Question:
(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.
Transcribed Image Text:
y' = x' – y у(0) — 1
> (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.
> Solve the differential equation. 2 yey2y' = 2x + 3√x
> (a). What is a differential equation? (b). What is the order of a differential equation? (c). What is an initial condition?
> (a). A direction field for the differential equation y' = x2 – y2 is shown. Sketch the solution of the initial-value problem Use your graph to estimate the value of y (0.3). (b). Use Euler’s method with step size 0
> (a). Sketch a direction field for the differential equation y' = x/y. Then use it to sketch the four solutions that satisfy the initial conditions y (0) = 1, y (0) = -1, y (2) = 1, and y (-2) = 1. (b). Check your work in part (a) by solving the different
> 1. All solutions of the differential equation y' = -1 – y4 are decreasing functions. 2. The function f (x) = (ln x)/x is a solution of the differential equation x2y' + xy = 1. 3. The equation y' = x + y is separable. 4. The equation y'
> If water (or other liquid) drains from a tank, we expect that the flow will be greatest at first (when the water depth is greatest) and will gradually decrease as the water level decreases. But we need a more precise mathematical description of how the f
> In this project we explore three of the many applications of calculus to baseball. The physical interactions of the game, especially the collision of ball and bat, are quite complex and their models are discussed in detail in a book by Robert Adair, The
> Suppose you throw a ball into the air. Do you think it takes longer to reach its maximum height or to fall back to earth from its maximum height? We will solve the problem in this project but, before getting started, think about that situation and make a
> 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. Зп? + Зп + 1 (n? + n)3
> Suppose ∑an and ∑bn are series with positive terms and ∑bn is known to be convergent. (a). If an > bn for all n, what can you say about ∑an? Why? (b). If an < bn for all n, what can you say about ∑an? Why?
> Show that y = 2/3 ex + e-2x is a solution of the differential equation y' = + 2y = 2ex.
> Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. y' = x + y', (0, 0)
> Match the differential equations with the solution graphs labeled I–IV. Give reasons for your choices. I y II y. III IV у. (a) y' = 1 + x² + y² (b) y' = xe-y 1 (c) y' = 1+ e (d) y' = sin(ry) cos(ry)
> Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. у — ху — х', (О, 1) y' = xy
> 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 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.
> 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?
> 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
