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mathematics
calculus 4th

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

Find the vertices and foci of the conic section. 2 x-4 (ਜਾਂ) 3 + 3 6 5 2 = 1
Find the area of the shaded region in Figure 12. Note that θ varies from 0 to π/2. 8 r= 0² +40 2 -X
(a) Describe the path of an ant that is crawling along the plane according to c1(t) = (ƒ(t), ƒ(t)), where ƒ(t) is an increasing function.(b) Compare that path to the path of a second ant crawling according to c2(t) = ƒ(2t), ƒ(2t).
If the straight line segment given by c(t) = (t, 3) for 0 ≤ t ≤ 2 is rotated around the x-axis, what surface area results?
Find the vertices and foci of the conic section. x-4 5 2 + (y + 3) 6 2 || 1
Find three different parametrizations of the graph of y = x3.
Which interval of θ-values corresponds to the shaded region in Figure 13? Find the area of the region. 2- r=3-0 3 X+
Find the equation of the ellipse obtained by translating (as indicated) the ellipseTranslated with center at the origin -8 6 + (y+4) 3 2 = 1
Match the derivatives with a verbal description:(a) dx/dt(b) dy/dt(c) dy/dx(i) Slope of the tangent line to the curve(ii) Vertical rate of change with respect to time(iii) Horizontal rate of change with respect to time
Find the total area enclosed by the cardioid in Figure 14. -2 -1 -X
Find the area of the shaded region in Figure 14. -2 -X
Find the equation of the ellipse obtained by translating (as indicated) the ellipseTranslated with center at (−2, −12) -8 6 + (y+4) 3 2 = 1
Find the area of one leaf of the “four-petaled rose” r = sin 2θ (Figure 15). Then prove that the total area of the rose is equal to one-half the area of the circumscribed circle. y r = sin 20 & -X
Find the equation of the ellipse obtained by translating (as indicated) the ellipseTranslated to the right 6 units -8 6 + (y+4) 3 2 = 1
Find the area enclosed by one loop of the lemniscate with equation r2 = cos 2θ (Figure 16). Choose your limits of integration carefully. - 1 -X
Find the equation of the ellipse obtained by translating (as indicated) the ellipseTranslated down 4 units -8 6 + (y+4) 3 2 = 1
Find the area of the intersection of the circles r = 2 sin θ and r = 2 cos θ.
Find the area of the shaded region in Figure 18 enclosed by the circle r = 12 and a petal of the curve r = cos 3θ. Compute the area of both the petal and the region inside the petal and outside the circle. r 11 r = cos 30 X
Find the area of region A in Figure 17. r=1 A 1 2 r = 4 cos 0 14 -X
Find the equation of the given ellipse.Vertices (±3, 0) and (0, ±5)
Find the area of the intersection of the circles r = sin θ and r = √3 cos θ.
Find the equation of the given ellipse.Foci (±6, 0) and focal vertices (±10, 0)
Find the equation of the given ellipse.Foci (0, ±10) and eccentricity e = 3/5
Find the equation of the given ellipse.Vertices (4, 0), (28, 0) and eccentricity e = 2/3
Use Eqs. (1) and (3) to find the moments and center of mass of the lamina S of constant density ρ = 2 g/cm2 occupying the region between y = x2 and y = 9x over [0, 3]. Sketch S , indicating the location of the center of mass. My = p of*x (length of vertical cut) dx = p a = p [₁² x ( a x(fi(x)
Find the centroid of the region lying underneath the graph of the function over the given interval.ƒ(x) = 4x, [0, 1]
Find the centroid of the region lying underneath the graph of the function over the give interval.ƒ(x) = 6 − 2x, [0, 3]
Calculate the fluid force on one side of the plate (an isosceles triangle) shown in Figure 15(B). Water surface f(y). 3 60⁰ (B) y 10 Vertical change Ay
Find the centroid of the region.Semicircle of radius r with center at the origin
Use Separation of Variables to find the general solution. dy dt 20te = 0
Which of the following differential equations is a logistic differential equation? dy dt dy (c) = dt (a) = 2y(1 - y²) = 2y (1-4) (b) (d) dt dy dt = 2y (1 - 1) 3 = = 2y(1-3y)
Find the general solution of the logistic equationThen find the particular solution satisfying y(0) = 2. dy dt = 3x(1-5)
Verify that the given function is a solution of the differential equation. y' - 8x = 0, y = 4x²
Write a solution to y' = 4(y − 5) that tends to − ∞ as t → ∞.
Which of the following are first-order linear equations? (a) y + x²y = 1 (c) x³y² + y = ex (b) y + xy² = 1 (d) x³y + y = el
Figure 9 shows the slope field for dy/dt = sin y sin t. Sketch the graphs of the solutions with initial conditions y(0) = 1 and y(0) = −1. Show that y(t) = 0 is a solution and add its graph to the plot. 3 2 1 0 -1 -2 -3 بنا y -3 -2 -1 0 1 2 3
Which of the following differential equations are linear? Determine the order of each equation. (a) y' = y³ - 3x^y (c) y=y"" - 3x √y (b) y = x³ - 3x¹y (d) sinx y'=y-1
Find the general solution of y' = 2(y − 10). Then find the two solutions satisfying y(0) = 25 and y(0) = 5, and sketch their graphs.
Consider y'+ x−1y = x3.(a) Verify that α(x) = x is an integrating factor.(b) Show that when multiplied by α(x), the differential equation can be written (xy)' = x4.(c) Conclude that xy is an antiderivative of x4 and use this information to find the general solution.(d) Find the particular
What is the slope of the segment in the slope field for dy/dt = ty + 1 at the point (2, 3)?
What are the constant solutions to dy/dt = ky(1 − y/A)?
Find the solution of dy/dt = 2y(3 − y), y(0) = 10.
Verify that the given function is a solution of the differential equation.yy' + 4x = 0, y = √12 − 4x2
Does y' = −4(y − 5) have a solution that tends to ∞ as t → ∞?
If α(x) is an integrating factor for y'+ A(x)y = B(x), then α'(x) is equal to (choose the correct answer):(a) B(x) (b) α(x)A(x)(c) α(x)A'(x) (d) α(x)B(x)
Consider dy/dt + 2y = e−3t.(a) Verify that α(t) = e2t is an integrating factor.(b) Use Eq. (4) to find the general solution.(c) Find the particular solution with initial condition y(0) = 1. 1 y = -a(x) (fa(x)Q(x) dx)
Verify directly that y = 12 + Ce−3t satisfies y' = −3(y − 12) for all C. Then find the two solutions satisfying y(0) = 20 and y(0) = 0, and sketch their graphs.
Figure 10 shows the slope field for dy/dt = y2 − t2. Sketch the integral curve passing through the point (0, −1), the curve through (0, 0), and the curve through (0, 2). Is y(t) = 0 a solution? 3 2 1 0 ترا -1 -2 -3 11\// T -3-2-1 y 1 1117 11117- 0 1 2 3 t $3 2 1 0 -1 -2 -3 11\// T -3-2-1 y 1 1117 11117- 0 1 2 3 t$
What is the equation of the isocline of slope c = 1 for dy/dt = y2 − t?
For each of (a)–(c), give the solution y(t) satisfying the initial condition. The general solution formula for the logistic equation, Eq. (5), applies when a solution is not constant.(a) y(0) = 6 (b) y(0) = 4 (c) y(4) = 0 dy dt = ky y = A 1- e-ki/B
Find a value of c such that y = x − 2 + ecx is a solution of 2y' + y = x.
Is the logistic equation separable?
Verify that the given function is a solution of the differential equation.y' + 4xy = 0, y = 25e−2x2
True or false? If k > 0, then all solutions of y' = −k(y − b) approach the same limit as t →∞.
For what function P is the integrating factor α(x) equal to x?
Solve y '= 4y + 24 subject to y(0) = 5.
For which of the following differential equations are the slopes at points on a vertical line t = C all equal? (a) dy dt = In y (b) dy dt = In t
Let α(x) = ex2. Verify the identity (α (x) y)'= α(x)(y' + 2xy) and explain how it is used to find the general solution ofy'+ 2xy = x
Show that ƒ(t) = 1/2 (t − 1/2) is a solution to dy/dt = t − 2y. Sketch the four solutions with y(0) = ±0.5, ±1 on the slope field in Figure 11. The slope field suggests that every solution approaches ƒ(t) as t → ∞. Confirm this by showing that y = ƒ(t) + Ce−2t is the general
Solve using Separation of Variables. dy = 1²y-³ dt
For each of (a)–(c), give the solution y(t) satisfying the initial condition. The general solution formula for the logistic equation, Eq. (5), applies when a solution is not constant.(a) y(0) = 6 (b) y(0) = 8(c) y(0) = −2 dy dt = ky y = A 1- e-k¹/B
Verify that the given function is a solution of the differential equation.(x2 − 1)y' + xy = 0, y = 4(x2 − 1)−1/2
As an object cools, its rate of cooling slows. Explain how this follows from Newton’s Law of Cooling.
For what function P is the integrating factor α(x) equal to ex?
Solve y' + 6y = 12 subject to y(2) = 10.
Find the solution of y' − y = e2x, y(0) = 1.
One of the slope fields in Figures 12(A) and (B) is the slope field for dy/dt = t2. The other is for dy/dt = y2. Identify which is which. In each case, sketch the solutions with initial conditions y(0) = 1, y(0) = 0, and y(0) = −1. 2 1 0 ندی -3 -2 -1 0 (A) 1 2 3 + 3 2 1 0 -2 -1 0 (B) 1 2 3
Let y(t) be the solution to dy/dt = F(t, y) with y(1) = 3. How many iterations of Euler’s Method are required to approximate y(3) if the time step is h = 0.1?
Solve using Separation of Variables. dy x - 2y = 3 dx
Solve using Separation of Variables.xyy' = 1 − x2
A population of squirrels lives in a forest with a carrying capacity of 2000. Assume logistic growth with growth constant k = 0.6 yr−1.(a) Find a formula for the squirrel population P(t), assuming an initial population of 500 squirrels.(b) How long will it take for the squirrel population to
Verify that the given function is a solution of the differential equation.y" − 2xy + 8y = 0, y = 4x4 − 12x2 + 3
A hot anvil with cooling constant k = 0.02 s−1 is submerged in a large pool of water whose temperature is 10°C. Let y(t) be the anvil’s temperature t seconds later.(a) What is the differential equation satisfied by y(t)?(b) Find a formula for y(t), assuming the object’s initial temperature
Consider the differential equation dy/dt = t − y. (a) Sketch the slope field of the differential equation =ty in the range -1 ≤ ≤ 3, -1 ≤ y ≤ 3. As an dy dt aid, observe that the isocline of slope c is the line t - y = c, so the segments have slope c at points on the line y = t -
Find the general solution of the first-order linear differential equation.xy' + y = x
Solve using Separation of Variables. y' xy² 1 I + zx
The population P(t) of mosquito larvae growing in a tree hole increases according to the logistic equation with growth constant k = 0.3 day−1 and carrying capacity A = 500.(a) Find a formula for the larvae population P(t), assuming an initial population of P0 = 50 larvae.(b) After how many days
Verify that the given function is a solution of the differential equation.y" − 2y' + 5y = 0, y = ex sin 2x
Frank’s automobile engine runs at 100◦C. On a day when the outside temperature is 21°C, he turns off the ignition and notes that 5 minutes later, the engine has cooled to 70°C.(a) Determine the engine’s cooling constant k.(b) What is the formula for y(t)?(c) When will the engine cool to
Find the general solution of the first-order linear differential equation.xy' − y = x2 − x
Show that the isoclines of dy/dt = 1/y are horizontal lines. Sketch the slope field for −2 ≤ t ≤ 2, −2 ≤ y ≤ 2 and plot the solutions with initial conditions y(0) = 0 and y(0) = 1.
Solve the Initial Value Problem using Separation of Variables. y = cos²x, y(0) = KIT π 4
Sunset Lake is stocked with 2000 rainbow trout, and after 1 year the population has grown to 4500. Assuming logistic growth with a carrying capacity of 20000, find the growth constant k (specify the units) and determine when the population will increase to 10000.
The following differential equations appear similar but have very different solutions:Solve both subject to the initial condition y(1) = −1. dy xp = 0, dy dx = 0.001
At 10:30 am, detectives discover a dead body in a room and measure its temperature at 26°C. One hour later, the body’s temperature had dropped to 24.8°C. Determine the time of death (when the body temperature was a normal 37°C), assuming that the temperature in the room was held constant at
Find the general solution of the first-order linear differential equation.3xy' − y = x−1
Sketch the slope field for dy/dt = y + t for −2 ≤ t ≤ 2, −2 ≤ y ≤ 2.
Solve the Initial Value Problem using Separation of Variables. y' = cos y, y(0) = π 4
A rumor spreads through a small town. Let y(t) be the fraction of the population that has heard the rumor at time t and assume that the rate at which the rumor spreads is proportional to the product of the fraction y of the population that has heard the rumor and the fraction 1 − y that has not
The following differential equations appear similar but have very different solutions:Solve both subject to the initial condition y(1) = 2. dy dx = X, dy dx = y
A cup of coffee with cooling constant k = 0.09 min−1 is placed in a room at temperature 20°C.(a) How fast is the coffee cooling (in degrees per minute) when its temperature is T = 80°C?(b) Use the Linear Approximation to estimate the change in temperature over the next 6 s when T = 80°C.(c) If
Find the general solution of the first-order linear differential equation.y' + xy = x
Sketch the slope field for dy/dt = t/y for −2 ≤ t ≤ 2, −2 ≤ y ≤ 2.
A rumor spreads through a school with 1000 students. At 8 am, 80 students have heard the rumor, and by noon, half the school has heard it. Using the logistic model of Exercise 8, determine when 90% of the students will have heard the rumor.
Verify that x2y' + e−y = 0 is separable. (a) Write it as e dy = -x²² dx. (b) Integrate both sides to obtain e = x¹ + C. (c) Verify that y = ln(x¹+C) is the general solution. (d) Find the particular solution satisfying y(2) = 4.
Solve the Initial Value Problem using Separation of Variables.y' = 6xy2, y(1) = 4
Find the general solution of the first-order linear differential equation.y' + 3x−1y = x + x−1
Show that the isoclines of dy/dt = t are vertical lines. Sketch the slope field for −2 ≤ t ≤ 2, −2 ≤ y ≤ 2 and plot the integral curves passing through (0, −1) and (0, 1).
A simpler model for the spread of a rumor assumes that the rate at which the rumor spreads is proportional (with factor k) to the fraction of the population that has not yet heard the rumor.(a) Compute the solutions to this model and the model of Exercise 8 with the values k = 0.9 and y0 = 0.1.(b)
Consider the differential equation y3y' − 9x2 = 0. (a) Write it as y³ dy = 9x² dx. (b) Integrate both sides to obtain y4 = 3x³ + C. (c) Verify that y = (12x³ + C)¹/4 is the general solution. (d) Find the particular solution satisfying y(1) = 2.

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