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mathematics
calculus 4th

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

Solve the Initial Value Problem using Separation of Variables.xyy'= 1, y(3) = 2
Use Newton’s Law of Cooling.When a hot object is placed in a water bath whose temperature is 25°C, it cools from 100°C to 50°C in 150 seconds. In another bath, the same cooling occurs in 120 s. Find the temperature of the second bath.
Sketch the slope field of dy/dt = ty for −2 ≤ t ≤ 2, −2 ≤ y ≤ 2. Based on the sketch, determine where y(t) is a solution with y(0) > 0. lim y(t), where y(t) 1-00
Find the general solution of the first-order linear differential equation.y' + x−1y = cos(x2)
Figure 1 shows the slope field for dy/dt = sin y + ty. Sketch the graphs of the solutions with the initial conditions y(0) = 1 , y(0) = 0, and y(0) = −1. 2 1 0 -1 -2 y / Л 1 I I -2 -1 0 1 2
Let k = 1 and A = 1 in the logistic equation.(a) Find the solutions satisfying y1(0) = 10 and y2(0) = −1.(b) Find the time t when y1(t) = 5.(c) When does y2(t) become infinite?
Use Separation of Variables to find the general solution. 0 = ET واح }/
Match each differential equation with its slope field in Figure 13(A)–(F). (i) | 7 (D) || || 4 (9) (vi) " = = = 民那 F+² =t
Use Newton’s Law of Cooling.Objects A and B are placed in a warm bath at temperature T0 = 40Q°C. Object A has initial temperature −20°C and cooling constant k = 0.004 s−1. Object B has initial temperature 0°C and cooling constant k = 0.002 s−1. Plot the temperatures of A and B for 0 ≤
A tissue culture grows until it has a maximum area of M square centimeters. The area A(t) of the culture at time t may be modeled by the differential equationwhere k is a growth constant.(a) Show that if we set A = u2, thenThen find the general solution using Separation of Variables.(b) Show that
Find the general solution of the first-order linear differential equation.xy'= y − x
Sketch the slope field for dy/dt = t2y for −2 ≤ t ≤ 2, −2 ≤ y ≤ 2. 2 1 0 -2 77 -2 - - - -1 0 |||| // w 1 - - 2
Use Separation of Variables to find the general solution.y' + 4xy2 = 0
In Newton’s Law of Cooling, the constant τ = 1/k is called the characteristic time. Show that τ is the time required for the temperature difference (y − T0) to decrease by the factor e−1 ≈ 0.37. For example, if y(0) = 100°C and T0 = 0°C, then the object cools to 100/e ≈ 37°C in time
Find the general solution of the first-order linear differential equation.xy'= x−2 − 3y/x
In the model of Exercise 12, let A(t) be the area at time t (hours) of a growing tissue culture with initial size A(0) = 1 cm2, assuming that the maximum area is M = 16 cm2 and the growth constant is k = 0.1.(a) Find a formula for A(t). The initial condition is satisfied for two values of the
Sketch the solution of dy/dt = ty2 satisfying y(0) = 1 in the appropriate slope field of Figure 13(A)–(F).Then show, using Separation of Variables, that if y(t) is a solution such that y(0) > 0, then y(t) tends to infinity as t → √2/y(0).
Sketch the slope field for dy/dt = y sin t for −2π ≤ t ≤ 2π, −2 ≤ y ≤ 2.
Use Separation of Variables to find the general solution.y'+ x2y = 0
Use Eq. (3) as a model for free-fall with air resistance.A 60-kg skydiver jumps out of an airplane. What is her terminal velocity, in meters per second, assuming that k = 10 kg/s for free-fall (no parachute)? V = k m mg k (+a).
Find the general solution of the first-order linear differential equation.y' + y = ex
Show that if a tissue culture grows according to Eq. (7), then the growth rate reaches a maximum when A = M/3. A(t) = M J (Cek/ √M) - 1 Cek/VM)1 + 1
Which of the equations (i)–(iii) corresponds to the slope field in Figure 2? dy dt dy dt dy (iii) = y dt (i) (ii) = 1-y² =1+y² 2
(a) Sketch the slope field of dy/dt = t/y in the region −2 ≤ t ≤ 2, −2 ≤ y ≤ 2.(b) Check that y = ± √t2 + C is the general solution.(c) Sketch the solutions on the slope field with initial conditions y(0) = 1 and y(0) = −1.
Use Eq. (3) as a model for free-fall with air resistance.Find the terminal velocity of a skydiver of weight w = 192 pounds if k = 1.2 lb-s/ft. How long does it take him to reach half of his terminal velocity if his initial velocity is zero? Mass and weight are related by w = mg, and Eq. (3) becomes
Use Separation of Variables to find the general solution.y' − ex+y = 0
Find the general solution of the first-order linear differential equation.y' − y = ex
In 1751, Benjamin Franklin predicted that the U.S. population P(t) would increase with growth constant k = 0.028 year−1. According to the census, the U.S. population was 5 million in 1800 and 76 million in 1900. Assuming logistic growth with k = 0.028, find the predicted carrying capacity for the
Sketch the slope field of dy/dt = t2 − y in the region −3 ≤ t ≤ 3, −3 ≤ y ≤ 3 and sketch the solutions satisfying y(1) = 0, y(1) = 1, and y(1) = −1.
An 80-kg skydiver jumps out of an airplane (with zero initial velocity). Assume that k = 12 kg/s with a closed parachute and k = 70 kg/s with an open parachute. What is the skydiver’s velocity at t = 25 s if the parachute opens after 20 s of free-fall?
Find the general solution of the first-order linear differential equation.y' + (tan x)y = cos x
Consider the following logistic equation (with k, B > 0): dP dt = : -KP(1 – B) ||
Let F(t, y) = t2 − y and let y(t) be the solution of dy/dt = F(t, y) satisfying y(2) = 3. Let h = 0.1 be the time step in Euler’s Method, and set y0 = y(2) = 3.(a) Calculate y1 = y0 + hF(2, 3).(b) Calculate y2 = y1 + hF(2.1, y1).(c) Calculate y3 = y2 + hF(2.2, y2) and continue computing
Find the general solution of the first-order linear differential equation.y' + (sec x)y = cos x
Let y(t) be the solution of 4 dy/dt = y2 + t satisfying y(2) = 1. Carry out Euler’s Method with time step h = 0.05 for n = 6 steps.
Use Separation of Variables to find the general solution.t3y'+ 4y2 = 0
Does a heavier or a lighter skydiver reach terminal velocity more quickly?
Use Euler’s Method to approximate the given value of y(t) with the time step h indicated. y(0.5); dy dt = y +t, y(0) = 1, h = 0.1
Let y(t) be the solution to dy/dt = te−y satisfying y(0) = 0.(a) Use Euler’s Method with time step h = 0.1 to approximate y(0.1), y(0.2), . . . , y(0.5).(b) Use Separation of Variables to find y(t) exactly.(c) Compute the errors in the approximations to y(0.1) and y(0.5).
Use Separation of Variables to find the general solution.yy' = x
Find polar coordinates for each of the seven points plotted in Figure 17. A B C y 4 D I E G (x, y) = (2√3,2) 4 X
Plot the points with polar coordinates: (a) (2,7) (b) (4,3) (c) (3,-5) (d) (0,5)
Which of the following are possible polar coordinates for the point P with rectangular coordinates (0, −2)? (a) (2,3 2.2 (0) (-2-21) (с) (-2,-2) 7П (b) (2, 2 (d) -2, (-2,7/17) (0) (2,-727)
Convert from polar to rectangular coordinates: (a) (3,7) (b) (6, ³7) (c) (0, 3) (d) (5,-2)
Describe each tan shaded sector in Figure 18 by inequalities in r and θ. (A) -45° 35 x (B) X D (C) x
Convert from rectangular to polar coordinates:(a) (1, 0) (b) (3,√3) (c) (−2, 2) (d) (−1,√3)
Convert from rectangular to polar coordinates using a calculator (make sure your choice of θ gives the correct quadrant):(a) (2, 3) (b) (4, −7) (c) (−3, −8) (d) (−5, 2)
Describe each green shaded sector in Figure 18 by inequalities in r and θ. (A) -45⁰ 35 x (B) x D (C) x
Find an equation in polar coordinates of the line through the origin with slope √1/3.
Find an equation in polar coordinates of the line through the origin with slope 1 − √2.
What is the slope of the line θ = 3π/5?
One of r = 2 sec θ and r = 2 csc θ is a horizontal line, and the other is a vertical line. Convert each to rectangular coordinates to show which is which.
Convert to an equation in rectangular coordinates.r = 7
Convert to an equation in rectangular coordinates.r = sin θ
Convert to an equation in rectangular coordinates.r = 2 sin θ
Convert to an equation in rectangular coordinates. r = cos 1 sin e
Convert to an equation in rectangular coordinates.r = 2 csc θ − sec θ
Convert to an equation in rectangular coordinates. r = 1 2- cos 0
Convert to an equation in polar coordinates of the form r = ƒ(θ).x2 + y2 = 5
Convert to an equation in polar coordinates of the form r = ƒ(θ).x = 5
Convert to an equation in polar coordinates of the form r = ƒ(θ).y = x2
Convert to an equation in polar coordinates of the form r = ƒ(θ).xy = 1
Convert to an equation in polar coordinates of the form r = ƒ(θ).e√x2+y2 = 1
Convert to an equation in polar coordinates of the form r = ƒ(θ).ln x = 1
Find the values of θ in the plot of r = 4 cos θ corresponding to points A, B, C, D in Figure 19. Then indicate the portion of the graph traced out as θ varies in the following intervals: (a) 0 ≤ 0 < (b) {
Match each equation with its description:(a) r = 2 (i) Vertical line(b) θ = 2 (ii) Horizontal line(c) r = 2 sec θ (iii) Circle(d) r = 2 csc θ (iv) Line through origin
Suppose that P = (x, y) has polar coordinates (r, θ). Find the polar coordinates for the points:(a) (x, −y) (b) (−x, −y) (c) (−x, y) (d) (y, x)
Match each equation in rectangular coordinates with its equation in polar coordinates: (a) x² + y² = 4 (b) x² + (y- 1)² = 1 (c) x² - y² = 4 (d) x + y = 4 (i) r²(1-2 sin²0) = 4 (ii) r(cos + sin 8) = 4 (iii) r = 2 sin 0 (iv) r = 2
Show that the circle with its center at (1/2, 1/2) in Figure 20 has polar equation r = sin θ + cos θ and find the values of θ between 0 and π corresponding to points A, B, C, and D. A B • (-/---/-) D C -X
What are the polar equations of the lines parallel to the line r cos (θ − π/3) = 1?
Sketch r = 3 cos θ − 1 (see Example 9). EXAMPLE 9 Symmetry About the x-Axis Sketch the limaçon curve r = 2 cos 0 - 1.
Figure 21 displays the graphs of r = sin 2θ in r versus θ rectangular coordinates and in polar coordinates, where it is a “rose with four petals.” Identify:(a) The points in (B) corresponding to points A–I in (A).(b) The parts of the curve in (B) corresponding to the angle intervals [0,
Show that the cardioid of Exercise 33 has equation (x2 + y2 − x)2 = x2 + y2 in rectangular coordinates.Data From Exercise 33Sketch the cardioid curve r = 1 + cos θ.
Sketch the curve r = sin 3θ by filling in the table of r-values below and plotting the corresponding points of the curve. Notice that the three petals of the curve correspond to the angle intervals [0, π/3], [π/3, 2π/3], and [π/3, π]. Then plot r = sin 3θ in rectangular coordinates and label
(a) Plot the curve r = 1/π−θ for 0 ≤ θ ≤ 2π.(b) With r as in (a), compute the limitsExplain how the limits in (b) show that the curve approaches a horizontal asymptote as θ approaches π, both from the left and from the right. What is the asymptote? lim rcos 0, lim r cos 0, lim r sin 0,
Plot the cissoid r = 2 sin θ tan θ and show that its equation in rectangular coordinates is ܐ = 3 2-x
Prove that r = 2a cos θ is the equation of the circle in Figure 22 using only the fact that a triangle inscribed in a circle with one side a diameter is a right triangle. 0 2a -X
Show that r = a cos θ + b sin θ is the equation of a circle passing through the origin. Express the radius and center (in rectangular coordinates) in terms of a and b and write the equation in rectangular coordinates.
Use the previous exercise to write the equation of the circle of radius 5 and center (3, 4) in the form r = a cos θ + b sin θ.
Use the identity cos 2θ = cos2θ − sin2θ to find a polar equation of the hyperbola x2 − y2 = 1.
Find an equation in rectangular coordinates for the curve r2 = cos 2θ.
Use the addition formula for the cosine to show that the line L with polar equation r cos(θ − α) = d has the equation in rectangular coordinates (cos α)x + (sin α)y = d. Show that L has slope m = − cot α and y-intercept d/sin α.
Find an equation in polar coordinates of the line L with the given description.The point on L closest to the origin has polar coordinates 2, π9.
Find an equation in polar coordinates of the line L with the given description.The point on L closest to the origin has rectangular coordinates (−2, 2).
Show that every line that does not pass through the origin has a polar equation of the formwhere b ≠ 0. r = b sin 8-a cos
Show that cos 3θ = cos3 θ − 3 cos θ sin2 θ and use this identity to find an equation in rectangular coordinates for the curve r = cos 3θ.
By the Law of Cosines, the distance d between two points (Figure 23) with polar coordinates (r, θ) and (r0, θ0) isUse this distance formula to show thatis the equation of the circle of radius 9 whose center has polar coordinates (5, π/4). d² = r² + r-2rro cos(8 - 00)
Show that a polar curve r =ƒ(θ) has parametric equations x = f(0) cos 0, Then apply Theorem 1 of Section 11.1 to prove dy dx where f'(0) = df/de. = y = f(0) sin 0 f(0) cos 0 + f'(0) sin -f(0) sin 0+ f'(0) cos 0
For a > 0, a lemniscate curve is the set of points P such that the product of the distances from P to (a, 0) and (−a, 0) is a2. Show that the equation of the lemniscate is (x 0. + y2)2 = 2a2(x2 − y2)Then find the equation in polar coordinates. To obtain the simplest form of the equation, use
Let c be a fixed constant. Explain the relationship between the graphs of:(a) y = ƒ(x + c) and y = ƒ(x) (rectangular).(b) r = ƒ(θ + c) and r = ƒ(θ) (polar).(c) y = ƒ(x) + c and y = ƒ(x) (rectangular).(d) r = ƒ(θ) + c and r = ƒ(θ) (polar).
Use Eq. (2) to find the slope of the tangent line to r = sin θ at θ = π/3. dy dx f(0) cose + f'(0) sin -f(0) sin 0 + f'(0) cos0
Use Eq. (2) to find the slope of the tangent line to r = θ at θ = π/2 and θ = π. dy dx f(0) cos 6 + f'(0) sin -f(0) sin 0 + f'(0) cos 0
Find the polar coordinates of the points on the lemniscate r2 = cos 2θ in Figure 24 where the tangent line is horizontal. r² = cos (20) ·x
Find the equation in rectangular coordinates of the tangent line to r = 4 cos 3θ at θ = π/6.
Find the polar coordinates of the points on the cardioid r = 1 + cos θ where the tangent line is horizontal (see Figure 25(A)). 1; +X 1 2 3 (A) r= 1 + cos (
Use Eq. (2) to show that for r = sin θ + cos θ,Then calculate the slopes of the tangent lines at points A, B,C in Figure 20. dy dx || cos 20 + sin 20 cos 20-sin 20
Use a graphing utility to convince yourself that the polar equations r = ƒ1(θ) = 2 cos θ − 1 and r = ƒ2(θ) = 2 cos θ + 1 have the same graph. Then explain why. Show that the points (ƒ1(θ + π), θ + π) and (ƒ2(θ), θ) coincide.
We investigate how the shape of the limac¸on curve r = b + cos θ depends on the constant b (see Figure 25).(a) Argue as in Exercise 63 to show that the constants b and −b yield the same curve.(b) Plot the limac¸on for b = 0, 0.2, 0.5, 0.8, 1 and describe how the curve changes.(c) Plot the
In Exercises 17 and 18, let y(t) be a solution of the logistic equationwhere A > 0 and k > 0.(a) Differentiate Eq. (9) with respect to t and use the Chain Rule to show that dt = ky(1 A
Find the general solution of the first-order linear differential equation.ex y'= 1 + 2exy

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