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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Uranium-238 is a radioactive material with a half-life of 4.468 billion years. With t in billions of years, let M(t) be the mass in grams of a sample of uranium-238 that initially consisted of 100 g. Set up and solve an Initial Value Problem for determining M(t).
Let A and B be constants. Prove that if A > 0, then all solutions of dy/dt + Ay = B approach the same limit as t→∞.
Consider a series circuit (Figure 4) consisting of a resistor of R ohms, an inductor of L henries, and a variable voltage source of V(t) volts (time t in seconds). The current through the circuit I(t) (in amperes) satisfies the differential equationSolve for I(t), assuming that R = 500 ohms, L = 4
Let a, b, r be constants. Show that is a general solution of y = Cekt + a + bk dy dt = k sin rt-r cos rt k² + r² -k(y - a - b sinrt)
Solve the Initial Value Problem.y' = y2 sin x, y(π) = 2
At time t = 0, a tank of height 5 m in the shape of an inverted pyramid whose cross section at the top is asquare of side 2 m is filled with water. Water flows through a hole at the bottom of area 0.002 m2. Determine the time required for the tank to empty.
Tank 1 in Figure 5 is filled with V1 liters of water containing blue dye at an initial concentration of c0 g/L. Water flows into the tank at a rate of R L/min, is mixed instantaneously with the dye solution, and flows out through the bottom at the same rate R. Let c1(t) be the dye concentration in
In Example 3, we calculated the thickness of a glacier assuming the friction at the base of the glacier is constant. In we consider cases where the friction varies along the length of the glacier.(a) Solve the glacier thickness differential equation [Eq. (2)] for T(x) with τ(x) = 75x N/m2 and
The trough in Figure 3 (dimensions in centimeters) is filled with water. At time t = 0 (in seconds), water begins leaking through a hole at the bottom of area 4 cm2. Let y(t) be the water height at time t. Find a differential equation for y(t) and solve it to determine when the water level
Continuing with the previous exercise, let tank 2 be another tank filled with V2 gallons of water. Assume that the dye solution from tank 1 empties into tank 2 as in Figure 5, mixes instantaneously, and leaves tank 2 at the same rate R. Let c2(t) be the dye concentration in tank 2 at time t.(a)
Find the solutions of the logistic equation dy/dt = y(4 − y) satisfying the initial conditions:(a) y(0) = 1 (b) y(0) = 4 (c) y(0) = 6
In Example 3, we calculated the thickness of a glacier assuming the friction at the base of the glacier is constant. In we consider cases where the friction varies along the length of the glacier.(a) Solve the glacier thickness differential equation [Eq. (2)] for T(x) with τ(x) = 0.3x(1000 − x)
Let y(t) be the solution of dy/dt = 0.3y(2 − y) with y(0) = 1. Determine without solving for yexplicitly. lim y(t) 1-80
Use the differential equation for a leaking container, Eq. (3).Water leaks through a hole of area B = 0.002 m2 at the bottom of a cylindrical tank that is filled with water and has height 3 m and a base of area 10 m2. How long does it take (a) for half of the water to leak out and (b) for the tank
Use the differential equation for a leaking container, Eq. (3).At t = 0, a conical tank of height 300 cm and top radius 100 cm [Figure 11(A)] is filled with water. Water leaks through a hole in the bottom of area B = 3 cm2. Let y(t) be the water level at time t.(a) Show that the tank’s
Suppose that y' = ky(1 − y/8) has a solution satisfying y(0) = 12 and y(10) = 24. Find k.
Show that y = sin(tan−1 x + C) is the general solution of y' = √1 − y2/(1 + x2). Then use the addition formula for the sine function to show that the general solution may be written y = (cos C)x+ sin C √1 + x²
Use the Fundamental Theorem of Calculus and the Product Rule to verify directly that for any x0, the functionis a solution of the Initial Value Problemwhere α(x) is an integrating factor [a solution to Eq. (3)]. f(x)= a(x)-¹ - Lane a(t)Q(t) dt
Use the differential equation for a leaking container, Eq. (3).A tank has the shape of the parabola y = x2, revolved around the y-axis. Water leaks from a hole of area B = 0.0005 m2 at the bottom of the tank. Let y(t) be the water level at time t. How long does it take for the tank to empty if it
A rabbit population on an island increases exponentially with growth rate k = 0.12 months−1. When the population reaches 300 rabbits (say, at time t = 0), wolves begin eating the rabbits at a rate of r rabbits per month.(a) Find a differential equation satisfied by the rabbit population P(t).(b)
Let α(x) be an integrating factor for y' + P(x)y = Q(x). The differential equation y' + P(x)y = 0 is called the associated homogeneous equation.(a) Show that y = 1/α(x) is a solution of the associated homogeneous equation.(b) Show that if y = ƒ(x) is a particular solution of y' + P(x)y = Q(x),
A lake has a carrying capacity of 1000 fish. Assume that the fish population grows logistically with growth constant k = 0.2 day−1. How many days will it take for the population to reach 900 fish if the initial population is 20 fish?
Use the differential equation for a leaking container, Eq. (3).The tank in Figure 11(B) is a cylinder of radius 4 m and height 15 m. Assume that the tank is half-filled with water and that water leaks through a hole in the bottom of area B = 0.001 m2. Determine the water level y(t) and the time te
Assume that the outside temperature varies as T(t) = 15 + 5 sin(πt/12) where t = 0 is 12 noon. A house is heated to 25°C at t = 0 and after that, its temperature y(t) varies according to Newton’s Law of Cooling (Figure 6):Data From Exercise 45Let a, b, r be constants. Show that dy = dt =
Use the differential equation for a leaking container, Eq. (3).A tank has the shape of the parabola y = ax2 (where a is a constant) revolved around the y-axis. Water drains from a hole of area B m2 at the bottom of the tank.(a) Show that the water level at time t iswhere y0 is the water level at
Suppose the circuit described by Eq. (9) is driven by a sinusoidal voltage source V(t) = V sinωt (where V and ω are constant).(a) Show that dl + dt L R -I = -V (1) L
A tank is filled with 300 liters of contaminated water containing 3 kg of toxin. Pure water is pumped in at a rate of 40 L/min, mixes instantaneously, and is then pumped out at the same rate. Let y(t) be the quantity of toxin present in the tank at time t.(a) Find a differential equation satisfied
A cylindrical tank filled with water has height h and a base of area A. Water leaks through a hole in the bottom of area B.(a) Show that the time required for the tank to empty is proportional to A√h/B.(b) Show that the emptying time is proportional to Vh−1/2, where V is the volume of the
At t = 0, a tank of volume 300 liters is filled with 100 L of water containing salt at a concentration of 8 g/L. Fresh water flows in at a rate of 40 L/min, mixes instantaneously, and exits at the same rate. Let c1(t) be the salt concentration at time t.(a) Find a differential equation satisfied by
When cane sugar is dissolved in water, it converts to invert sugar over a period of several hours. The amount A(t) of unconverted cane sugar at time t (in hours) satisfies A' = −0.2A. If there are initially 500 g of unconverted cane sugar, how much unconverted cane sugar remains after 5 h? After
The outflow of the tank in Exercise 51 is directed into a second tank containing V liters of fresh water where it mixes instantaneously and exits at the same rate of 40 L/min. Determine the salt concentration c2(t) in the second tank as a function of time in the following two cases:(a) V = 200 (b)
A certain RNA molecule replicates every 3 minutes. Find the differential equation for the number N(t) of molecules present at time t (in minutes). How many molecules will be present after 1 hour if there is one molecule at t = 0?
Assume that during periods of job growth, the rate of increase of the number employed is proportional to the number who are employed. At the outset of a job growth period, the number employed in a certain country grew from 11.3 million to 11.7 million in 10 weeks. Let N(t) represent the number
Bismuth-210 decays at a rate proportional to the amount present. A sample of Bismuth-210 that initially had a mass of 1000 mg decayed 500 mg in 5 days. Let M(t) be the mass of Bismuth-210 in milligrams t days after the initial sample of 1000 mg began to decay. Set up and solve an Initial Value
(a) With y(t) = y0ekt, at what value of t (in terms of p and k) is y(t) = py0?(b) If y(t) = y0e1.5t, with t in hours, how long does it take for y to double? To triple? To increase 10-fold?
Let y(t) be the drug concentration (in micrograms per kilogram) in a patient’s body at time t. The initial concentration is y(0) = L. Additional doses that increase the concentration by an amount d are administered at regular time intervals of length T. In between doses, y(t) decays
Let v(t) be the velocity of an object of mass m in free-fall near the earth’s surface. If we assume that air resistance is proportional to v2, then v satisfies the differential equation m dv/dt = −g + kv2 for some constant k > 0. (a) Set a = (g/k)¹/2 and rewrite the differential equation
A 50-kg model rocket lifts off by expelling fuel downward at a rate of k = 4.75 kg/s for 10 s. The fuel leaves the end of the rocket with an exhaust velocity of b = −100 m/s. Let m(t) be the mass of the rocket at time t. From the law of conservation of momentum, we find the following differential
Show that the differential equations y' = 3y/x and y' = −x/3y define orthogonal families of curves; that is, the graphs of solutions to the first equation intersect the graphs of the solutions to the second equation in right angles (Figure 15). Find these curves explicitly. y
Captain Quint is standing at point B on a dock and is holding a rope of length ℓ attached to a boat at point A [Figure 14(A)]. As the captain walks along the dock, holding the rope taut, the boat moves along a curve called a tractrix (from the Latin tractus, meaning “pulled”). The segment
Figure 13 shows a circuit consisting of a resistor of R ohms, a capacitor of C farads, and a battery of voltage V. When the circuit is completed, the amount of charge q(t) (in coulombs) on the plates of the capacitor varies according to the differential equation (t in seconds)where R, C, and V are
The general solution to dy/dt = ky is y = Dekt. Here we prove that every solution to the differential equation, defined on an interval, is given by the general solution.(a) Show that if y(t) satisfies the differential equation, then d/dt (ye−kt) = 0.(b) Assume that y(t) satisfies the
A spherical tank of radius R is half-filled with water. Suppose that water leaks through a hole in the bottom of area B. Let y(t) be the water level at time t (seconds). dy √2gB √y (a) Show that dt л(2Ry - y²) (b) Show that for some constant C, = 2π (10Ry3/2-3y5/2) = C-t 15B √2g (c) Use
Show that y = Cerx is a solution of y" + ay' + by = 0 if and only if r is a root of P(r) = r2 + ar + b. Then verify directly that y = C1e3x + C2e−x is a solution of y" − 2y' − 3y = 0 for any constants C1,C2.
A basic theorem states that a linear differential equation of order n has a general solution that depends on n arbitrary constants. The following examples show that, in general, the theorem does not hold for nonlinear differential equations.(a) Show that (y')2 + y2 = 0 is a first-order equation
In Section 6.2, we computed the volume V of a solid as the integral of cross-sectional area. Explain this formula in terms of differential equations. Let V(y) be the volume of the solid up to height y, and let A(y) be the cross-sectional area at height y as in Figure 17.(a) Explain the following
If a bucket of water spins about a vertical axis with constant angular velocity ω (in radians per second), the water climbs up the side of the bucket until it reaches an equilibrium position (Figure 16). Two forces act on a particle located at a distance x from the vertical axis: the gravitational
Calculate the arc length over the given interval. y = x/2x¹/2, [2,8]
Fluid pressure is proportional to depth. What is the factor of proportionality?
If a rectangular plate that is 1 by 2 m is dipped into a pool of water so that initially its top edge of length 1 is even with the surface of the water, and then it is lowered so that its top edge is at a depth of 1 m, calculate the increase in fluid force on one side of it.
Find a constant C such that p is a probability density function on the given interval, and compute the probability indicated. p(x) = C √1-x² on (-1,1); P(-1< X < / )
Which exponential probability density has mean μ = 1/4?
The center of mass of a lamina of total mass 5 has coordinates (2, 1). What are the lamina’s x- and y-moments?
Find the arc length of y = 1/12 x3 + x−1 for 1 ≤ x ≤ 2. Show that 1 + (v²)² = (x² + x^²) ².
Four particles are located at points (1, 1), (1, 2), (4, 0), and (3, 1).(a) Find the moments Mx and My and the center of mass of the system, assuming that the particles have equal mass m.(b) Find the center of mass of the system, assuming the particles have masses 3, 2, 5, and 7, respectively.
If 0 ≤ ƒ(x) ≤ g(x) for x in the interval [a, b], can the surface obtained by rotating the graph of y = g(x) around the x-axis over the interval have less surface area than the surface obtained by rotating the graph of y = ƒ(x) around the x-axis over the same interval?
Find a constant C such that p(x) = Cx3e−x2 is a probability density over the domain [0,∞) and compute P(0 ≤ X ≤ 1).
When fluid force acts on the side of a submerged object, in which direction does it act?
A plate in the shape of an isosceles triangle with base 1 m and height 2 m is submerged vertically in a tank of water so that its vertex touches the surface of the water (Figure 7).(a) Show that the width of the triangle at depth y is ƒ(y) = 12/y.(b) Consider a thin strip of thickness Δy at depth
Why is fluid pressure on a surface calculated using thin horizontal strips rather than thin vertical strips?
Find a constant C such that p is a probability density function on the given interval, and compute the probability indicated. p(x) Ce-* 1 + e-2x on (-∞0,00); P(X ≤ -4)
Explain how the Symmetry Principle is used to conclude that the centroid of a rectangle is the center of the rectangle.
Find the center of mass for the system of particles of masses 4, 2, 5, and 1 located at (1, 2), (−3, 2), (2, −1), and (4, 0).
Find the arc length of Show that 1 + (y')2 is a perfect square. y = X 1 + over [1,4]. 2x²
Use the formula for arc length to show that for any constant C, the graphs y = ƒ(x) and y = ƒ(x) + C have the same length over every interval [a, b]. Explain geometrically.
The interval between patient arrivals in an emergency department is a random variable with exponential density function p(t) = 0.125e−0.125t (t in minutes). What is the average time between patient arrivals? What is the probability of two patients arriving within 3 min of each other?
Repeat Exercise 4, but assume that the top of the triangle is located 3 m below the surface of the water.Data From Exercise 4A plate in the shape of an isosceles triangle with base 1 m and height 2 m is submerged vertically in a tank of water so that its vertex touches the surface of the water
If a thin plate is submerged horizontally, then the fluid force on one side of the plate is equal to pressure times area. Is this true if the plate is submerged vertically?
Find a constant C such that p is a probability density function on the given interval, and compute the probability indicated.p(x) = C sin x on [0, π]; P(π/4 ≤ X ≤ 3π/4)
Give an example of a plate such that its center of mass does not occur at any point on the plate.
Point masses of equal size are placed at the vertices of the triangle with coordinates (a, 0), (b, 0), and (0, c). Show that the center of mass of the system of masses has coordinates (1/3 (a + b), 1/3 c).
The plate R in Figure 8, bounded by the parabola y = x2 and y = 1, is submerged vertically in water (distance in meters). (a) Show that the width of R at height y is f(y) = 2√y and the fluid force on a side of a horizontal strip of thickness Ay at height y is approximately (pg)2y¹/2(1-y)Ay. (b)
Calculate the arc length over the given interval.y = 3x + 1, [0, 3]
Calculate the following probabilities, assuming that X is normally distributed with mean μ = 40 and σ = 5.(a) P(X ≥ 45) (b) P(0 ≤ X ≤ 40)
Find a constant C such that p is a probability density function on the given interval, and compute the probability indicated.p(x) = C ln x on [1, e]; P(1 ≤ X ≤ 2)
Draw a plate such that its center of mass occurs on its boundary. (You do not need to verify this fact. It should just be believable from the drawing.)
Point masses of mass m1, m2, and m3 are placed at the points (−1, 0), (3, 0), and (0, 4).(a) Suppose that m1 = 6. Find m2 such that the center of mass lies on the y-axis.(b) Suppose that m1 = 6 and m2 = 4. Find the value of m3 such that y̅ = 2.
Let F be the fluid force on a side of a semicircular plate of radius r meters, submerged vertically in water so that its diameter is level with the water’s surface (Figure 9).(a) Show that the width of the plate at depth y is 2 r2 − y2.(b) Calculate F as a function of r using Eq. (2).
Calculate the arc length over the given interval.y = 9 − 3x, [1, 3]
According to kinetic theory, the molecules of ordinary matter are in constant random motion. The energy E of a molecule is a random variable with density function p(E) = 1/kT e−E/(kT), where T is the temperature (in kelvins) and k is Boltzmann’s constant. Compute the mean kinetic energy E in
Sketch the lamina S of constant density ρ = 3 g/cm2 occupying the region beneath the graph of y = x2 for 0 ≤ x ≤ 3.(a) Use Eqs. (1) and (2) to compute Mx and My.(b) Find the area and the center of mass of S. My = p x(length of vertical cut) dx = pf x (fi(x) - f2(x)) dx Р a
Find a constant C such that p is a probability density function on the given interval, and compute the probability indicated.p(x) = C√1 − x2 on (−1, 1); P(− 1/2 ≤ X ≤ 1)
Calculate the arc length over the given interval. y = 10 + -3 X 6 [1,2]
Calculate the arc length over the given interval.y = x3/2, [1, 2]
Calculate the force on one side of a circular plate with radius 2 m, submerged vertically in a tank of water so that the top of the circle is tangent to the water surface.
Find a constant C such that p is a probability density function on the given interval, and compute the probability indicated.p(x) = Ce−xe−e−x on (−∞,∞); P(−4 ≤ X ≤ 4) This function, called the Gumbel density, is used to model extreme events such as floods and earthquakes.
Calculate the arc length over the given interval.y = ex/2 + e−x/2, [0, 2]
Calculate the arc length over the given interval.y = 4x − 2, [−2, 2]
A semicircular plate of radius r meters, oriented as in Figure 9, is submerged in water so that its diameter is located at a depth of m meters. Calculate the fluid force on one side of the plate in terms of m and r.
Verify that p(x) = 3x−4 is a probability density function on [1,∞) and calculate its mean value.
Calculate the arc length over the given interval. =x²-lnx, [1,2e] In y=
Find the moments and center of mass of the lamina of uniform density ρ occupying the region underneath y = x3 for 0 ≤ x ≤ 2.
Calculate Mx (assuming ρ = 1) for the region underneath the graph of y = 1 − x2 for 0 ≤ x ≤ 1 in two ways, first using Eq. (2) and then using Eq. (3). My = p x(length of vertical cut) dx = pf * x (fi(x) – f2(x)) dx Р a
Calculate the arc length over the given interval.y = x2/3, [1, 8]
A plate extending from depth y = 2 m to y = 5 m is submerged in a fluid of density ρ = 850 kg/m3. The horizontal width of the plate at depth y is ƒ(y) = 2(1 + y2)−1. Calculate the fluid force on one side of the plate.
Show that the density function p(x) = 2/π(x2 + 1) on [0,∞) has infinite mean.
Calculate the arc length over the given interval.y = ln(cos x), [0, π/4]
Show that the arc length of y = 2 √x over [0, a] is equal to √a(a + 1) + ln(√a + √a + 1). Apply the substitution x = tan2 θ to the arc length integral.
Compute the trapezoidal approximation T5 to the arc length s of y = tan x over [0, π/4].
Use the Theorem of Pappus to find the volume of the solid of revolution obtained by rotating the region in the first quadrant bounded by y = x2 and y = √x about the y-axis.

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