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study help
mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
Describe the set of all points P = (x, y, z) satisfying x2 + y2 ≤ 4 in both cylindrical and spherical coordinates.
Convert the point (ρ, θ, ϕ) = (3, π/6, π/3) from spherical to cylindrical coordinates.
Convert (r, θ, z) = (3, π/6, 4) from cylindrical to spherical coordinates.
Convert (x, y, z) = (3, 4, −1) from rectangular to cylindrical and spherical coordinates.
Describe the traces of the surfacein the three coordinate planes. (H) + z-2 = 1
Determine the type of the quadric surface ax2 + by2 − z2 = 1 if:(a) a < 0, b < 0(b) a > 0, b > 0(c) a > 0, b < 0
Determine the type of the quadric surface. ()()-22 = 1 3
Determine the type of the quadric surface. () - () - 2 = 0
Determine the type of the quadric surface. 2 (5) - () - 3 - 2z = 0
Determine the type of the quadric surface. () + () - - 2z = 0
Determine the type of the quadric surface. (5) - ()- + 2z = 1
Determine the type of the quadric surface. 2 2 (37) + ( ) + + 2z = 1
Find the intersection of the planes x + y + z = 1 and 3x − 2y + z = 5.
Find the trace of the plane 3x − 2y + 5z = 4 in the xy-plane.
Find the intersection of the line r(t) = (3t + 2, 1, −7t) and the plane 2x − 3y + z = 5.
Find the plane through P = (4, −1, 9) containing the line r(t) = (1, 4, −3) + t(2, 1, 1).
Find all the planes parallel to the plane passing through the points (1, 2, 3), (1, 2, 7), and (1, 1, −3).
Write the equation of the plane P with vector equation (1, 4, −3) · (x, y, z) = 7 in the formYou must find a point P = (x0, y0, z0) on P. a (x xo) + b (y - yo) + c(z-Zo) = 0 -
Find an equation of the plane through (1, −3, 5) with normal vector n = (2, 1, −4).
Use the identityto prove that ux (v x W) = (uw) v (u v) w .
Prove with a diagram the following: If e is a unit vector orthogonal to v, then e × (v × e) = (e × v) × e = v.
Show that the equation (1, 2, 3) × v = (−1, 2, a) has no solution for a ≠ −1.
Find the area of the parallelogram spanned by vectors v and w such that ΙΙvΙΙ = ΙΙwΙΙ = 2 and v · w = 1.
Find ΙΙe − 4fΙΙ, assuming that e and f are unit vectors such that ΙΙe + fΙΙ= √3.
Show that if the vectors v, w are orthogonal, then ΙΙv + wΙΙ2 = ΙΙvΙΙ2 + ΙΙwΙΙ2.
Calculate ΙΙv × wΙΙ if ΙΙvΙΙ = 2, v · w = 3, and the angle between v and w is π/6.
Use the cross product to find the area of the triangle whose vertices are (1, 3, −1), (2, −1, 3), and (4, 1, 1).
We compute the cross product as the following determinant:v · (u × w)
We compute the cross product as the following determinant: u det v W
Let v = (1, 2, 4), u = (6, −1, 2), and w = (1, 0, −3). Calculate the given quantity.w × u
Let v = (1, 2, 4), u = (6, −1, 2), and w = (1, 0, −3). Calculate the given quantity.v × w
Let v, w, and u be the vectors in R3. Which of the following is a scalar?(a) v × (u + w)(b) (u + w) · (v × w)(c) (u × w) + (w − v)
A 50-kg wagon is pulled to the right by a force F1 making an angle of 30° with the ground. At the same time, the wagon is pulled to the left by a horizontal force F2.(a) Find the magnitude of F1 in terms of the magnitude of F2 if the wagon does not move.(b) What is the maximal magnitude of F1 that
Calculate the magnitude of the forces on the two ropes in Figure 2. A 30 Rope 1 P 10 kg 45 B Rope 2
Calculate the component of v = (− 2, 1/2, 3) along w = (1, 2, 2).
Let v = (1, −1, 3) and w = (4, −2, 1). (a) Find the decomposition (b) Find the decomposition v=Vw + Vw with respect to w. w = W + Wiy with respect to v.
Use vectors to prove that the line connecting the midpoints of two sides of a triangle is parallel to the third side.
Find all the vectors orthogonal to both v and w.
Let v = (1, 3, −2 and w = 2, −1, 4).Find the volume of the parallelepiped spanned by v, w, and u = (1, 2, 6).
Let v = (1, 3, −2 and w = 2, −1, 4).Find the area of the parallelogram spanned by v and w.
Let v = (1, 3, −2 and w = 2, −1, 4).Compute v × w.
Let v = (1, 3, −2 and w = 2, −1, 4).Compute the angle between v and w.
Let v = (1, 3, −2 and w = 2, −1, 4).Compute v · w.
Sketch the sums v1 + v2 + v3, v1 + 2v2, and v2 − v3 for the vectors in Figure 1(B). V3 V2 (B) X
Sketch the vector sum v = v1 − v2 + v3 for the vectors in Figure 1(A). V3 (A) -x
Find a such that the lines r1 = (1, 2, 1) + t(1, −1, 1) and r2 = (3, −1, 1) + t(a, 4, −2) intersect.
Find a and b such that the lines r1 = (1, 2, 1) + t(1, −1, 1) and r2 = (3, −1, 1) + t(a, b, −2) are parallel.
Let r1(t) = v1 + tw1 and r2(t) = v2 + tw2 be parametrizations of lines L1 and L2. For each statement (a)–(e), provide a proof if the statement is true and a counterexample if it is false. (a) If L : = (b) If L : = (c) If L = L2, then V V and W = W. L2 and v V2, then w = W. = L2 and W W2, then V =
Find a parametrization r1(t) of the line passing through (1, 4, 5) and (−2, 3, −1). Then find a parametrization r2(t) of the line parallel to r1 passing through (1, 0, 0).
Let v = 3i − j + 4k. Find the length of v and the vector 2v + 3 (4i − k).
Let w = (2, −2, 1) and v = (4, 5, −4). Solve for u if v + 5u = 3w − u.
Let P = (1, 4, −3). (a) Find the point Q such that PO is equivalent to (3,-1,5). (b) Find a unit vector e equivalent to PO.
Find the value of β for which w = (−2, β) is parallel to v = (4, −3).
Calculate 3 (i − 2j) − 6 (i + 6j).
Find the vector with length 3 making an angle of 7π/4 with the positive x-axis.
Let A = (2, −1), B = (1, 4), and P = (2, 3). Find the point Q such that PO is equivalent to AB. Sketch PO and AB.
If P = (1, 4) and Q = (−3, 5), what are the components of Po? What is the length of PO?
Let v = (−2, 5) and w = (3, −2).Find a scalar α such that v + αw = 6.
Let v = (−2, 5) and w = (3, −2).Express i as a linear combination rv + sw.
Let v = (−2, 5) and w = (3, −2).Find the length of v + w.
Let v = (−2, 5) and w = (3, −2).Find the unit vector in the direction of v.
Let v = (−2, 5) and w = (3, −2).Sketch v, w, and 2v − 3w.
Let v = (−2, 5) and w = (3, −2).Calculate 5w − 3v and 5v − 3w.
For which value of ϕ0 is ϕ = ϕ0 a plane? Which plane?
Which statement about cylindrical coordinates is correct?(a) If θ = 0, then P lies on the z-axis.(b) If θ = 0, then P lies in the xz-plane.
Describe the surfaces r = R in cylindrical coordinates and ρ = R in spherical coordinates.
What is the definition of a parabolic cylinder?
A surface is called bounded if there exists M > 0 such that every point on the surface lies at a distance of at most M from the origin. Which of the quadric surfaces are bounded?
Use Exercise 71 to compute the orthogonal distance from the plane x + 2y + 3z = 5 to the origin.Data From Exercise 71In this exercise, we show that the orthogonal distance D from the plane P with equation ax + by + cz = d to the origin O is equal to (Figure 11) D = |d| a + b + c
Exercises 12 and 13 refer to the data in Example 1. Approximate the derivative with the symmetric difference quotient (SDQ) approximation: (a) At what t does the SDQ approximation give the fastest rate of increase of temperature? What is the rate of change?(b) At what t does the SDQ approximation
Evaluate the limit. lim sin 2ti + costj + tan 4tk 1
Evaluate the limit. 1 (F, 41, 7 ) lim
Evaluate the limit. limei+ln(t + 1)j + 4k 1-0
Evaluate the limit. 1 lim 10t+1' e - 1 t 4t
Evaluate the limit. Evaluate lim h0 r(t + h) - r(t) h for r(t) = (t, sint, 4).
Evaluate the limit. r(t) Evaluate lim for r(t) = (sint, 1 - cost, -2t). 1-0 t
Compute the derivative.r(t) = (t, t2, t3)
Compute the derivative.r(t) = (7 − t, 4√t, 8)
Compute the derivative. b(t) = (e-4, e6, (t + 1)-) 6-1
Compute the derivative.r(s) = (e3s, e−s, s4)
Compute the derivative. c(t) = t−1i − e2t k
Compute the derivative.a(θ) = (cos 3θ)i + (sin2 θ)j + (tan θ)k
Calculate r'(t) and r'(t) for r(t) = (t, t2, t3).
Sketch the curve parametrized by r(t) = (1 − t2, t) for −1 ≤ t ≤ 1. Compute the tangent vector at t = 1 and add it to the sketch.
Sketch the curve parametrized by r1(t) =(t, t2) together with its tangent vector at t = 1. Then do the same for r2(t) = (t3, t6).
Sketch the cycloid r(t) = (t − sin t, 1 − cos t) together with its tangent vectors at t = π/3 and 3π/4.
Evaluate the derivative by using the appropriate Product Rule, where r(t) = (,1,t), r(t) = (e, e, e)
Evaluate the derivative by using the appropriate Product Rule, where r(t) = (,1,t), r(t) = (e, e, e)
Determine the value of t between 0 and 2π such that the tangent vector to the cycloid r(t) = (t − sin t, 1 − cos t) is parallel to (√3, 1).
Determine the values of t between 0 and 2π such that the tangent vector to the cycloid r(t) = (t − sin t, 1 − cos t) is a unit vector.
Evaluate the derivative by using the appropriate Product Rule, where r(t) = (,1,t), r(t) = (e, e, e)
Let r1(t) = (t2, 1, 2t), r2(t) = (1, 2, et) d Computer1(1) r2(1) in two ways: dt (a) Calculate r(t) r(t) and differentiate. (b) Use the Dot Product Rule.
Let r1(t) = (t2, 1, 2t), r2(t) = (1, 2, et) d Computer(1) x r() in two ways: (a) Calculate r (t) x r(t) and differentiate. (b) Use the Cross Product Rule.
Evaluate d/dt r(g(t)) using the Chain Rule.r(t) =(t2, 1 − t), g(t) = et
Evaluate d/dt r(g(t)) using the Chain Rule.r(t) =(t2, t3), g(t) = sin t
Evaluate d/dt r(g(t)) using the Chain Rule.r(t) = (et, e2t, 4), g(t) = 4t + 9
Evaluate d/dt r(g(t)) using the Chain Rule.r(t) = (4 sin 2t, 6 cos 2t), g(t) = t2
Let r(t) = (t2, 1 − t, 4t). Calculate the derivative of r(t) · a(t) at t = 2, assuming that a(2) = (1, 3, 3) and a'(2) = (−1, 4, 1).
Let v(s) = s2i + 2sj + 9s−2k. Evaluate d/ds v(g(s)) at s = 4, assuming that g(4) = 3 and g '(4) = −9.
Find a parametrization of the tangent line at the point indicated.r(t) = (t2, t4), t = −2
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