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calculus

Questions and Answers of Calculus

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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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’
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
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
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).
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
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
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
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
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
(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
(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
(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) =
(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
(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
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
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.
(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)
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
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
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
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
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,
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
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
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
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
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)
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
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
(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)
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
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
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)
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
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
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
(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
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
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
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
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
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
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 <
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
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
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
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
(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

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