When money is spent on goods and services, those who receive the money also spend some of it. The people receiving some of the twice-spent money will spend some of that, and so on. Economists call this chain reaction the multiplier effect. In a hypothetical isolated community, the local government begins the process by spending D dollars. Suppose that each recipient of spent money spends 100c% and saves 100%c of the money that he or she receives. The values c and s are called the marginal propensity to consume and the marginal propensity to save and, of course, c + s = 1. (a). Let Sn be the total spending that has been generated after n transactions. Find an equation for Sn. (b). Show that limn→∞ Sn = KD, where k = 1/s. The number k is called the multiplier. What is the multiplier if the marginal propensity to consume is 80%? Note: The federal government uses this principle to justify deficit spending. Banks use this principle to justify lending a large percentage of the money that they receive in deposits.
> If $1000 is invested at 6% interest, compounded annually, then after years the investment is worth an = 1000 (1.06)n dollars. (a). Find the first five terms of the sequence {an}. (b). Is the sequence convergent or divergent? Explain.
> Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? (-1)" Σ R-1 n 5* (lerror|< 0.0001)
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. 1.3. 5.. an - (2n – 1) (2n)"
> List the first nine terms of the sequence {cos(nπ/3)}. Does this sequence appear to have a limit? If so, find it. If not, explain why.
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. n' cos n a, 1+ n? 2
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. 34 + 5a
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. 3 + 2n? V 8n? + n =
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. a, = /n sin(7//n)
> Determine whether the sequence converges or diverges. If it converges, find the limit. (-3)" n!
> Determine whether the sequence converges or diverges. If it converges, find the limit. а, — In(2n? + 1) — In(n? + 1)
> Determine whether the sequence converges or diverges. If it converges, find the limit. (In n)? de
> Determine whether the sequence converges or diverges. If it converges, find the limit. {0, 1, 0, 0, 1, 0, 0, 0, 1, ...}
> Find the radius of convergence and interval of convergence of the series. 00 (х — 2)" 2 A-0 n + 1
> Determine whether the sequence converges or diverges. If it converges, find the limit. (2л — 1)! in (2n + 1)!
> Determine whether the sequence converges or diverges. If it converges, find the limit. 1+3a
> Determine whether the sequence converges or diverges. If it converges, find the limit. an 1 +
> Determine whether the sequence converges or diverges. If it converges, find the limit. {n cos nT}
> Determine whether the sequence converges or diverges. If it converges, find the limit. cos'n 2"
> Determine whether the sequence converges or diverges. If it converges, find the limit. {arctan 2n}
> Determine whether the sequence converges or diverges. If it converges, find the limit. {n°e"}
> Determine whether the sequence converges or diverges. If it converges, find the limit. a, = cos(2/n)
> Determine whether the sequence converges or diverges. If it converges, find the limit. [e^ + e-^] - 1 2n e
> (a). What is a convergent sequence? Give two examples. (b). What is a divergent sequence? Give two examples.
> Find the radius of convergence and interval of convergence of the series. (2n)! Σ 00 2"
> Determine whether the sequence converges or diverges. If it converges, find the limit. (-1)*-'n n? + 1
> Determine whether the sequence converges or diverges. If it converges, find the limit. n + 1 an 9n + 1
> Determine whether the sequence converges or diverges. If it converges, find the limit. 2nt tan 1+ 8n
> Determine whether the sequence converges or diverges. If it converges, find the limit. 3a+2 a. 5"
> Determine whether the sequence converges or diverges. If it converges, find the limit. a, = el/a
> Determine whether the sequence converges or diverges. If it converges, find the limit. п+ 1
> In the figure there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the circles.
> Consider the series∑∞n-1n/ (n + 1)!. (a). Find the partial sums s1, s2, s3 and s4. Do you recognize the denominators? Use the pattern to guess a formula for sn. (b). Use mathematical induction to prove your guess. (c). Show that the given infinite series
> (a). A sequence {an} is defined recursively by the equation an = ½ (an-1 + an-2) for n > 3, where a1 and a2 can be any real numbers. Experiment with various values of a1 and a2 and use your calculator to guess the limit of the sequence. (b). Find limn→∞
> The Cantor set, named after the German mathematician Georg Cantor (1845–1918), is constructed as follows. We start with the closed interval [0, 1] and remove the open interval (1/3, 2/3). That leaves the two intervals [0, 1/3] and [2/3,
> (a). What is a sequence? (b). What does it mean to say that limn→∞ an = 8? (c). What does it mean to say that limn→∞ an = ∞?
> The Fibonacci sequence was defined in Section 8.1 by the equations Show that each of the following statements is true. fi = 1, fi= 1, fa= fa-1 + fa-2 %3D n> 3 1 (a) fa-1 fa+1 fa-1 fa fafa+1 1 1 1 = 1 -2 fa-1 fa+1 (b) E fa (c) E = 2 a-2 fa-1 fa+1
> Suppose that a series ∑ an has positive terms and its partial sums Sn satisfy the inequality Sn < 100 for all n. Explain why ∑an must be convergent.
> If ∑ an and ∑bn are both divergent, is ∑ (an + bn) necessarily divergent?
> If ∑ an is convergent and ∑bn is divergent, show that the series ∑(an + bn) is divergent. [Hint: Argue by contradiction.]
> A sequence is defined recursively by Find the first eight terms of the sequence {an}. What do you notice about the odd terms and the even terms? By considering the odd and even terms separately, show that {an} is convergent and deduce that This give
> The size of an undisturbed fish population has been modeled by the formula where Pn is the fish population after n years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is p0 >
> A right triangle ABC is given with ∠A = θ and |AC| = b. CD is drawn perpendicular to AB, DE, is drawn perpendicular to BC, EF ⊥ AB, and this process is continued indefinitely, as shown in the figure. Find t
> The figure shows two circles C and D of radius 1 that touch at P. Tis a common tangent line; C1 is the circle that touches C, D, and T; C2 is the circle that touches C, D, and C1; C3 is the circle that touches C, D, and C2. This procedure can be continue
> Graph the curves y = xn, 0 00 Σ Ei n(n + 1) = 1
> Show that the sequence defined by is increasing and an a, = 1 az+1 = 3 - an
> Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain w
> A sequence {an} is given by a1 = √2, an+1 = √2 + an. (a). By induction or otherwise, show that is increasing and bounded above by 3. Apply the Monotonic Sequence Theorem to show that lim n→∞ an exists. (b). Find limn lim n→∞ an.
> Suppose you know that {an} is a decreasing sequence and all its terms lie between the numbers 5 and 8. Explain why the sequence has a limit. What can you say about the value of the limit?
> A certain ball has the property that each time it falls from a height h onto a hard, level surface, it rebounds to a height rh, where 0 < r < h. Suppose that the ball is dropped from an initial height H of meters. (a). Assuming that the ball continues to
> Find the radius of convergence and interval of convergence of the series. (-1)ª-'x" Σ 00 3
> Find the radius of convergence and interval of convergence of the series. (-1)"x" 00 -о п + 1 A-0
> Find the radius of convergence and interval of convergence of the series. x" Σ 00
> (a). What is the radius of convergence of a power series? How do you find it? (b). What is the interval of convergence of a power series? How do you find it?
> To control an agricultural pest called the medfly (Mediterranean fruit fly), N sterilized male flies are released into the general fly population every day. If is the proportion of these sterilized flies that survive a given day, then Nsk will survive fo
> What is a power series?
> Explain how Euler’s method works.
> (a). Use Euler’s method with step size 0.2 to estimate y (0.4), where y (x) is the solution of the initial-value problem (b). Repeat part (a) with step size 0.1. (c). Find the exact solution of the differential equation and compare th
> What can you say about the solutions of the equation y' = x2 + y2 just by looking at the differential equation?
> A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells. (a). Find the number of bacteria after hours. (b). Find the number of bacteria after 4 hours. (c).
> The Brentano-Stevens Law in psychology models the way that a subject reacts to a stimulus. It states that if R represents the reaction to an amount of stimulus, then the relative rates of increase are proportional: where is a positive constant. Find R as
> The von Bertalanffy growth model is used to predict the length L (t) of a fish over a period of time. If L∞ is the largest length for a species, then the hypothesis is that the rate of growth in length is proportional to L∞ - L, the length yet to be achi
> (a). Write a differential equation that expresses the law of natural growth. What does it say in terms of relative growth rate? (b). Under what circumstances is this an appropriate model for population growth? (c). What are the solutions of this equation
> (a). Write the solution of the initial-value problem and use it to find the population when t = 20. (b). When does the population reach 1200? dP P 0.1P(1 P(0) 100 dt 2000
> A cup of hot chocolate has temperature 800Cin a room kept at 200C. After half an hour the hot-chocolate cools to 600C. (a). What is the temperature of the chocolate after another half hour? (b). When will the chocolate have cooled to 400C?
> A patient is prescribed a drug and is told to take one 100-mg pill every eight hours. After eight hours, about 5% of the drug remains in the body. (a). What quantity of the drug remains in the body after the patient takes three pills? (b). What quantity
> Let C (t) be the concentration of a drug in the bloodstream. As the body eliminates the drug, C (t) decreases at a rate that is proportional to the amount of the drug that is present at the time. Thus C (t) = -kC (t), where is a positive number called th
> Cobalt-60 has a half-life of 5.24 years. (a). Find the mass that remains from a 100-mg sample after 20 years. (b). How long would it take for the mass to decay to 1 mg?
> A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at th
> Find the orthogonal trajectories of the family of curves. y = e kx
> Find the orthogonal trajectories of the family of curves. y = kex
> Solve the initial-value problem. (1 + cos x)y' = (1 + e-y) sin x, y (0) = 0
> Solve the initial-value problem. dr/dt + 2tr = r, r (0) = 5
> Solve the differential equation. dx/dt = 1 – t + x - rx
> Find the curve y = f (x) such that f (x) > 0, f (0) = 0, f (1) = 1, and the area under the graph of f from 0 to x is proportional to the (n + 1) power of f (x).
> Find all functions f that satisfy the equation (f (x)dx) (f 1/f (x) dx) = -1
> (a). If lima→∞ an = L, what is the value of lima→∞ an+1? (b). A sequence {an} is defined by Find the first ten terms of the sequence correct to five decimal places. Does it appear
> Let f be a function with the property that f (0) = 1, f'(0) = 1, and f (a + b) = f (a) f (b) for all real numbers a and b. Show that f'(x) = f (x) for all and deduce that f (x) = ex.
> A student forgot the Product Rule for differentiation and made the mistake of thinking that (fg)' = f'g'. However, he was lucky and got the correct answer. The function f that he used was f (x) = ex2 and the domain of his problem was the interval (1/2, ∞
> (a). A direction field for the differential equation y' = y (y – 1) (y – 4) is shown. Sketch the graphs of the solutions that satisfy the given initial conditions. (b). If the initial condition is y (0) = c, for what
> Barbara weighs 60 kg and is on a diet of 1600 calories per day, of which 850 are used automatically by basal metabolism. She spends about 15 cal/kg/day times her weight doing exercise. If 1 kg of fat contains 10,000 cal and we assume that the storage of
> Suppose the model of Exercise 22 is replaced by the equations (a). According to these equations, what happens to the insect population in the absence of birds? (b). Find the equilibrium solutions and explain their significance. (c). The figure at the r
> Populations of birds and insects are modeled by the equations (a). Which of the variables, x or y, represents the bird population and which represents the insect population? Explain. (b). Find the equilibrium solutions and explain their significance. (
> The transport of a substance across a capillary wall in lung physiology has been modeled by the differential equation where h is the hormone concentration in the bloodstream, t is time, R is the maximum transport rate, V is the volume of the capillary, a
> A tank contains 100 L of pure water. Brine that contains 0.1 kg of salt per liter 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 after 6 minutes?
> One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the we
> Find all functions f such that f' is continuous and [f(x)]? = 100 + {LSMI° + [f'O1*} dt for all real x
> (a). Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nt
> What is a separable differential equation? How do you solve it?
> (a). The population of the world was 5.28 billion in 1990 and 6.07 billion in 2000. Find an exponential model for these data and use the model to predict the world population in the year 2020. (b). According to the model in part (a), when will the world
> (a). Write Lotka-Volterra equations to model populations of food fish (F) and sharks (S). (b). What do these equations say about each population in the absence of the other?
> Find all curves with the property that if the normal line is drawn at any point P on the curve, then the part of the normal line between P and the x-axis is bisected by the y-axis.
> Recall that the normal line to a curve at a point P on the curve is the line that passes through P and is perpendicular to the tangent line at P. Find the curve that passes through the point (3, 2) and has the property that if the normal line is drawn at
> Find the curve that passes through the point (3, 2) and has the property that if the tangent line is drawn at any point P on the curve, then the part of the tangent line that lies in the first quadrant is bisected at P.
> (a). Write the logistic equation. (b). Under what circumstances is this an appropriate model for population growth?
> (a). Suppose that the dog in Problem 9 runs twice as fast as the rabbit. Find a differential equation for the path of the dog. Then solve it to find the point where the dog catches the rabbit. (b). Suppose the dog runs half as fast as the rabbit. How clo
> A dog sees a rabbit running in a straight line across an open field and gives chase. In a rectangular coordinate system (as shown in the figure), assume: (i). The rabbit is at the origin and the dog is at the point (L, 0) at the instant the dog first s
> Snow began to fall during the morning of February 2 and continued steadily into the afternoon. At noon a snowplow began removing snow from a road at a constant rate. The plow traveled 6 km from noon to 1 PM but only 3 km from 1 PM to 2 PM. When did the s
> Use the partial fraction command on your CAS to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using the CAS to sum the series directly. 1 Σ A-2 n
> A peach pie is taken out of the oven at 5:00 PM. At that time, it is piping hot, 1000C. At 5:10 PM its temperature is 800C; at 5:20 PM it is 650C. What is the temperature of the room?
> A subtangent is a portion of the x-axis that lies directly beneath the segment of a tangent line from the point of contact to the x-axis. Find the curves that pass through the point (c, 1) and whose subtangents all have length c.
