The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample. (a). Find the mass that remains after years. (b). How much of the sample remains after 100 years? (c). After how long will only 1 mg remain?
> Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. у — у — 2х, (1, 0)
> A function satisfies the differential equation dy/dt = y4 – 6y3 + 5y2 (a). What are the constant solutions of the equation? (b). For what values of y is increasing? (c). For what values of y is decreasing?
> A direction field for the differential equation y' = x cos Ï€y is shown. (a). Sketch the graphs of the solutions that satisfy the given initial conditions. (b). Find all the equilibrium solutions. ~~ーー|/ン //|\~~~+N ~ \|/ンノン ///|\\\ ////
> A sphere with radius 1 m has temperature 150C. It lies inside a concentric sphere with radius 2 m and temperature 250C. The temperature T (r) at a distance r from the common center of the spheres satisfies the differential equation If we let S = dT/dr,
> In contrast to the situation of Exercise 40, experiments show that the reaction H2 + Br2 → 2HBr satisfies the rate law and so, for this reaction the differential equation becomes where x = [HBr] and a and b are the initial concentr
> In an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product C: A + B → C. The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A
> Find the first 40 terms of the sequence defined by and a1 = 11. Do the same if a1 = 22. Make a conjecture about this type of sequence. Sta. if a, is an even number as+1 3a, + 1 if a, is an odd number
> In Exercise 15 in Section 7.1 we formulated a model for learning in the form of the differential equation dP/dt = k (M – P) where P (t) measures the performance of someone learning a skill after a training time t, M is the maximum level of performance, a
> In Exercise 28 in Section 7.2 we discussed a differential equation that models the temperature of a cup of coffee in a room. Solve the differential equation to find an expression for the temperature of the coffee at time t.
> Solve the initial-value problem in Exercise 27 in Section 7.2 to find an expression for the charge at time t. Find the limiting value of the charge.
> Find a function f such that f (3) = 2 and (t2 + 1) f'(t) + [f (t)]2 + 1 = 0, t ≠ 1 [Hint: Use the addition formula for on Reference Page 2.]
> An integral equation is an equation that contains an unknown function f (x) and an integral that involves y (x). Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.] y(x) = 4 + 2tvy(1) dt
> An integral equation is an equation that contains an unknown function f (x) and an integral that involves y (x). Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.] dt y(x) = 2 + o x>0 ty(t)'
> An integral equation is an equation that contains an unknown function f (x) and an integral that involves y (x). Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.] y(x) = 2 + [t – ty(1)] dt
> Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. y = x/1 + kx
> Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. y = k/x
> Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. .3 y? = kx %3D
> A fish farmer has 5000 catfish in his pond. The number of catfish increases by 8% per month and the farmer harvests 300 catfish per month. (a). Show that the catfish population Pn after n months is given recursively by (b). How many catfish are in the
> Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. x² + 2y? = k?
> In Exercise 14 in Section 7.1 we considered a 950C cup of coffee in a 200C room. Suppose it is known that the coffee cools at a rate of 10C per minute when its temperature is 700C. (a). What does the differential equation become in this case? (b). Sketch
> (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
> (a). Program your computer algebra system, using Euler’s method with step size 0.01, to calculate y (2), where y is the solution of the initial-value problem (b). Check your work by using the CAS to draw the solution curve. y' = x
> (a). Program a calculator or computer to use Euler’s method to compute y (1), where y (x) is the solution of the initial value problem (b). Verify that y = 2 + e-x3 is the exact solution of the differential equation. (c). Find the er
> (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 y' = x + y2, y (0) = 0. (b). Repeat part (a) with step size 0.1.
> (a). Solve the differential equation y' = 2x√1 – y2. (b). Solve the initial-value problem y' = 2x√1 – y2, y (0) = 0, and graph the solution. (c). Does the initial-value problem y' = 2x√1 – y2, y (0) = 2, have a solution? Explain.
> Let c be a positive number. A differential equation of the form dy/dt = ky1+c where is a positive constant, is called a doomsday equation because the exponent in the expression is larger than the exponent 1 for natural growth. (a). Determine the solution
> Consider a population P = P (t) with constant relative birth and death rates α and β, respectively, and a constant emigration rate m, where α, β, and m are positive constants. Assume that α
> (a). How long will it take an investment to double in value if the interest rate is 6% compounded continuously? (b). What is the equivalent annual interest rate?
> Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula William Gosper used this series in 1985 to compute the first 17 million digits of π. (a). Verify that the series is convergent. (b). How many correct decimal pl
> (a). If $3000 is invested at 5% interest, find the value of the investment at the end of 5 years if the interest is compounded (i) annually, (ii) semiannually, (iii) monthly, (iv) weekly, (v) daily, and (vi) continuously. (b). If A (T) is the amount of t
> (a). If $1000 is borrowed at 8% interest, find the amounts due at the end of 3 years if the interest is compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) weekly, (v) daily, (vi) hourly, and (vii) continuously. (b). Suppose $1000 is borrowed an
> The rate of change of atmospheric pressure P with respect to altitude h is proportional to P, provided that the temperature is constant. At 150C the pressure is 101.3kPa at sea level and 87.14 kPa at h = 1000m. (a). What is the pressure at an altitude of
> A freshly brewed cup of coffee has temperature 950C in a 200C room. When its temperature is 700C, it is cooling at a rate of 10C per minute. When does this occur?
> When a cold drink is taken from a refrigerator, its temperature is 50C. After 25 minutes in a 200C room its temperature has increased to 100C. (a). What is the temperature of the drink after 50 minutes? (b). When will its temperature be 150C?
> Find the solution of the differential equation that satisfies the given initial condition. xy sin x y' y +1 y(0) = 1 '
> A roast turkey is taken from an oven when its temperature has reached 1850F and is placed on a table in a room where the temperature is 750F. (a). If the temperature of the turkey is 1500F after half an hour, what is the temperature after 45 minutes? (b
> A curve passes through the point (0, 5) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve?
> (a). Show that if P satisfies the logistic equation (1), then d2P/dt2 = k2P (1 – P/M) (1 – 2P/M). (b). Deduce that a population grows fastest when it reaches half its carrying capacity.
> Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 10,000. The number of fish tripled in the first year. (a). Assuming that the size of the fish population
> (a). Show that ∑∞n=0 xn/n! converges for all x. (b). Deduce that limn→∞ xn/n! = 0 for all x.
> In Example 1(b) we showed that the rabbit and wolf populations satisfy the differential equation By solving this separable differential equation, show that where C is a constant. It is impossible to solve this equation for W, as an explicit function
> Graphs of populations of two species are shown. Use them to sketch the corresponding phase trajectory. y species 1 1200- 1000 800 600 400 species 2 200 5 10 15
> Graphs of populations of two species are shown. Use them to sketch the corresponding phase trajectory. y. species 1 200+ species 2 150+ 100+ 50 1
> Solve the differential equation. du/dr = 1 + √r/1 + √u
> Suppose a population grows according to a logistic model with initial population 1000 and carrying capacity 10,000. If the population grows to 2500 after one year, what will the population be after another three years?
> Suppose a population P (t) satisfies dP/dt = 0.4P – 0.001P2, P (0) = 50 where t is measured in years. (a). What is the carrying capacity? (b). What is P'(0)? (c). When will the population reach 50% of the carrying capacity?
> The Pacific halibut fishery has been modeled by the differential equation where y (t) is the biomass (the total mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be M = 8 Ã&#
> A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 60 cells. (a). Find the relative growth rate. (
> A population of protozoa develops with a constant relative growth rate of 0.7944 per member per day. On day zero the population consists of two members. Find the population size after six days.
> For which positive integers is the following series convergent? ∑∞n=1 (n!)2/(kn)!
> Strontium-90 has a half-life of 28 days. (a). A sample has a mass of 50 mg initially. Find a formula for the mass remaining after days. (b). Find the mass remaining after 40 days. (c). How long does it take the sample to decay to a mass of 2 mg? (d). Sk
> Experiments show that if the chemical reaction N2O5 → 2NO2 + 1/2O2 takes place at 450C, the rate of reaction of dinitrogen pentoxide is proportional to its concentration as follows: (a). Find an expression for the concentration [N2O5
> Solve the differential equation. (y + sin y) y' = x + x3
> A bacteria culture grows with constant relative growth rate. The bacteria count was 400 after 2 hours and 25,600 after 6 hours. (a). What is the relative growth rate? Express your answer as a percentage. (b). What was the intitial size of the culture? (c
> A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420. (a). Find an expression for the number of bacteria after hours. (b). Find the number of bacteria after 3 hour
> Scientists can determine the age of ancient objects by the method of radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, 14C, with a half-life of about 5730 years. Vegetation ab
> A sample of tritium-3 decayed to 94.5% of its original amount after a year. (a). What is the half-life of tritium-3? (b). How long would it take the sample to decay to 20% of its original amount?
> Solve the differential equation. du/dt = 2 + 2u + t + tu
> Solve the differential equation. dy/dx = ey sin2θ/y sec θ
> Solve the differential equation. dy/dt = tet/y √1 + y2
> Let limn→∞ n√|an| = L The Root Test says the following: (i). If L (ii). If l > 1 (or L = ∞), then ∑an is divergent. (iii). If L = 1, then the Root Test is inconc
> Solve the differential equation. (y2 + xy2) y' = 1
> Solve the differential equation. (x2 + 1) y' = xy
> Solve the differential equation by making xy' = y + xey/x the change of variable v = y/x.
> Solve the differential equation y' = x + y by making the change of variable u =x + y.
> Find the function f such that f'(x) = f (x) (1 – f (x)) and f (0) = 1/2.
> Solve the differential equation. dy/dx = xe-y
> Find an equation of the curve that passes through the point (0, 1) and whose slope at (x, y) is xy.
> Find the solution of the differential equation that satisfies the given initial condition. dL = = -1 kL² In t, L(1) dt
> Find the solution of the differential equation that satisfies the given initial condition. y' tan x 3D а + у, у(п/3) — а, 0<x<п/2
> Find the solution of the differential equation that satisfies the given initial condition. dP VPt, P(1) = 2 dt %3D
> Let limn→∞ n√|an| = L The Root Test says the following: (i). If L (ii). If l > 1 (or L = ∞), then ∑an is divergent. (iii). If L = 1, then the Root Test is inconc
> Find the solution of the differential equation that satisfies the given initial condition. x In x = y(1 + 3 + y? )y', y(1) = 1
> Find the solution of the differential equation that satisfies the given initial condition. du 2t + sec?t u(0) = -5 dt 2u
> Find the solution of the differential equation that satisfies the given initial condition. dy In x y(1) = 2 dx ху
> Find the solution of the differential equation that satisfies the given initial condition. dy dx y(0) = -3 y ||
> Solve the differential equation. dz/dx + et+z = 0
> Solve the differential equation. dy/dx = xy2
> Sketch a direction field for the differential equation. Then use it to sketch three solution curves. y' = 1/2 y
> Use the direction field labeled IV (above) to sketch the graphs of the solutions that satisfy the given initial conditions. (а) у(0) — — 1 (b) у(0) — 0 (с) у(0) — 1 %3D %3D
> Use the direction field labeled II (above) to sketch the graphs of the solutions that satisfy the given initial conditions. (а) у(0) — 1 (b) у(0) — 2 (с) у(0) — —1 %3!
> Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer. y' = sin x sin y II y4 у. IV y4 -- III -- + 2 X -2
> For which of the following series is the Ratio Test inconclusive (that is, it fails to give a definite answer)? (a) (b) (-3)ª-1 ( d) Σ -i 1 + n? - |
> Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer. y' = x + y - 1 II y4 у. IV y4 -- III -- + 2 X -2
> Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer. y' = x (2 - y) II y4 у. IV y4 -- III -- + 2 X -2
> A population is modeled by the differential equation dP/dt = 1.2P (1 – P/4200) (a). For what values of P is the population increasing? (b). For what values of P is the population decreasing? (c). What are the equilibrium solutions?
> (a). Show that every member of the family of functions y = (ln x + C)/x is a solution of the differential equation x2y' + xy = 1. (b). Illustrate part (a) by graphing several members of the family of solutions on a common screen. (c). Find a solution of
> Which of the following functions are solutions of the differential equation y" + y = sin x? (a) y = sin x (b) y = cosx (c) y = }x sin x (d) y = -x cos x
> 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
