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calculus

Calculus Early Transcendentals 9th Edition James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin - Solutions

For a fixed value of k (say k = 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 change when varies?
Let€™s modify the logistic differential equation of Example 1 as follows:(a) Suppose P(t) represents a fish population at time , where t is measured in weeks. Explain the meaning of the term €“ 15.(b) Draw a direction field for this differential equation.(c) What are the
Consider the differential equation as a model for a fish population, where is measured in weeks and 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 c for which there is at least one equilibrium
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 equation by introducing the factor (1 €“
Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation dP/dt = c In (K/P) P where is a constant and K is the carrying capacity. (a) Solve this differential equation. (b) Compute lim t→∞
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 seasonal-growth model where k, r, and Φ are
Suppose we alter the differential equation in Exercise 15 as follows:(a) Solve this differential equation with the help of a table of integrals or a CAS.(b) Graph the solution for several values of k, r, and Φ. How do the values of k, r, and Φ affect the solution? What can you say about lim
Graphs of logistic functions (Figures 2 and 4) look suspiciously similar to the graph of the hyperbolic tangent function (Figure 3 in Section 3.9). Explain the similarity by showing that the logistic function given by Equation 4 can be written as where c = (In A)/ k.Thus, the logistic function is
Solve the differential equation and use a graphing calculator or computer to graph several members of the family of solutions. How does the solution curve change as C varies?
A Bernoulli differential equation (named after James Bernoulli) is of the form dy/dx + P(x) y = Q(x)yn Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, show that the substitution u = y1-n transforms the Bernoulli equation into the linear equation
Use the method of Exercise 23 to solve the differential equation.
In the circuit shown in Figure 4, a battery supplies a constant voltage of 40 V, the inductance is 2 H, the resistance is 10Ω, and l(0) = 0. (a) Find l(t). (b) Find the current after 0.1s.
In the circuit shown in Figure 4, a generator supplies a voltage of E(t) = 40 sin 60t volts, the inductance is H, the resistance is 20Ω, and l(0) = 1 A. (a) Find l(t). (b) Find the current after 0.1 s. (c) Use a graphing device to draw the graph of the current function.
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 coulombs),so in this case Kirchhoff€™s Law gives Rl +
In the circuit of Exercise 29, R = 2Ω, C = 0.01 F, Q (0) = 0, and E(t) = 10 sin 60t. Find the charge and the current at time t.
Let P (t) be the performance level of someone learning a skill as a function of the training time t. The graph of P is called a learning curve. In Exercise 13 in Section 9.1 we proposed the differential equation dP/dt = k [M – P (t)] as a reasonable model for learning, where is a positive
Two new workers were hired for an assembly line. Jim processed 25 units during the first hour and 45 units during the second hour. Mark processed 35 units during the first hour and 50 units the second hour. Using the model of Exercise 31 and assuming that P (0) = 0, estimate the maximum number of
In Section 9.3 we looked at mixing problems in which the volume of fluid remained constant and saw that such problems give rise to separable equations. (See Example 6 in that section.) If the rates of flow into and out of the system are different, then the volume is not constant and the resulting
A tank with a capacity of 400 L is full of a mixture of water and chlorine with a concentration of 0.05 g of chlorine per liter. In order to reduce the concentration of chlorine, fresh water is pumped into the tank at a rate of 4 L/s. The mixture is kept stirred and is pumped out at a rate of
An object with mass m is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If s (t) is the distance dropped after seconds, then the speed is v = s’ (t) and the acceleration is a = v’ (t). If g is the acceleration due to gravity, then the
If we ignore air resistance, we can conclude that heavier objects fall no faster than lighter objects. But if we take air resistance into account, our conclusion changes, use the expression for the velocity of a falling object in Exercise 35(a) to find dv/dm and show that heavier objects do fall
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 feed only on the prey or do they have
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 cooperation and explain why it is a reasonable
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.
Graphs of populations of two species are shown. Use them to sketch the corresponding phase trajectory.
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 is a constantIt is impossible to solve this equation for W as an explicit function of R (or vice versa). If you have a computer algebra
Populations of aphids and ladybugs are modeled by the equations dA/dt = 2A €“ 0.01AL dL/dt = €“ 0.5L + 0.0001AL(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
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) Find all the equilibrium solutions and explain their
In Exercise 8 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.(c) Find an expression for dL/dA.(d) Use a computer
(a) What is a differential equation?(b) What is the order of a differential equation?(c) What is an initial condition?
What can you say about the solutions of the equation y’ = x2 + y2 just by looking at the differential equation?
What is a separable differential equation? How do you solve it?
What is a first-order linear differential equation? How do you solve it?
(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 logistic equation.(b) Under what circumstances is this an appropriate model for population growth?
(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?
(a) A direction field for the differential equation y€™ = y(y €“ 2) (y €“ 4) is shown. Sketch the graphs of the solutions that satisfy the given initial conditions.(i) y (0) = €“ 0.3(ii) y (0) = 1(iii) y (0) = 3(iv) y (0) = 4.3(b) If the initial condition is
(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 differential equation explicitly. What type of
(a) A direction field for the differential equation y€™ = x2 €“ y2 is shown. Sketch the solution of the initial-value problem y€™ = x2 €“ y2 y (0) = 1Use your graph to estimate the value of y (0.3).(b) Use Euler€™s method with step size 0.1 to
(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' = 2xy2 y (0) = 1(b) Repeat part (a) with step size 0.1.(c) Find the exact solution of the differential equation and compare the value at 0.4 with the approximations in
A bacteria culture starts with 1000 bacteria and the growth rate is proportional to the number of bacteria. After 2 hours the population is 9000.(a) Find an expression for the number of bacteria after hours.(b) Find the number of bacteria after 3 h.(c) Find the rate of growth after 3 h.(d) How long
An isotope of strontium, 90Sr, has a half-life of 25 years.(a) Find the mass of 90Sr that remains from a sample of 18 mg after years.(b) How long would it take for the mass to decay to 2 mg?
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 the elimination constant of the drug.(a)
(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 population exceed 10 billion?(c) Use the data in
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 achieved. (a) Formulate and solve 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 week and 1200 have it at the end of the week.
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:1/R dR/dt = k/S dS/dt where is a positive constant. Find R as a function of S.
The transport of a substance across a capillary wall in lung physiology has been modeled by the differential equation dh/dt = – R/V (h/ k + h) where is the hormone concentration in the bloodstream, is time, R is the maximum transport rate, V is the volume of the capillary, and is a positive
Populations of birds and insects are modeled by the equations dx/dt = 0.4x = 0.002xy dy/dt = €“0.2y + 0.000008xy(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
Suppose the model of Exercise 24 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 shows the phase trajectory that starts with 100 birds and
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 calories in the form of fat 100% is
When a flexible cable of uniform density is suspended between two fixed points and hangs of its own weight, the shape y = f(x) of the cable must satisfy a differential equation of the form where is a positive constant. Consider the cable shown in the figure.(a) Let z = dy/dx in the differential
Find all functions f such that f€™ is continuous and
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 (½, ∞). What was the function g?
Let f be a function with the property that f(0) = 1, f’(0) = 1, and f (a + b) = f(a + b) = f(a) f (b) for all real numbers and . Show that f’(x) = f(x) for all x and deduce that f(x) = ex.
Find all functions f that satisfy the equation
A peach pie is taken out of the oven at 5:00 P.M. At that time it is piping hot: 100oC. At 5:10 P.M. its temperature is 80oC; at 5:20 P.M. it is 65oC. What is the temperature of the room?
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 P.M. but only 3 km from 1 P.M. to 2 P.M. When did the snow begin to fall? [Hints: To get
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 sees the rabbit.(ii) The rabbit runs up the -axis
(a) Suppose that the dog in Problem 7 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 close does the dog get to the rabbit? What are
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 the top of the silo. The silo is a cylinder 100
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.
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 any point on the curve, then the y-intercept
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 and P the x-axis is bisected by the y-axis.
Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as increases.
(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.(b) Eliminate the parameter to find a Cartesian equation of the curve.
Describe the motion of a particle with position (x, y)as t varies in the given interval.
Suppose a curve is given by the parametric equations x = f(t), y = g(t), where the range of f is [1, 4] and the range of g is [2, 3]. What can you say about the curve?
Match the graphs of the parametric equations x = f (t) and y = g (t) in (a)€“(d) with the parametric curves labeled I€“IV. Give reasons for your choices.
Use the graphs of x = f (t) and y = g (t) to sketch the parametric curve x = f (t), y = g (t). Indicate with arrows the direction in which the curve is traced as increases.
Match the parametric equations with the graphs labeled I€“VI. Give reasons for your choices. (Do not use a graphing device.)
Graph the curve x = y - 3y3 + y5.
Graph the curves y = x5 and x = y(y – 1)2 and find their points of intersection correct to one decimal place.
(a) Show that the parametric equations x = x1 + (x2 – x1)t y = y1 + (y2 – y1) t where 0 < t < 1, describe the line segment that joins the points P1(x1, y1) and P2(x2, y2).(b) Find parametric equations to represent the line segment from (– 2, 7) to (3, –1).
Use a graphing device and the result of Exercise 31(a) To draw the triangle with vertices A (1, 1), B (4, 2) and C (1, 5)
Find parametric equations for the path of a particle that moves along the circle x2 + (y – 1)2 = 4 in the manner described(a) Once around clockwise, starting at (2, 1)(b) Three times around counterclockwise, starting at (2, 1)(c) Halfway around counterclockwise, starting at (0, 3)
(a) Find parametric equations for the ellipse x2/a2 + y2/b2 = 1.(b) Use these parametric equations to graph the ellipse when a = 3 and b = 1, 2, 4, and 8.(c) How does the shape of the ellipse change as b varies?
Find three different sets of parametric equations to represent the curve y = x3, x ε R.
Derive Equations 1 for the case π/2 < θ < π.
Let P be a point at a distance from the center of a circle of radius r. The curve traced out by P as the circle rolls along a straight line is called a trochoid. (Think of the motion of a point on a spoke of a bicycle wheel.) The cycloid is the special case of a trochoid with d = r. Using the same
If a and b are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle θ as the parameter. Then eliminate the parameter and identify the curve.
If and are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle θ as the parameter. The line segment AB is tangent to the larger circle.
A curve, called a witch of Maria Agnesi, consists of all possible positions of the point P in the figure. Show that parametric equations for this curve can be written as x = 2a cot θ y = 2a sin2θ Sketch the curve.
Find parametric equations for the curve that consists of all possible positions of the point P in the figure, where | OP | = | AB |. Sketch the curve. (This curve is called the cissoids of Diocles after the Greek scholar Diocles, who introduced the cissoids as a graphical method for constructing
Suppose that the position of one particle at time is given by x1 = 3 sin t y1 = 2 cos t 0 < t < 2π and the position of a second particle is given by x2 = – 3 + cos t y2 = 1 + sin t 0 < t < 2π (a) Graph the paths of both particles. How many points of intersection are there? (b) Are any
If a projectile is fired with an initial velocity of vo meters per second at an angle α above the horizontal and air resistance is assumed to be negligible, then its position after seconds is given by the parametric equations x = (vo cos α) t y = (vo sin α) t – ½ gt2 where is the
Investigate the family of curves defined by the parametric equations x = t2, y = t3 – ct. How does the shape change as c increases? Illustrate by graphing several members of the family.
The swallowtail catastrophe curves are defined by the parametric equations x = 2ct – 4t3, y = – ct2 + 3t4. Graph several of these curves. What features do the curves have in common? How do they change when increases?
The curves with equations x = a sin nt, y = b cos t are called Lissajous figures. Investigate how these curves vary when a, b, and n vary. (Take n to be a positive integer.)
Investigate the family of curves defined by the parametric equations x = sin t (c – sin t) y = cos t (c – sin t) How does the shape change as c changes? In particular, you should identify the transitional values of c for which the basic shape of the curve changes.
Find an equation of the tangent to the curve at the given point by two methods:(a) Without eliminating the parameter and(b) By first eliminating the parameter.
Find an equation of the tangent(s) to the curve at the given point. Then graph the curve and the tangent(s).
Use a graph to estimate the coordinates of the leftmost point on the curve x = t4 – t2, y = t + in t. Then use calculus to find the exact coordinates.
Try to estimate the coordinates of the highest point and the leftmost point on the curve x = tet, y = te–t . Then find the exact coordinates. What are the asymptotes of this curve?
Graph the curve in a viewing rectangle that displays all the important aspects of the curve.
Show that the curve x = cos t, y = sin t cos t has two tangents at (0, 0) and find their equations. Sketch the curve.
At what point does the curve x = 1 – 2 cos2t, y = (tan t) (1 – 2 cos2t) cross itself? Find the equations of both tangents at that point.
(a) Find the slope of the tangent line to the trochoid x = r θ – d sin θ, y = r – d cos θ in terms of θ. (See Exercise 38 in Section 10.1.) (b) Show that if d < r, then the trochoid does not have a vertical tangent.
(a) Find the slope of the tangent to the asteroid x = a cos3 θ, y = a sin3 θ in terms of θ. (Asteroids are explored in the Laboratory Project on page 659.) (b) At what points is the tangent horizontal or vertical? (c) At what points does the tangent have slope 1 or – 1?

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