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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Sketch the graph of the cylindrical equation z = 2r cos θ and write the equation in rectangular coordinates.
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
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
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
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
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)
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).

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