(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 differential equation y' = -y2 that is not a member of the family in part (b)? (d). Find a solution of the initial-value problem y' = -y2, y (0) = 0.5
> Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer. II y4 у. IV y4 -- III -- + 2 X -2 y' = 2 – y
> (a). For what values of does the function y = cos kt satisfy the differential equation 4y" = -25? (b). For those values of k, verify that every member of the family of functions y = A sin kt + B cos kt is also a solution.
> The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of C farads (F), and a resistor with a resistance of R ohms (Ω). The voltage drop across the capacitor is Q/C, where Q is the charge (in coul
> Use Euler’s method with step size 0.1 to estimate y (0.5), where y (x) is the solution of the initial-value problem y' = y + xy, y (0) = 1.
> Use Euler’s method with step size 0.2 to estimate y (1), where y (x) is the solution of the initial-value problem y' = xy – x2, y (0) = 1.
> Let p and q be real numbers with p (а) (р, q) (с) [р. 9) (b) (р, q] (d) [р, q]
> Use Euler’s method with step size 0.5 to compute the approximate y-values y1, y2, y3 and y4 of the solution of the initial-value problem y' = y – 2x, y (1) = 0.
> A direction field for a differential equation is shown. Draw, with a ruler, the graphs of the Euler approximations to the solution curve that passes through the origin. Use step sizes h = 1 and h = 0.5. Will the Euler estimates be under - estimates or ov
> A direction field for the differential equation y' = tan (1/2 Ï€y) is shown. (a). Sketch the graphs of the solutions that satisfy the given initial conditions. (b). Find all the equilibrium solutions. -1 ノーー \| らす///ーGー\\|*///ーロ \||| |I
> (a). Use Euler’s method with each of the following step sizes to estimate the value of y (0.4), where is the solution of the initial-value problem y' = y, y (0) = 1. (b). We know that the exact solution of the initial-value problem in
> (a). For what values of does the function y = erx satisfy the differential equation 2y" + y' – y = 0? (b). If r1 and r2 are the values of r that you found in part (a), show that every member of the family of functions y = aer1x + ber2x is also a solution
> Use a computer algebra system to draw a direction field for the differential equation y' = y3 – 4y. Get a printout and sketch on its solutions that satisfy the initial condition y (0) = c for various values of c. For what values of c does limx→∞ y (t) ex
> Verify that y = -t cos t – t is a solution of the initial-value problem dy d =y +t°sin t y(7) = 0
> According to Newton’s Law of Universal Gravitation, the gravitational force on an object of mass m that has been projected vertically upward from the earth’s surface is Where x = x (t) is the objectâ€
> Let A (t) be the area of a tissue culture at time t and let M be the final area of the tissue when growth is complete. Most cell divisions occur on the periphery of the tissue and the number of cells on the periphery is proportional to √A (t). So, a reas
> Homeostasis refers to a state in which the nutrient content of a consumer is independent of the nutrient content of its food. In the absence of homeostasis, a model proposed by Sterner and Elser is given by where and represent the nutrient content of t
> Let x = 0.99999… (a). Do you think that x < 1 or x = 4? (b). Sum a geometric series to find the value of x. (c). How many decimal representations does the number 1 have? (d). Which numbers have more than one decimal representation?
> Find the radius of convergence and interval of convergence of the series. 10"x" Σ 00 A-1
> Allometric growth in biology refers to relationships between sizes of parts of an organism (skull length and body length, for instance). If L1 (t) and L2 (t) are the sizes of two organs in an organism of age t, then L1 and L2 satisfy an allometric law if
> An object of mass m is moving horizontally through a medium which resists the motion with a force that is a function of the velocity; that is, Where v = v (t) and s = s (t) represents the velocity and position of the object at time t, respectively. Fo
> When a raindrop falls, it increases in size and so its mass at time t is a function of t, namely m (t). The rate of growth of the mass is km (t) for some positive constant k. When we apply New ton’s Law of Motion to the raindrop, we get (mv)' = gm, where
> A tank contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 L/min. The solution is kept thorough
> A vat with 500 gallons of beer contains 4% alcohol (by volume). Beer with 6% alcohol is pumped into the vat at a rate of 5 gal/min and the mixture is pumped out at the same rate. What is the percentage of alcohol after an hour?
> The air in a room with volume 180m3 contains 0.15% carbon dioxide initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of 2 m3/min and the mixed air flows out at the same rate. Find the percentage of carbon dioxide in the r
> A tank contains 1000 L of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank (a) after minutes and (b) after
> A certain small country has $10 billion in paper currency in circulation, and each day $50 million comes into the country’s banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency
> A glucose solution is administered intravenously into the bloodstream at a constant rate r. As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate that is proportional to the concentration at that time.
> For a fixed value of M (say M = 10), the family of logistic functions given by Equation 4 depends on the initial value P0 and the proportionality constant k. Graph several members of this family. How does the graph change when P0 varies? How does it chan
> Suppose that the radius of convergence of the power series ∑cnxn is R. What is the radius of convergence of the power series ∑cnx2n?
> Explain why the functions with the given graphs can’t be solutions of the differential equation dy = e'(y – 1)? dt (а) у (b) у. 1+ 1+ 1
> Sketch a direction field for the differential equation. Then use it to sketch three solution curves. y' = x – y + 1
> 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
> 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)
> We know that lim n→∞ (0.8)n = 0 [from (7) with r = 0.8]. Use logarithms to determine how large has to be so that (0.8)n < 0.000001.
> 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
