91 1990(50) 159 2000(60) 256 Source: hflps:IIwww.savingcranes,o-g/imagesI soriesIsite_imagesIconservationIwtooping_crane/ pdtsThastoric__numbers.pdt 204. Find the equation and parameters r and T that best fit the data for the threshold logistic equation.Below is a table of the populations of whooping cranes in the wild from 1940 to 2000. The population rebounded from near extinction after conservation efforts began. The following problems consider applying population models to fit the data. Assume a carrying capacity of 10,000 cranes. Fit the data assuming years since 1940 (so your initial population at time 0 would be 22 cranes). Year (years since conservation began) Whooping Crane Population 1940(0) 22 1950(10) 31 1960(20) 36 1970(30) 57 I90(40) 91 1990(50) 159 2000(60) 256 Source: hflps:IIwww.savingcranes,o-g/imagesI soriesIsite_imagesIconservationIwtooping_crane/ pdtsThastoric__numbers.pdt 205. Find the equation and parameter a that best fit the data for the Gompertz equation.Below is a table of the populations of whooping cranes in the wild from 1940 to 2000. The population rebounded from near extinction after conservation efforts began. The following problems consider applying population models to fit the data. Assume a carrying capacity of 10,000 cranes. Fit the data assuming years since 1940 (so your initial population at time 0 would be 22 cranes). Year (years since conservation began) Whooping Crane Population 1940(0) 22 1950(10) 31 1960(20) 36 1970(30) 57 I90(40) 91 1990(50) 159 2000(60) 256 Source: hflps:IIwww.savingcranes,o-g/imagesI soriesIsite_imagesIconservationIwtooping_crane/ pdtsThastoric__numbers.pdt 206. Graph all three solutions and the data on the same graph. Which model appears to be most accurate?Below is a table of the populations of whooping cranes in the wild from 1940 to 2000. The population rebounded from near extinction after conservation efforts began. The following problems consider applying population models to fit the data. Assume a carrying capacity of 10,000 cranes. Fit the data assuming years since 1940 (so your initial population at time 0 would be 22 cranes). Year (years since conservation began) Whooping Crane Population 1940(0) 22 1950(10) 31 1960(20) 36 1970(30) 57 I90(40) 91 1990(50) 159 2000(60) 256 Source: hflps:IIwww.savingcranes,o-g/imagesI soriesIsite_imagesIconservationIwtooping_crane/ pdtsThastoric__numbers.pdt 207. Using the three equations found in the previous problems, estimate the population in 2010 (year 70 after conservation). The real population measured at that time was 437. Which model is most accurate?Are the following differential equations linear? Explain your reasoning. 208. dydt=x2y+sinx Are the following differential equations linear? Explain your reasoning. 209. dydt=ty Are the following differential equations linear? Explain your reasoning. 210. dydt+y2=x Are the following differential equations linear? Explain your reasoning. 211. y=x3+ex Are the following differential equations linear? Explain your reasoning. 212. y=y+ey Write the following first-order differential equations in standard form. 213. y=x3y+sinx Write the following first-order differential equations in standard form. 214. y+3yInx=0 Write the following first-order differential equations in standard form. 215. xy=(3x+2)y+xex Write the following first-order differential equations in standard form. 216. dydt=4y+ty+tant Write the following first-order differential equations in standard form. 217. dydt=yx(x+1)What are the integrating factors for the following differential equations? 218. y=xy+3 What are the integrating factors for the following differential equations? 219. y+exy=sinx What are the integrating factors for the following differential equations? 220. y=x1n(x)y+3x What are the integrating factors for the following differential equations? 221. dydx=tanh(x)y+1 What are the integrating factors for the following differential equations? 222. dydt+3ty=ety Solve the following differential equations by using integrating factors. 223. y=3y+2 Solve the following differential equations by using integrating factors. 224. y=2yx2 Solve the following differential equations by using integrating factors. 225. xy=3y6x2 Solve the following differential equations by using integrating factors. 226. (x+2)y=3x+y Solve the following differential equations by using integrating factors. 227. y=3x+xy Solve the following differential equations by using integrating factors. 228. xy=x+y Solve the following differential equations by using integrating factors. 229. sin(x)y=y+2x Solve the following differential equations by using integrating factors. 230. y=y+ex Solve the following differential equations by using integrating factors. 231. xy=3y+x2 Solve the following differential equations by using integrating factors. 232. y+1nx=yx Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution? 233. [T] (x+2)y'=2y-1 Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution? 234. [T] y'=3et/32y Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution? 235. [T] xy+y2=sin(3t)Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution? 236. [T] xy=2cosxx3y Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution? 237 [T] (x+1)y=3y+x2+2x+1 Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution? 238. [T] sin(x)y+cos(x)y=2x Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution? 239. [T] x2+1y=y+2 Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution? 240. [T] x3y+2x2y=x+1 Solve the following initial-value problems by using integrating factors. 241. y '+y=x, y(0)= 3 Solve the following initial-value problems by using integrating factors. 242. y' = y + 2x2, y(0) = 0 Solve the following initial-value problems by using integrating factors. 243. xy' =y 3x3, y(l) = 0 Solve the following initial-value problems by using integrating factors. 244. x2y' = xy 1nx,y(1) = 1 Solve the following initial-value problems by using integrating factors. 245. (1 + x2)y' = y 1, v(0) = 0 Solve the following initial-value problems by using integrating factors. 246. xy' = y + 2x1nx, y(1) = 5 Solve the following initial-value problems by using integrating factors. 247. (2 + x)y' = y + 2 + x, y(0) = 0 Solve the following initial-value problems by using integrating factors. 248. y' = xy + 2xex, y(0) = 2 Solve the following initial-value problems by using integrating factors. 249. xy=y+2x,y(0)=1 Solve the following initial-value problems by using integrating factors. 250. y=2y+xex,y(0)=1 A falling object of mass m can reach terminal velocity when the drag force is proportional to its velocity, with proportionality constant k. Set up the differential equation and solve for the velocity given an initial velocity of 0.Using your expression from the preceding problem, what is the terminal velocity? (Hint: Examine the limiting behavior; does the velocity approach a value?)[T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall 5000 meters if the mass is 100 kilograms, the acceleration due to gravity is 9.8 m/s2and the proportionality constant is 4?A more accurate way to describe terminal velocity is that the drag force is proportional to the square of velocity, with a proportionality constant k. Set up the differential equation and solve for the velocity.Using your expression from the preceding problem, what is the terminal velocity? (Hint: Examine the limiting behavior: Does the velocity approach a value?)[T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall 5000 meters if the mass is 100 kilograms, the acceleration due to gravity is 9.8 m/s2 and the proportionality constant is 4? Does it take more or less time than your initial estimate?For the following problems, determine how parameter a affects the solution. 257. Solve the generic equation y' = ax + y. How does varying a change the behavior?For the following problems, determine how parameter a affects the solution. 258. Solve the generic equation y' = ax + y. How does varying a change the behavior?For the following problems, determine how parameter a affects the solution. 259. Solve the generic equation y' = ax + xy. How does varying a change the behavior?For the following problems, determine how parameter a affects the solution. 260. Solve the generic equation y' = x + ax. How does varying a change the behavior?For the following problems, determine how parameter a affects the solution. 261. Solve y' y = ektwith the initial condition y(0) = 0. As k approaches 1, what happens to your formula?True or False? Justify your answer with a proof or a counterexample. 262. The differential equation y' = 3x2y cos(x)y" is linear.True or False? Justify your answer with a proof or a counterexample. 263. The differential equation y' = x y is separable.True or False? Justify your answer with a proof or a counterexample. 264. You can explicitly solve all first-order differential equations by separation or by the method of integrating factors.True or False? Justify your answer with a proof or a counterexample. 265. You can determine the behavior of all first-order differential equations using directional fields or Euler’s method.For the following problems, find the general solution to the differential equations. 266. y=x2+3ex2x For the following problems, find the general solution to the differential equations. 267. y=2x+cos1x For the following problems, find the general solution to the differential equations. 268. y=y(x2+1)For the following problems, find the general solution to the differential equations. 269. y=eysinx For the following problems, find the general solution to the differential equations. 270. y=3x2y For the following problems, find the general solution to the differential equations. 271. y=y1ny For the following problems, find the solution to the initial value problem. 272. y' = 8x 1nx 3x4, y(1) = 5 For the following problems, find the solution to the initial value problem. 273. y' = 3x cos x + 2, y(0) = 4 For the following problems, find the solution to the initial value problem. 274. x y'= y(x 2), y(1) = 3 For the following problems, find the solution to the initial value problem. 275. y' = 3y2(x + cos x), y(0) = 2 For the following problems, find the solution to the initial value problem. 276. (x 1)y' = y 2, y(0) = 0 For the following problems, find the solution to the initial value problem. 277. y' = 3 x + 6x2, y(0) = 1 For the following problems, draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field. 278. y=2yy2 For the following problems, draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field. 279. y=1x+1nx-y, for x > 0 For the following problems, use Euler’s Method with n = 5 steps over the interval t = [0, 1]. Then solve the initial-value problem exactly. How close is your Euler’s Method estimate? 280. y' = 4yx, y(0) = 1 For the following problems, use Euler’s Method with n = 5 steps over the interval t = [0, 1]. Then solve the initial-value problem exactly. How close is your Euler’s Method estimate? 281. y' = 3x 2y, y(0) = 0 For the following problems, set up and solve the differential equations. 282. A car drives along a freeway, accelerating according to a = 5 sin( t ), where t represents time in minutes. Find the velocity at any time t, assuming the car starts with an initial speed of 60 mph.For the following problems, set up and solve the differential equations. 283. You throw a ball of mass 2 kilograms into the air with an upward velocity of 8 m/s. Find exactly the time the ball will remain in the air, assuming that gravity is given by g=9.8m/s2.For the following problems, set up and solve the differential equations. 284. You drop a ball with a mass of 5 kilograms out an airplane window at a height of 5000 m. How long does it take for the ball to reach the ground?For the following problems, set up and solve the differential equations. 285. You drop the same ball of mass 5 kilograms out of the same airplane window at the same height. except this time you assume a drag force proportional to the balls velocity, using a proportionality constant of 3 and the ball reaches terminal velocity. Solve for the distance fallen as a function of time. How long does it take the ball to reach the ground?For the following problems, set up and solve the differential equations. 286. A drug is administered to a patient every 24 hours and is cleared at a rate proportional to the amount of drug left in the body, with proportionality constant 0.2. If the patient needs a baseline level of 5 mg to be in the bloodstream at all times, how large should the dose be?For the following problems, set up and solve the differential equations. 287. A 1000-liter tank contains pure water and a solution of 0.2 kg salt/L is pumped into the tank at a rate of I L/ min and is drained at the same rate, Solve for total amount of salt in the tank a time g.For the following problems, set up and solve the differential equations. 288. You boil water to make tea. When you pour the water into your teapot, the temperature is 100°C. After 5 minutes in your 15°C room, the temperature of the tea is 85°C. Solve the equation to determine the temperatures of the tea at time t. How long must you wait until the tea is at a drinkable temperature (72°C)?For the following problems, set up and solve the differential equations. 289. The human population (in thousands) of Nevada in 1950 was roughly 160. If the carrying capacity is estimated at 10 million individuals, and assuming a growth rate of 2% per year. develop a logistic growth model and solve for the population in Nevada at any time (use 1950 as time 0). What population does your model predict for 2000? How close is your prediction to the true value of 1.998.257?For the following problems, set up and solve the differential equations. 290. Repeat the previous problem but use Gompertz growth model. Which is more accurate?The Fibonacci numbers are defined recursively by the sequence {Fn} where F0= 0. F1= I and for is n2 . Fn=Fn1+Fn2 Here we look at properties of the Fibonacci numbers. 1. Write out the first twenty Fibonacci numbers.The Fibonacci numbers are defined recursively by the sequence {Fn} where F0= 0. F1= I and for is n2 .Fn=Fn1+Fn2Here we look at properties of the Fibonacci numbers.The Fibonacci numbers are defined recursively by the sequence {Fn} where F0= 0. F1= I and for is n2 .Fn=Fn1+Fn2Here we look at properties of the Fibonacci numbers. 3. Use the answer in 2c. to show that limnFn+1Fn=1+52. The number =(1+5)/2 is known as the golden ratio (Figure 5.8 and Figure 5.9) Figure 5.8 The seeds in a sunflower exhibit spiral patterns curving to the left and to the right. The number of spirals in each direction is always a Fibonacci number—.always. (credit: modification of work by Esdras Calderan, Wikixnedia Commons) Figure 5.9 The propoition of the golden ratio appears in many famous examples of art and architecture. The ancient Greek temple known as the Panhenon was designed s1th these proportions, and the ratio appears again in many of the smaller details. (credit: modification of bodc by TravelingOner, Flickr)Find the first six terms of each of the following sequences, starting with n=1 . an=1+(1)nforn1 Find the first six terms of each of the following sequences, starting with n=1 . 2. an=n11forn1 Find the first six terms of each of the following sequences, starting with n=1 . 3. a1=1 and an=an1+nforn2 Find the first six terms of each of the following sequences, starting with n=1 . 4. a1=1,a2=1andan+2=an+an+1 for n1 Find the first six terms of each of the following sequences, starting with n=1 . 5. Find an explicit formula for an where an=1 and an1an=17 for n1 Find the first six terms of each of the following sequences, starting with n=1 . 6. Find a formula an for the n th term of the arithmetic sequence whose term is a1=1 such that an1an=17 for n1 Find the first six terms of each of the following sequences, starting with n=1 . 7. Find a formula an for the nth term of the arithmetic sequence whose first term is a1=3 such that an1an=4 for n1 .Find the first six terms of each of the following sequences, starting with n=1 . 8. Find a formula an for the nth term of the geometric sequence whose first term is a1=1 such that an+1an=1/10 for n1 Find the first six terms of each of the following sequences, starting with n=1 . 9. Find a formula an for the nth term of the geometric sequence whose first term is a1=3 such that an+1an=1/10 for n1 Find the first six terms of each of the following sequences, starting with n = 1. 10. Find an explicit formula for the nth term of the sequence whose first several terms (0. 3, 8, 15, 24. 35. 48. 63. 80. 99,...). (Hint: First add one to each term.)Find the first six terms of each of the following sequences, starting with n = 1. 11. Find an explicit formula for the nth term of the sequence satisfying a1=0 and an=2an1+1 for n2 .Find a formula for the general term a of each of the following sequences. 12. (1,0. —1.0, 1,0, —1.0,...) (Hint: Find where sin x takes these values)Find a formula for the general term anof each of the following sequences. 13. 1,1/3,1/5,1/7,...Find a function f(n) that identifies the nth term an of the following recursively defined sequences, as an=f(n). 14. a1=1andan+1=anforn1 Find a function f(n) that identifies the nth term an of the following recursively defined sequences, as an=f(n). 14. a1=1andan+1=anforn1 Find a function f(n) that identifies the nth term an of the following recursively defined sequences, as an=f(n). 16. a1=1andan+1=(n+1)anforn1 Find a function f(n) that identifies the nth term an of the following recursively defined sequences, as an=f(n). 17. a1=2andan+1=(n+1)an/2forn1 Find a function f(n) that identifies the nth term an of the following recursively defined sequences, as an=f(n). 18. a1=1andan+1=an/2nforn1 Plot the first N terms of each sequence. State whether the graphical evidence suggests that the sequence converges or diverges. 19. [T]a1=1,a2=2,andforn2,an=12(an1+an2);N=30 Plot the first N terms of each sequence. State whether the graphical evidence suggests that the sequence converges or diverges. 20. [T]a1=1,a2=2,a3=3andforn2,an=13(an1+an2+an3);N=30 Plot the first N terms of each sequence. State whether the graphical evidence suggests that the sequence converges or diverges. 21. [T]a1=1,a2=2,andforn2,an=an1an2;N=30 Plot the first N terms of each sequence. State whether the graphical evidence suggests that the sequence converges or diverges. 22. [T]a1=1,a2=2,a3=3,andforn4,an=an1an2an3;N=30 Suppose that limnan=1 , limnbn=1 , and 0bnan , for all n. Evaluate each of the following limits, or state that the limit does not exist, or state that there is not enough information to determine whether the limit exists. 23. limn3an4bn Suppose that limnan=1 , limnbn=1 , and 0bnan , for all n. Evaluate each of the following limits, or state that the limit does not exist, or state that there is not enough information to determine whether the limit exists. 24. limn12bn11an Suppose that limnan=1 , limnbn=1 , and 0bnan , for all n. Evaluate each of the following limits, or state that the limit does not exist, or state that there is not enough information to determine whether the limit exists. 25. limnan+bnanbn Suppose that limnan=1 , limnbn=1 , and 0bnan , for all n. Evaluate each of the following limits, or state that the limit does not exist, or state that there is not enough information to determine whether the limit exists. 26. limnanbnan+bn Find the limit of each of the following sequences, using L’ Hôpital ‘s rule when appropriate. 27. n22n Find the limit of each of the following sequences, using L’ Hôpital’s rule when appropriate. 28. (n1)n(n+1)2 Find the limit of each of the following sequences, using L’ H ôpital’s rule when appropriate. 29. nn+1 Find the limit of each of the following sequences, using L’ Hôpital’s ruile when appropriate. 30. n1/n(Hint:n1/n=e1nlnn)For each of the following sequences, whose nth terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing. 31. n/2n,n2 For each of the following sequences, whose nth terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing. 32. ln(1+1n)For each of the following sequences, whose nth terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing. 33. Sin n For each of the following sequences, whose nth terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing. 34. cos(n2)For each of the following sequences, whose nth terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing. 35. n1/n,n3 For each of the following sequences, whose nth terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing. 36. n1/n,n3 For each of the following sequences, whose nth terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing. 37. tann Determine whether the sequence defined as follows has a limit. If it does, find the limit. a1=2,a2=22,a3=222 etc.Determine whether the sequence defined as follows has a limit. If it does, find the limit. a1=3,an=2an1,n=2,3,...Use the Squeeze Theorem to find the limit of each of the following sequences. 40. n sin (1/n)Use the Squeeze Theorem to find the limit of each of the following sequences. 41. cos(1/n)11/n Use the Squeeze Theorem to find the limit of each of the following sequences. 42. an=n!nn Use the Squeeze Theorem to find the limit of each of the following sequences. 42. an=n!nn For the following sequences, plot the first 25 terms of the sequence and state whether the graphical evidence suggests that the sequence converges or diverges. 44. [T]an=sinn For the following sequences, plot the first 25 terms of the sequence and state whether the graphical evidence suggests that the sequence converges or diverges. 45. [T]an=cosn Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. 46. an=tan1(n2)Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. 47. an=(2n)1/nn1/n Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. 48. an=ln(n2)ln(2n)Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. 49. an=(12n)n Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. 50. an=ln(n+2n23)Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. 51. an=2n+3n4n Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. 52. an=(1000)nn!Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. 53. an=(n!)2(2n)!Newton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0 and successively defines a sequence zn+1=xnf(xn)f(xn) . For the given choice of f and x0. write out the formula for xn+1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places. 54. [T]f(x)=x22,x0=1 Newton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0 and successively defines a sequence zn+1=xnf(xn)f(xn) . For the given choice of f and x0. write out the formula for xn+1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places. 55. [T]f(x)=(x1)22,x0=2 New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0and successively defines a sequence zn+1=xnf(xn)f(xn) . For the given choice of f and x0. write out the formula for xn+1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places. 56. [T]f(x)=ex2,x0=1 New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0and successively defines a sequence zn+1=xnf(xn)f(xn) . For the given choice of f and x0. write out the formula for xn+1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places. 57. [T]f(x)=lnx1,x0=2 New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0and successively defines a sequence zn+1=xnf(xn)f(xn) . For the given choice of f and x0. write out the formula for xn+1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places. 58. [T] Suppose you start with one liter of vinegar and repeatedly remove 0.1 L. replace with water, mix, and repeat. a. Find a formula for the concentration after n steps. b. After how many steps does the mixture contain less than 10% vinegar?New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0and successively defines a sequence zn+1=xnf(xn)f(xn) . For the given choice of f and x0. write out the formula for xn+1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places. 59. [T] A lake initially contains 2000 fish. Suppose that in the absence of predators or other causes of removal, the fish population increases by 6% each month. However, factoring in all causes, 150 fish ate lost each month. a. Explain why the fish population after ii months is modeled by Pn= 1 .06P n— — 150 with P0= 2000. b. How many fish will be in the pond after one year?New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0and successively defines a sequence zn+1=xnf(xn)f(xn) . For the given choice of f and x0. write out the formula for xn+1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places. 60. [T] A bank account earns 5% interest compounded monthly. Suppose that S 1000 is initially deposited into the account, but that $ 1 0 is withdrawn each month. a. Show that the amount in the account after n months is An=(1.05/12)An110; A0=1000 b. How much money will be in the account after I year? c. Is the amount increasing or decreasing? d. Suppose that instead of $10. a fixed amount d dollars is withdrawn each month. Find a value of d such that the amount in the account after each month remains $1000. e. What happens if d is greater than this amount?New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0and successively defines a sequence zn+1=xnf(xn)f(xn) . For the given choice of f and x0. write out the formula for xn+1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places. 61. [T] A student takes out a college loan of $10000 at an annual percentage rate of 6%. compounded monthly. a. If the student makes payments of $100 per month, how much does the student owe after 1 2 months? b. After how many months will the loan be paid off?New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0and successively defines a sequence zn+1=xnf(xn)f(xn) . For the given choice of f and x0. write out the formula for xn+1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places.New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0and successively defines a sequence zn+1=xnf(xn)f(xn) . For the given choice of f and x0. write out the formula for xn+1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places.Euler’s Constant We have shown that the harmonic series n=11n diverges. Here we investigate the behavior of the partial Sksums, as k . In particular, we show that they behave like the natural logarithm function by showing that there exists a constant such that n=1knlnkask.‘This constant is known as Euler’s constant. Let Tk=n=1k1nlnk . Evaluate Tk for various of k.Euler’s Constant We have shown that the harmonic series n=11n diverges. Here we investigate the behavior of the partial Sksums, as k . In particular, we show that they behave like the natural logarithm function by showing that there exists a constant such that n=1knlnkask. ‘This constant is known as Euler’s constant. 2. For Tkas defined In part 1. show that the sequence [Tk]converges by using the following step. a. Show that the sequence [Tk]is monotone decreasing. (Hint: Show that In(1 + 1lk>1/(k + 1)) b. Show that the sequence [Tk]is bounded below by zero. (Hint: Express Ink as a definite integral.) C. Use the Monotone Convergence Theorem o conclude that the sequence [Tk] converges. The limit is Euler’s constant.Euler’s Constant We have shown that the harmonic series n=11n diverges. Here we investigate the behavior of the partial Sksums, as k . In particular, we show that they behave like the natural logarithm function by showing that there exists a constant such that n=1knlnkask. ‘This constant is known as Euler’s constant. 3. Now estimate how far Tkis from for a given integer k. Prove that for k 1. 0 < Tk — k by using the following Steps.Using sigma notation, write the following expressions as infinite series. 67. 1+12+13+14+...Using sigma notation, write the following expressions as infinite series. 68. 1-1+1-1+…Using sigma notation, write the following expressions as infinite series. 69. 112+1314+...Using sigma notation, write the following expressions as infinite series. 70. sin1+sin1/2+sin1/3+sin1/4+…Compute the first four partial sums S1,...,S4 for the series having nth term an starting with n = 1 as follows. 71. an=n Compute the first four partial sums S1,...,S4 for the series having nth term anstarting with n = 1 as follows. 72. an=1/n Compute the first four partial sums S1,...,S4 for the series having nth term anstarting with n = 1 as follows. 73. an=sin(n/2)Compute the first four partial sums S1,...,S4 for the series having nth term anstarting with n = 1 as follows. 74. an=(1)n In the following exercises, compute the general term a of the series with the given partial sum Sn. If the sequence of partial sums converges, find its limit S. 75. Sn=11n,n2 In the following exercises, compute the general term anof the series with the given partial sum Sn. If the sequence of partial sums converges, find its limit S. 76. Sn=n(n+1)2,n1 In the following exercises, compute the general term anof the series with the given partial sum Sn. If the sequence of partial sums converges, find its limit S. 77. Sn=n,n2 In the following exercises, compute the general term anof the series with the given partial sum Sn. If the sequence of partial sums converges, find its limit S. 78. Sn=2(n+2)/2n,n1 For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. 79. n=1nn+2 For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. 80. n=1(1( 1)n)For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. 81. n=11(n+1)(n+2) (Hint: Use a partial fraction decomposition like that for n=11n(n+1) ).For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. 82. n=112n+1 (Hint: Follow the reasoning for n=11n .)Suppose that n=1an=1 that n=1bn=1 that a1=2 , and b1=3 . Find the sum of the indicated series. 83. n=1(an+bn)Suppose that n=1an=1 that n=1bn=1 that a1=2 , and b1=3 . Find the sum of the indicated series. 84. n=1(an2bn)Suppose that n=1an=1 that n=1bn=1 that a1=2 , and b1=3 . Find the sum of the indicated series.. 85. n=2(anbn)Suppose that n=1an=1 that n=1bn=1 that a1=2 , and b1=3 . Find the sum of the indicated series. 86. n=1(3an+14bn+1)State whether the given series converges and explain why. 87. n=11n+1000 (Hint: Rewrite using a change of index.)State whether the given series converges and explain why. 88. n=11n+1080 (Hint: Rewrite using a change of index.)State whether the given series converges and explain why. 89. 1+110+1100+11000+...State whether the given series converges and explain why. 90. 1+e+e24+e33+...State whether the given series converges and explain why. 91. 1+e+2e4+3e6+4e8+...State whether the given series converges and explain why. 92. 13+29327+...For anas follows, write the sum as a geometric series of the form n=1arn. State whether the series converges and if it does, find the value of an. 93. a1=1andan/an+1=5forn1 For anas follows, write the sum as a geometric series of the form n=1arn. State whether the series converges and if it does, find the value of an. 94. a1=2andan/an+1=1/2forn1 For anas follows, write the sum as a geometric series of the form n=1arn. State whether the series converges and if it does, find the value of an. 95. a1=10andan/an+1=10forn1 For anas follows, write the sum as a geometric series of the form n=1arn. State whether the series converges and if it does, find the value of an. 96. a1=1/10andan/an+1=10forn1 Use the identity 11y=n=0yn to express the function as a geometric series in the indicated term. 97. x1+xinx Use the identity 11y=n=0yn to express the function as a geometric series in the indicated term. 98. x1x3/2inx Use the identity 11y=n=0yn to express the function as a geometric series in the indicated term. 99. 11+sin2xinsinx Use the identity 11y=n=0yn to express the function as a geometric series in the indicated term. 100. sec2xinsinx Evaluate the following telescoping series or state whether the series diverges. 101. n=121/n21/(n+1)Evaluate the following telescoping series or state whether the series diverges. 102. n=11n131( n+1)13 Evaluate the following telescoping series or state whether the series diverges. 103. n=1(nn+1)Evaluate the following telescoping series or state whether the series diverges. 104. n=1(sinnsin(n+1))Express the following series as a telescoping sum and evaluate its nth partial sum. 105. n=1ln(nn+1)Express the following series as a telescoping sum and evaluate its nth partial sum. 106. n=1ln2n+1( n 2+n)2 (Hint: Factor denominator and use partial fractions.)Express the following series as a telescoping sum and evaluate its nth partial sum. 107. n=1ln(1 + n 1)lnnln(n+1)Express the following series as a telescoping sum and evaluate its nth partial sum. 108. n=1(n+2)n(n+1)2n+1 (Hint: Look at 1/(n2n).)A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms. 109. Let an=f(n)2f(n+1)+f(n2) in which f(n)0 as n . Find n=1an A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms. 110. an=f(n)f(n+1)f(n+2)+f(n+3)A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms. 111. Suppose that an=c0f(n)+f(n+1)+c2f(n+2)+c3f(n+3)+c4f(n+4) . Where f(n)0 as n . Find a condition on the coefficients c0,...,c4 that make this a general telescoping series.A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms. 112. Evaluate n=11n(n+1)(n+2) (Hint: 1n(n+1)(n+2)=12n1n+1+12(n+2))A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms. 113. Evaluate n=22n3n.A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms. 114. Find a formula for n=11n(n+N) Where N is a positive integer.A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms. 115. [T] Define a sequence tk=n=1k1(1/k)lnk . Use the graph of 1/x to verify that tkis increasing. Plot tk is increasing. Plot tk for k=1…100 and state where it appears that the sequence converges.[T] Suppose that N equal uniform rectangular blocks are stacked one on top of the other, allowing for some overhang. Archimedes’ law of the lever implies that the stack of N blocks is stable as long as the center of mass of the top (N1) blocks lies at the edge of the bottom block. Let x denote the position of the edge of the bottom block, and think of its position as relative to the center of the next-to—bottom block. This implies that (N1)x=(12x) or x=1/(2N) . Use this expression to compute the maximum overhang (the position of the edge of the top block over the edge of the bottom block.) See the following figure.Each of the following infinite series converges to the given multiple of or 1/ . In each case, find the minimum value of N such that the Nth partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and give the desired approximate value. Up to place, = 3.14 1592653589793.... 117. [T]=3+n=1n2nn!2(2n),error0.0001 Each of the following infinite series converges to the given multiple of or 1/ . In each case, find the minimum value of N such that the Nth partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and give the desired approximate value. Up to place, = 3.14 1592653589793.... 118. [T]2=k=0k=0k!(2k+1)!= 2 kk!2( 2k+1),error10-4 Each of the following infinite series converges to the given multiple of or 1/ . In each case, find the minimum value of N such that the Nth partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and give the desired approximate value. Up to place, = 3.14 1592653589793.... 119. [T]98012=49801k=0(4k)!(1303+26390k)(3k)!(k!)6403203k+3/2,error1015 Each of the following infinite series converges to the given multiple of or 1/ . In each case, find the minimum value of N such that the Nth partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and give the desired approximate value. Up to place, = 3.14 1592653589793.... 120. 112=k=0( 1)k(6k)!(13591409+545140134k)(3k)!( k!)36403203k+3/2error10-5 [T] A fair coin is one that has probability 1/2 of coming up heads when flipped. a. That is the probability that a fair coin will come up tails ii times in a row? b. Find the probability that a coin comes up heads for the first time after an even number of coin flips.[TI Find the probability that a fair coin is flipped a multiple of three times before coming up heads.[T] Find the probability that a fair coin will come up heads for the second time after an even number of flips.[T] Find a series that expresses the probability that a fair coin will come up heads for the second time on a multiple of three flips.[T] The expected number of times that a fair coin will come up heads is defined as the sum over n = 1, 2,... of ii times the probability that the coin will come up heads exactly n times in a row, or n/2n+1 • Compute the expected number of consecutive times that a fair coin will come up heads.[T] A person deposits $10 at the beginning of each quarter into a bank account that earns 4% annual interest compounded quarterly (four times a year). a. Show that the interest accumulated after ii quarters is$10(1.01n+110.01n) b. Find the first eight terms of the sequence. c. How much interest has accumulated after 2 years?[T] Suppose that the amount of a drug in a patient’s system diminishes by a multiplicative factor r < 1 each hour. Suppose that a new dose is administered every N hours. Find an expression that gives the amount A(n) in the patient’s system after n hours for each n in terms of the dosage d and the ratio r. (Hint: Write n = mN + k. where 0kN. and sum over values from the different doses administered.)[T] A certain drug is effective for an average patient only if there is at least I mg per kg in the patient’s system, while it is safe only if there is at most 2 mg per kg in an average patient’s system. Suppose that the amount in a patient’s system diminishes by a multiplicative factor of 0.9 each hour after a dose is administered. Find the maximum interval N of hours between doses, and corresponding dose range d (in mg/kg) for this N that will enable use of the drug to be both safe and effective in the long term.Suppose that an0 is a sequence of numbers. Explain why the sequence of partial sums of anis increasing.[T] Suppose that an is a sequence of positive numbers and the sequence Sn of partial sums of anis bounded above. Explain why n=1an converges. Does the conclusion remain true if we remove the hypothesis an0 ?[T] Suppose that a1=s1=1 and that, for given numbers S>1 and 0 < k < I. one defines an+1=k(SSn)and Sn+1=an+1+Sn. Does Snconverge? If so, to what? (Hint: First argue that SnSfor all n and Snis increasing.)[T] A version of von Bertalanffy growth can be used to estimate the age of an individual in a homogeneous species from its length if the annual increase in year n + 1 satisfies , an+1=k(SSn). with Sn as the length at year n. S as a limiting length, and k as a relative growth constant. If S1=3 . S1=9 . and k = 1/2. numerically estimate the smallest value of ii such that Sn8 . Note that Sn+1=Sn+1 • Find the corresponding ii when k= 1/4.[T] Suppose that n=1an is a convergent series of positive terms. Explain why limNn=N+1an=0 .

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