Question: We know that lim n→∞ (0.8)

> The table gives the number of yeast cells in a new labora tory culture. (a). Plot the data and use the plot to estimate the carrying capacity for the yeast population. (b). Use the data to estimate the initial relative growth rate. (c). Find both an ex

> The table gives estimates of the world population, in millions, from 1750 to 2000. (a). Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950. Compare with the actual figures. (b). Use t

> Suppose that a population grows according to a logistic model with carrying capacity 6000 and k = 0.0015 per year. (a). Write the logistic differential equation for these data. (b). Draw a direction field (either by hand or with a com puter algebra syste

> Suppose that a population develops according to the logistic equation dP/dt = 0.05P – 0.0005P2 where t is measured in weeks. (a). What is the carrying capacity? What is the value of k? (b). A direction field for this equation is shown.

> Suppose you have just poured a cup of freshly brewed coffee with temperature 950C in a room where the temperature is 200C. (a). When do you think the coffee cools most quickly? What happens to the rate of cooling as time goes by? Explain. (b). Newton’s L

> Make a rough sketch of a direction field for the autonomous differential equation y' = f (y), where the graph of f is as shown. How does the limiting behavior of solutions depend on the value of y (0)? fy)A -2 -1 0 y

> Psychologists interested in learning theory study learning curves. A learning curve is the graph of a function P (t), the performance of someone learning a skill as a function of the training time t. The derivative dP/dt represents the rate at which perf

> Use a computer algebra system to draw a direction field for the given differential equation. Get a printout and sketch on it the solution curve that passes through (0, 1). Then use the CAS to draw the solution curve and compare it with your sketch. y' =

> Suppose the series ∑cnxn has radius of convergence 2 and the series ∑dnxn has radius of convergence 3. What is the radius of convergence of the series ∑(cn + dn)?

> Use a computer algebra system to draw a direction field for the given differential equation. Get a printout and sketch on it the solution curve that passes through (0, 1). Then use the CAS to draw the solution curve and compare it with your sketch. y' =

> In a murder investigation, the temperature of the corpse was 32.50C at 1:30 PM and an hour later. Normal body temperature is 37.00C and the temperature of the surroundings was 20.00C. When did the murder take place?

> Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. у — у+ ху, (0, 1)

> (a). Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation. (b). Solve the differential equation. (c). Use the CAS to draw

> The function with the given graph is a solution of one of the following differential equations. Decide which is the correct equation and justify your answer. A. y' = 1+ xy B. y' = -2xy C. y' = 1 - 2xy

> Solve the equation y' = x √x2 + 1/ (y ey) and graph several members of the family of solutions (if your CAS does implicit plots). How does the solution curve change as the constant C varies?

> Solve the initial-value problem y' = (sin x)/ sin y, y (0) = π/2, and graph the solution (if your CAS does implicit plots).

> Solve the equation e-yy' + cos x = 0 and graph several members of the family of solutions. How does the solution curve change as the constant C varies?

> (a). What can you say about the graph of a solution of the equation y = xy3 when is close to 0? What if is large? (b). Verify that all members of the family y = (c – x2)-1/2 are solutions of the differential equation y' = xy3. (c). Graph several members

> (a). What can you say about a solution of the equation y' = -y2 just by looking at the differential equation? (b). Verify that all members of the family y = 1/ (x + C) are solutions of the equation in part (a). (c). Can you think of a solution of the dif

> If f (x) =∑∞n=0 cnxn, where cn+4 = cn for all n > 0, find the interval of convergence of the series and a formula for f (x).

> The table gives the population of India, in millions, for the second half of the 20th century. (a). Use the exponential model and the census figures for 1951 and 1961 to predict the population in 2001. Compare with the actual figure. (b). Use the expon

> Suppose we alter the differential equation in Exercise 19 as follows: Exercise 19: In a seasonal-growth model, a periodic function of time is introduced to account for seasonal variations in the rate of growth. Such variations could, for example, be c

> In a seasonal-growth model, a periodic function of time is introduced to account for seasonal variations in the rate of growth. Such variations could, for example, be caused by seasonal changes in the availability of food. (a). Find the solution of the s

> Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation where c is a constant and M is the carrying capacity. (a). Solve this differential equation. (b). Compute l

> There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equatio

> Consider the differential equation as a model for a fish population, where is measured in weeks and c is a constant. (a). Use a CAS to draw direction fields for various values of c. (b). From your direction fields in part (a), determine the values of f

> Let’s modify the logistic differential equation of Example 1 as follows: (a). Suppose P (t) represents a fish population at time t, where is measured in weeks. Explain the meaning of the final term in the equation (-15). (b). Draw a d

> The table gives the midyear population of Spain, in thousands, from 1955 to 2000. Use a graphing calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy

> The table gives the midyear population of Japan, in thousands, from 1960 to 2005. Use a graphing calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy

> In Exercise 10 we modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows: (a). In the absence of ladybugs, what does the model predict about the aphids? (b). Find the equilibrium solutions.

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 00 2 5

> In Example 1 we used Lotka-Volterra equations to model populations of rabbits and wolves. Let’s modify those equations as follows: (a). According to these equations, what happens to the rabbit population in the absence of wolves? (b)

> Populations of aphids and ladybugs are modeled by the equations (a). Find the equilibrium solutions and explain their significance. (b). Find an expression for dL/dA. (c). The direction field for the differential equation in part (b) is shown. Use it t

> One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction of the population who have heard the rumor and the fraction who have not heard the rumor. (a). Write a differential equation that is satisfied b

> (a). Make a guess as to the carrying capacity for the US population. Use it and the fact that the population was 250 million in 1990 to formulate a logistic model for the US population. (b). Determine the value of in your model by using the fact that the

> The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. Let’s assume that the carrying capacity for world population is 100 billion.

> A phase trajectory is shown for populations of rabbits R and foxes (F). (a). Describe how each population changes as time goes by. (b). Use your description to make a rough sketch of the graphs of R and F as functions of time. F 160- + 1=0 120+ 80

> A phase trajectory is shown for populations of rabbits R and foxes (F). (a). Describe how each population changes as time goes by. (b). Use your description to make a rough sketch of the graphs of R and F as functions of time. FA 300 200 100+ t=0 4

> Flies, frogs, and crocodiles coexist in an environment. To survive, frogs need to eat flies and crocodiles need to eat frogs. In the absence of frogs, the fly population will grow exponentially and the crocodile population will decay exponentially. In th

> The system of differential equations is a model for the populations of two species. (a). Does the model describe cooperation, or competition, or a predator-prey relationship? (b). Find the equilibrium solutions and explain their significance. dx 0.

> Each system of differential equations is a model for two species that either compete for the same resources or cooperate for mutual benefit (flowering plants and insect pollinators, for instance). Decide whether each system describes competition or coope

> The function A defined by is called the Airy function after the English mathematician and astronomer Sir George Airy (1801–1892). (a). Find the domain of the Airy function. (b). Graph the first several partial sums on a common screen.

> For each predator-prey system, determine which of the variables, x or y, represents the prey population and which represents the predator population. Is the growth of the prey restricted just by the predators or by other factors as well? Do the predators

> A sequence that arises in ecology as a model for population growth is defined by the logistic difference equation Where pn measures the size of the population of the nth generation of a single species. To keep the numbers manageable, pn is a fraction o

> Determine whether the sequence converges or diverges. If it converges, find the limit. а, — 1 — (0.2)"

> Determine whether the sequence converges or diverges. If it converges, find the limit. n' + 1

> Determine whether the sequence converges or diverges. If it converges, find the limit. 3 + 5n2 a, n + n?

> Suppose that ∑∞n-1 an (an ≠ 0) is known to be a convergent series. Prove ∑∞n-1 1/an that is a divergent series.

> What is wrong with the following calculation? (Guido Ubaldus thought that this proved the existence of God because “something has been created out of nothing.”) 0 = 0 + 0 + 0 + · · . = (1 – 1) + (1 – 1) + (1 – 1)

> In Example 7 we showed that the harmonic series is divergent. Here we outline another method, making use of the fact that ex > 1 + x for any x > 0. (See Exercise 4.3.62.) If sn is the nth partial sum of the harmonic series, show that esn > n + 1. Why doe

> Find the value of c such that ∑∞n=0 enc = 10

> The function j1 defined by is called the Bessel function of order 1. (a). Find its domain. (b). Graph the first several partial sums on a common screen. (c). If your CAS has built-in Bessel functions, graph j1 on the same screen as the partial sums in

> Find the value of c if ∑∞n-2 (1 + c)-n = 2

> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? a, = n +

> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? а, — п(-1)"

> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 2n – 3 an Зп + 4

> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 1 An 2n + 3

> If the nth partial sum of a series ∑∞n-1 an is sn = 3 – n2-n, find an and ∑∞n-1 an.

> If the nth partial sum of a series ∑∞n-1 an is sn = n - / n + 1 find an and ∑∞n-1 an.

> Find all values of c for which the following series converges. 1 Σ ガー」\n n +1,

> Show that if an > 0 and limn→∞ nan ≠ 0 then ∑an is divergent.

> Find all positive values of b for which the series ∑∞n=1 b ln n converges.

> Graph the first several partial sums sn(x) of the series∑∞n=0xn, together with the sum function f (x)= 1/ (1 – x), on a common screen. On what interval do these partial sums appear to be converging to f (x)?

> If ∑ an is a convergent series with positive terms, is it true that ∑ sin (an) is also convergent?

> Show that if we want to approximate the sum of the series ∑∞n=1 n-1.001 so that the error is less than 5 in the ninth decimal place, then we need to add more than 1011.301 terms!

> A series ∑an is defined by the equations Determine whether ∑an converges or diverges 2 + cos n - an an+1 aj = 1 Vn

> Is it possible to find a power series whose interval of convergence is [0, ∞]? Explain.

> A function f is defined by f (x) = 1 + 2x + x2 + 2x3 + x4 + … that is, its coefficients are c2n = 1 and c2n+1 = 2 for all n > 0. Find the interval of convergence of the series and find an explicit formula for f (x).

> Determine whether the series is absolutely convergent. ∑∞n=1 sin4n/4n

> Determine whether the series is absolutely convergent. ∑∞n=1 102/(n + 1)42n+1

> Determine whether the series is absolutely convergent. ∑∞n=1 (-1)n-12nn4

> If k is a positive integer, find the radius of convergence of the series (n!)* 00 A-0 (kn)!

> Determine whether the series is absolutely convergent. ∑∞n=1 n2/2n

> Determine whether the series is convergent or divergent. ∑∞n=1 1/√n3 + 1

> Determine whether the series is absolutely convergent. ∑∞k=1 k(2/3)k

> Find the radius of convergence and interval of convergence of the series. 2)" (3x Σ n 34 00

> Find the radius of convergence and interval of convergence of the series. 00 (x + 1)ª

> Find the radius of convergence and interval of convergence of the series. 34(х + 4) Σ 00 in

> Find the radius of convergence and interval of convergence of the series. (x – 3)" E (-1)*. 2η +1 A-0

> Find the radius of convergence and interval of convergence of the series. (-2)"x" Σ In 00 1. オー

> Find the radius of convergence and interval of convergence of the series. 00 2a Σ (-1). (2n)! n-0

> Find the radius of convergence and interval of convergence of the series. n?x" 00 E (-1)*- 2" n-1

> Test the series for convergence or divergence. ∑∞n=1 (-1)n, n /√3 + 2

> Find the radius of convergence and interval of convergence of the series. 00 n!

> Suppose that ∑∞n=0 cnxn converges when x = -4 and diverges when x= 6. What can be said about the convergence or divergence of the following series? ( a ) Σ ca A-0 (b) E ca n-0 ( c) Σ C-3)" ( d) Σ (-1)'c, 9" A-0 A-

> Find the radius of convergence and interval of convergence of the series. 00 E yn x"

> Test the series for convergence or divergence. ∑∞n=1 (-1)n-1 / 2n + 1

> Test the series for convergence or divergence. -3/4 + 5/5 – 7/6 + 9/7 – 11/8 + …

> Test the series for convergence or divergence. 4/7 – 4/8 + 4/9 – 4/10 + 4/11 - ….

> Suppose f is a continuous positive decreasing function for x > 1 and an = f (n). By drawing a picture, rank the following three quantities in increasing order: 5 6 °f(x) dx Σ 2 ai i-l i-2

> Draw a picture to show that What can you conclude about the series? Σ 1.3 -2 n 1.3

> Test the series for convergence or divergence. ∑∞n=1 (-1)n+1 n/n2 + 9

> Test the series for convergence or divergence. ∑∞n=1 (-1)n-1 / 2n + 1

> Test the series for convergence or divergence. ∑∞n=1 (-1)n-1 / ln (n + 4)

> The terms of a series are defined recursively by the equations Determine whether ∑an converges or diverges. 5n + 1 aj = 2 An+1 an 4n + 3

> If ∑∞n=0 cn4n is convergent, does it follow that the following series are convergent? (a) 2 ca(-2)ª (b) Σ c.(-4)" A-0 A-0

> What can you say about the series ∑an in each of the following cases? An+1 an+1 (а) lim an = 8 (b) lim = 0.8 an+1 (c) lim

> Determine whether the series is absolutely convergent. 2/5 + 2 ∙ 6/5 ∙ 8 + 2 ∙ 6 ∙ 10/5 ∙ 8 ∙ 11 + 2 ∙ 6 ∙ 10 ∙ 14/5 ∙ 8 ∙ 11 ∙ 14 + ∙∙∙

> Determine whether the series is absolutely convergent. 1 – 1 ∙3/3! + 1∙3∙5/5! - 1∙2∙5∙7/7!+ …+ (-1)n-1 1∙3∙5∙∙∙∙(2n – 1)/(2n – 1)!+ ∙∙∙

> Determine whether the series is absolutely convergent. ∑∞n=1 (-2)n n!/(2n)!