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mathematics
calculus 4th

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

Prove that the length of a curve as computed using the arc length integral does not depend on its parametrization. More precisely, let C be the curve traced by r(t) for a ≤ t ≤ b. Let ƒ (s) be a differentiable function such that ƒ(s) > 0 and ƒ(c) = a and ƒ(d) = b. Then r1(s) = r(ƒ(s))
The curve known as the Bernoulli spiral (Figure 7) has parametrization r(t) = (et cos 4t, et sin 4t). (a) Evaluate s(1) = ||' (u) du. It is convenient to take lower limit -co because _lim_r(1) = (0,0). -8 (b) Use (a) to find an arc length parametrization of the spiral. -10 t = 2 t=0 20 X
Express the arc length s of y = x3 for 0 ≤ x ≤ 8 as an integral in two ways, using the parametrizations r1(t) =(t, t3) and r2(t) = (t3, t9). Do not evaluate the integrals, but use substitution to show that they yield the same result.
Find an arc length parametrization of the line y = mx for an arbitrary slope m.
Find an arc length parametrization of the cycloid with parametrization r(t) = t − sin t, 1 − cos t.
Find an arc length parametrization of the curve parametrized by r(t) =(t2, t3).
Find an arc length parametrization of the curve parametrized by r(t) = (et sin t, et cos t, et).
Find an arc length parametrization of the curve parametrized by with the parameter s measuring from (1, 0, 0). r(t)= (cost, sint, 13/2)
Find an arc length parametrization of the curve parametrized by ,with the parameters measuring from (0, 0, 0). r(1) = (1, 32, = 3/2
Find a path that traces the circle in the plane y = 10 with radius 4 and center (2, 10, −3) with constant speed 8.
Prove the Chain Rule for vector-valued functions.
Use FTC I from single-variable calculus to prove the first part of the Fundamental Theorem of Calculus for Vector-Valued Functions.
Two paths r1(t) and r2(t) intersect if there is a point P lying on both curves. We say that r1(t) and r2(t) collide if r1(t0) = r2(t0) at some time t0.Which of the following statements are true?(a) If r1(t) and r2(t) intersect, then they collide.(b) If r1(t) and r2(t) collide, then they
The sun revolves around the center of mass of the Milky Way galaxy in an orbit that is approximately circular, of radius a ≈ 2.8 × 1017 km and velocity v ≈ 250 km/s. Use the result of Exercise 2 to estimate the mass of the portion of the Milky Way inside the sun’s orbit (place all of this
The perihelion and aphelion are the points on the orbit closest to and farthest from the sun, respectively (Figure 8). The distance from the sun at the perihelion is denoted rper and the speed at this point is denoted vper. Similarly, we write rap and vap for the distance and speed at the aphelion.
The perihelion and aphelion are the points on the orbit closest to and farthest from the sun, respectively (Figure 8). The distance from the sun at the perihelion is denoted rper and the speed at this point is denoted vper. Similarly, we write rap and vap for the distance and speed at the aphelion.
The perihelion and aphelion are the points on the orbit closest to and farthest from the sun, respectively (Figure 8). The distance from the sun at the perihelion is denoted rper and the speed at this point is denoted vper. Similarly, we write rap and vap for the distance and speed at the aphelion.
The perihelion and aphelion are the points on the orbit closest to and farthest from the sun, respectively (Figure 8). The distance from the sun at the perihelion is denoted rper and the speed at this point is denoted vper. Similarly, we write rap and vap for the distance and speed at the aphelion.
Show that a planet in an elliptical orbit has total mechanical energy E = −GMm/2a, where a is the semimajor axis. Use Exercise 19 to compute the total energy at the perihelion.Data From Exercise 19The perihelion and aphelion are the points on the orbit closest to and farthest from the sun,
Prove that at any point on an elliptical orbit, where r = ΙΙrΙΙ, v is the velocity, and a is the semimajor axis of the orbit. M ( 7 - 1/) v = GM
Two space shuttles A and B orbit the earth along the solid trajectory in Figure 9. Hoping to catch up to B, the pilot of A applies a forward thrust to increase her shuttle’s kinetic energy. Use Exercise 20 to show that shuttle A will move off into a larger orbit as shown in the figure. Then use
Kepler’s Third Law. Figure 10 shows an elliptical orbit with polar equation r = P 1 + e cos 0
Kepler’s Third Law. Figure 10 shows an elliptical orbit with polar equation r = P 1 + e cos 0
According to Eq. (7), the velocity vector of a planet as a function of the angle θ is (2 Use this to explain the following statement: As a planet revolves around the sun, its velocity vector traces out a circle of radius k/J with its center at the terminal point of c (Figure 11). This beautiful
A particle located at (1, 1, 0) at time t = 0 follows a path whose velocity vector is v(t) = (1, t, 2t2). Find the particle’s location at t = 2.
Find the vector-valued function r(t) = (x(t), y(t)) in R2 satisfying r (t) = −r(t) with initial conditions r(0) = (1, 2).
Calculate r(t), assuming that r"(t) = (4 - 16t, 12t-1), r' (0) = (1, 0), r(0) = (0, 1)
Solve subject to the initial conditions r(0) = (1, 0, 0) and r '(0) = (−1, 1, 0) r' (t) = (-1,t + 1,
Compute the length of the path r(t) = (sin 2t, cos 2t, 3t-1) for 1 t3
Express the length of the path r(t) = (ln t, t, et) for 1 ≤ t ≤ 2 as a definite integral, and use a computer algebra system to find its value to two decimal places.
Find an arc length parametrization of a helix of height 20 cm that makes four full rotations over a circle of radius 5 cm.
Find the minimum speed of a particle with trajectory r(t) = (t, et−3, e4−t).
A projectile fired at an angle of 60° lands 400 m away. What was its initial speed?
A specially trained mouse runs counterclockwise in a circle of radius 0.6 m on the floor of an elevator with speed 0.3 m/s, while the elevator ascends from ground level (along the z-axis) at a speed of 12 m/s. Find the mouse’s acceleration vector as a function of time. Assume that the circle is
During a short time interval [0.5, 1.5], the path of an unmanned spy plane is described byA laser is fired (in the tangential direction) toward the yz-plane at time t = 1. Which point in the yz-plane does the laser beam hit? =(-100,7-1,40 140-2) r(t) =
A force F = (12t + 4, 8 − 24t) (in newtons) acts on a 2-kg mass. Find the position of the mass at t = 2 s if it is located at (4, 6) at t = 0 and has initial velocity (2, 3) in meters per second.
Find the unit tangent vector to r(t) = (sin t, t, cos t) at t = π.
Find the unit tangent vector to r(t) = (t2, tan−1 t, t) at t = 1.
A particle follows a path r(t) for 0 ≤ t ≤ T, beginning at the origin O. The vector called the average velocity vector. Suppose that v̅ = 0. Answer and explain the following:(a) Where is the particle located at time T if v̅ = 0?(b) Is the particle’s average speed necessarily equal to
A force F = (24t, 16 − 8t) (in newtons) acts on a 4-kg mass. Find the position of the mass at t = 3 s if it is located at (10, 12) at t = 0 and has zero initial velocity.
A constant force F = (5, 2) (in newtons) acts on a 10-kg mass. Find the position of the mass at t = 10 seconds if it is located at the origin at t = 0 and has initial velocity v0 = (2, −3) (in meters per second).
A soccer ball is kicked from ground level with (x, y)-coordinates (85, 20) on the soccer field shown in Figure 11 and with an initial velocity v0 = 10i − 5j + 25k ft/s. Assume an acceleration of 32 ft/s2 due to gravity and that the goal net has a height of 8 ft and a total width of 24 ft.(a)
Show that a projectile launched at an angle θ will hit the top of an h-meter tower located d meters away if its initial speed is Vo = g/2d sec 0 dtan 0-h
Show that a projectile launched at an angle θ with initial speed v0 travels a distance (v20/g) sin 2θ before hitting the ground. Conclude that the maximum distance (for a given v0) is attained at θ = π/4.
Golfer Judy Robinson hit a golf ball on the planet Priplanus with an initial speed of 50 m/s at an angle of 40°. It landed exactly 2 km away. What is the acceleration due to gravity on Priplanus?
Assume that astronaut Alan Shepard hit his golf shot on the moon (acceleration due to gravity = 1.6 m/s2) with a modest initial speed of 35 m/s at an angle of 30°. How far did the ball travel?
Find the initial velocity vector v0 of a projectile released with initial speed 100 m/s that reaches a maximum height of 300 m.
A projectile is launched from the ground at an angle of 45°. What initial speed must the projectile have in order to hit the top of a 120-m tower located 180 m away?
Find r(t) and v(t) given a(t) and the initial velocity and position.a(t) = cos tk, v(0) = i − j, r(0) = i
Find r(t) and v(t) given a(t) and the initial velocity and position.a(t) = tk, v(0) = i, r(0) = j
Use Eq. (5) to compute the curvature at the given point.(cosh s, s), s = 0 k(t): [x' (1)y"(t)- y' (t)x" (1)| (x'(1) + y(1)2) /2
Use Eq. (5) to compute the curvature at the given point. k(t): [x'(1)y"(t)- y' (t)x" (1)| (x'(1) + y'(1)2) /2
Use Eq. (5) and a parametrization x = t and y = ƒ(t) to prove that the curvature of the graph of y = ƒ(x) is given by K(X) = [f"(x)] (1 + f'(x)) 3/2
In the notation of Exercise 25, assume that a ≥ b. Show that b/a2 ≤ κ(t) ≤ a/b2 for all t.Data From Exercise 25Show that the curvature function of the parametrization r(t) = (a cos t, b sin t) of the ellipse (-). () + = 1 is
Use a sketch to predict where the points of minimal and maximal curvature occur on an ellipse. Then use Eq. (11) to confirm or refute your prediction. k(t) ab (b cos 1 + a sin1)3/2
Find the point of maximum curvature on y = ex.
Find the value(s) of α such that the curvature of y = eαx at x = 0 is as large as possible.
Show that curvature at an inflection point of a plane curve y = ƒ(x) is zero.
Show that the tractrix r(t) = (t − tanh t, sech t) has the curvature function κ(t) = sech t.
Find the curvature function κ(x) for y = sin x. Use a computer algebra system to plot κ(x) for 0 ≤ x ≤ 2π. Prove that the curvature takes its maximum at x = π/2 and 3π/2. As a shortcut to finding the max, observe that the maximum of the numerator and the minimum of the denominator of
Find the curvature of r(t) = (2 sin t, cos 3t, t) at t = π/3 and t = π/2.
Find an arc length parametrization of the circle in the plane z = 9 with radius 4 and center (1, 4, 9).
Let r(t) = w + tv be a parametrization of a line. (a) Show that the arc length function s(1) = [[|1r (1) du is given by s(1): t||v||. This shows that r(t) is an arc length parametrizaton if and only if v is a unit vector. (b) Find an arc length parametrization of the line with w = (1,2,3) and v =
Find an arc length parametrization of the line y = 4x + 9.
Let r(t) =(3t + 1, 4t − 5, 2t). (a) Evaluate the arc length integral s(1) = f( ||~ (1)|| du. (b) Find the inverse g(s) of s(t). (c) Verify that r(s) = r(g(s)) is an arc length parametrization.
Which of the following is an arc length parametrization of a circle of radius 4 centered at the origin? (a) r(t) = (4 sint, 4 cost) (b) r(t) =(4 sin 4t, 4 cos 4t) (c) r3(t) = (4 sin 1,4 cos 1)
The cycloid generated by the unit circle has parametrization r(t) = (t sint, 1 - cost) (a) Find the value of t in [0,27] where the speed is at a maximum. (b) Show that one arch of the cycloid has length 8. Recall the identity sin(1/2) = (1 - cost)/2.
Let(a) Show that r(t) parametrizes a helix of radius R and height h making N complete turns.(b) Guess which of the two springs in Figure 6 uses more wire.(c) Compute the lengths of the two springs and compare. 2Nt 2Nt r(1) = ( R cos (2#N"). R sin ( 2RN).s). h h 0th
The DNA molecule comes in the form of a double helix, meaning two helices that wrap around one another. Suppose a single one of the helices has a radius of 10Å (1 angstrom Å = 10−8 cm) and one full turn of the helix has a height of 34Å. 34t (a) Show that the helix can be parametrized by r(1) =
A bee with velocity vector r (t) starts out at the origin at t = 0 and flies around for T seconds. Where is the bee located at time Tif if for r' (u) du 0? What does the quantity = ||r' (u)|| du represent?
What is the velocity vector of a particle traveling to the right along the hyperbola y = x−1 with constant speed 5 cm/s when the particle’s location is (2, 1/2) ?
At an air show, a jet has a trajectory following the curve y = x2. If when the jet is at the point (1, 1), it has a speed of 500 km/h, determine its velocity vector at this point.
Find the speed at the given value of t. r(t) = (cosht, sinh t, t), t = 0
Kepler’s Third Law states that T2/a3 has the same value for each planetary orbit. Do the data in the following table support this conclusion? Estimate the length of Jupiter’s period, assuming that a = 77.8 × 1010 m. Planet a (10 m) T (years) Mercury 5.79 0.241 Venus Earth Mars 10.8 15.0 22.8
Using Kepler’s Third Law, show that if a planet revolves around a star with period T and semimajor axis a, then the mass of the star is M = 3 (+7) () G
Ganymede, one of Jupiter’s moons discovered by Galileo, has an orbital period of 7.154 days and a semimajor axis of 1.07 × 109 m. Use Exercise 2 to estimate the mass of Jupiter.Data From Exercise 2Using Kepler’s Third Law, show that if a planet revolves around a star with period T and
An astronomer observes a planet orbiting a star with a period of 9.5 years and a semimajor axis of 3 × 108 km. Find the mass of the star using Exercise 2.Data From Exercise 2 Using Kepler’s Third Law, show that if a planet revolves around a star with period T and semimajor axis a, then the mass
A satellite orbiting above the equator of the earth is geosynchronous if the period is T = 24 hours (in this case, the satellite stays over a fixed point on the equator). Use Kepler’s Third Law to show that in a circular geosynchronous orbit, the distance from the center of the earth is R ≈
Show that a planet in a circular orbit travels at constant speed. Use the facts that J is constant and that r(t) is orthogonal to'(t) for a circular orbit.
Verify that the circular orbit r(t) = (R cos wt, R sin wt)
Prove that if a planetary orbit is circular of radius R, then vT = 2πR, where v is the planet’s speed (constant by Exercise 7) and T is the period. Then use Kepler’s Third Law to prove that Data From Exercise 7Show that a planet in a circular orbit travels at constant speed. Use the facts that
Find the velocity of a satellite in geosynchronous orbit about the earth. Use Exercises 6 and 9.Data From Exercise 6A satellite orbiting above the equator of the earth is geosynchronous if the period is T = 24 hours (in this case, the satellite stays over a fixed point on the equator). Use
A communications satellite orbiting the earth has initial position r(0) = (29,000, 20,000, 0) (in kilometers) and initial velocity r'(0) = (1, 1, 1) (in kilometers per second), where the origin is the earth’s center. Find the equation of the plane containing the satellite’s orbit. Hint: This
Assume that the earth’s orbit is circular of radius R = 150 × 106 km (it is nearly circular with eccentricity e = 0.017). Find the rate at which the earth’s radial vector sweeps out area in units of square kilometers per second. What is the magnitude of the vector J = r × r for the earth (in
The perihelion and aphelion are the points on the orbit closest to and farthest from the sun, respectively (Figure 8). The distance from the sun at the perihelion is denoted rper and the speed at this point is denoted vper. Similarly, we write rap and vap for the distance and speed at the aphelion.
The perihelion and aphelion are the points on the orbit closest to and farthest from the sun, respectively (Figure 8). The distance from the sun at the perihelion is denoted rper and the speed at this point is denoted vper. Similarly, we write rap and vap for the distance and speed at the aphelion.
The perihelion and aphelion are the points on the orbit closest to and farthest from the sun, respectively (Figure 8). The distance from the sun at the perihelion is denoted rper and the speed at this point is denoted vper. Similarly, we write rap and vap for the distance and speed at the aphelion.
Determine the domains of the vector-valued functions. (a) r(t) = (t, (t+1)-, sint) (b) r(t) =(8-1, Int, e Vi)
Sketch the paths r1(θ) = (θ, cos θ) and r2(θ) = (cos θ, θ) in the xy-plane.
Find a vector parametrization of the intersection of the surfaces x2 + y4 + 2z3 = 6 and x = y2 in R3.
Find a vector parametrization using trigonometric functions of the intersection of the plane x + y + z = 1 and the elliptical cylinder (5) () + 3 8 = 1 in R.
Calculate the derivative at t = 3, assuming that r(3) = (1, 1,0), r(3) = (0, 0, 1), r2(3) = (1, 1,0) r(3) = (0, 2, 4)
Calculate the derivative at t = 3, assuming that r(3)= (1, 1,0), r(3) = (0, 0, 1), r2(3)= (1, 1,0) r(3) = (0,2, 4)
Calculate the derivative at t = 3, assuming that r(3)= (1, 1,0), r(3) = (0, 0, 1), r2(3) = (1, 1, 0) r(3) = (0,2, 4)
Calculate the derivative at t = 3, assuming that r(3)= (1, 1,0), r(3) = (0, 0, 1), r2(3) = (1, 1,0) r(3) = (0, 2, 4)
Calculate ∫30 (4t + 3, t2, − 4t3) dt.
Calculate∫π0 (sin θ, θ, cos 2θ) dθ.
How is the period T affected if the semimajor axis a is increased 4-fold?
Eq. (1) shows that r is proportional to r. Explain how this fact is used to prove Kepler’s Second Law. v(1) | = face a(t) dt = (sin t)i + e'j+k+ Co
Describe the relation between the vector J = r × r and the rate at which the radial vector sweeps out area.

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