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Final review problems
(1) Find the cosine of the angle between vectors u = 2i+3j+7k and v = 2j+k. ( √^13310 ) (2) Find the area of the parallelogram built on vectors a = 3i − 4 j − k and b = i − 2 j + 2k. (
(3) Find the volume of the parallelepiped whose vertices are A(0, 0 , 0), B(1, − 1 , 1), C(2, 1 , −2), D((− 1 , 2 , −1). (4) (4) Find the point of intersection of the lines x = 3 − t, y = 5 + 3t, z = − 1 − 4 t and x = 8 + 2t, y = − 6 − 4 t, z = 5 + t. ((4, 2 , 3)) (5) Find the equation of the plane through P (2, 1 , 3), Q(3, 3 , 5), and R(1, 3 , 6). (2x − 5 y + 4z − 11 = 0) (6) Find the parametric equations of the line through (2, 0 , −3) that is parallel to the line of intersection of the planes x+2y+3z+4 = 0 and 2x−y−z−5 =
- (x = 2 + t, y = 7t, z = − 3 − 5 t) (7) Identify and sketch the following surfaces:
- 2 x^2 + y^2 − 4 z^2 = 0; (the elliptic cone z^2 = x
2 2 +^
y^2 4 “along^ z^ axis” with vertex (0, 0 , 0))
- 9 x^2 + 4y^2 − 54 x − 16 y − 36 z + 277 = 0; (the elliptic paraboloid z − 5 = (x−3)^2 4 +^
(y−2)^2 9 “along^ z^ axis” with vertex (3,^2 ,^ 5))
- x^2 = y; (the parabolic cylinder “along z axis”)
- x^2 + 4y^2 + z^2 − 8 y = 0. (the ellipsoid x
2 4 + (y^ −^ 1)
(^2) + z^2 4 = 1 centered at (0, 1 , 0) with semi-axes 2, 1 , 2) (8) Transform r^2 cos 2θ = z^2 in cylindrical coordinates to rectangular coordi- nates. Name the resulting surface. (the circular cone x^2 = y^2 + z^2 “along x axis”) (9) Transform z = 14 (x^2 + y^2 ) from rectangular coordinates to spherical coor- dinates. (ρ = 4 cot φ csc φ) (10) Find parametric equations for r = (2 + cos 3t)i + (3 − sin 3t)j + 4tk, 0 ≤ t ≤ 2 π 3 , using arc length,^ s, as a parameter. Use the point on the curve where t = 0 as the reference point. (x = 2 + cos 35 s , y = 3 − sin 35 s , z = 45 s , 0 ≤ s ≤ 103 π ) (11) Find the unit tangent vector to r(t) = et^ sin ti + et^ cos tj at t = 0. (T (0) = √^1 2 i^ +^ √^1 2 j) (12) Use the chain rule to find dzdt if z =
x^2 + y^2 ; x = et, y = cos t. ( e
2 t−sin t cos t √ e^2 t+cos^2 t ) (13) Use the chain rule to find ∂z∂r and ∂z∂θ if z = (^) x 2 xy+y 2 ; x = r cos θ, y = r sin θ.
( ∂z∂r = 0, ∂z∂θ = cos 2θ) (14) Find the directional derivative of f (x, y, z) = x^2 y + xy^2 + z^2 if one leaves (1, 1 , 1) in the direction of (3, 1 , 2). ( √^85 )
(15) The temperature of a region in space is given by T (x, y, z) = x^2 yz^3. Find the maximum rate of increase in temperature at (2, 1 , −1). (
(16) Find the tangent plane to the surface z = sin x + sin(y + π 3 ) at (0, 0 ,
√ 3 2 ). (z =
√ 3 2 +^ x^ +^
1 2 y) (17) Locate all relative maxima, relative minima, and saddle points for f (x, y) = x^2 − xy + y^2 + 2x + 2y − 4. (a relative minimum at (− 2 , −2)) 1
2
(18) Use the Lagrange multiplier method to find the points on z^2 = x^2 + y^2 that are closest to (2, 2 , 0). ((1, 1 ,
- and (1, 1 , −
(19) Evaluate
∫^2
1
√x ∫ 0
y ln x^2 dy dx. (2 ln 2 − 34 )
(20) Sketch the region R and express
∫^1
0
(^2) ∫−y
1 −y
f (x, y) dx dy as a repeated integral
in the reversed order. (
∫^1
0
∫^1
1 −x
f (x, y) dy dx +
∫^2
1
(^2) ∫−x
0
f (x, y) dy dx)
(21) Find the volume of the solid in the first octant enclosed by y = x
2 4 , z^ = 0 , y = 4, x = 0, and x − y + 2z = 2. ( 23215 ) (22) Use polar coordinates to evaluate
R 2(x^ +^ y)^ dA^ where^ R^ is the region enclosed by x^2 + y^2 = 9 and x ≥ 0. (36) (23) Use a double integral in polar coordinates to find the volume enclosed by by the sphere x^2 +y^2 +z^2 = 16 and the cylinder (x−2)^2 +y^2 = 4. ( 1289 (3π −4)) (24) Evaluate
G
x dV where G is the solid in the first octant enclosed by
x + y + z = 3 and the coordinate planes. ( 278 ) (25) Use cylindrical coordinates to find the volume inside the the cylinder x^2 + y^2 = 4x, above the plane z = 0, and below the paraboloid 4z = x^2 + y^2. (6π) (26) Use spherical coordinates to find the mass of the sphere x^2 + y^2 + z^2 = 2z if its density is given by δ(x, y, z) =
x^2 + y^2 + z^2. ( 85 π ) (27) Find div F and curl F of the vector field F(x, y, z) = x^2 yi + xy^2 j + xyzk. (div F = 5xy, curl F = xzi − yzj + (y^2 − x^2 )k) (28) Find the area of the surface extending upward from the semicircle√ y = 25 − x^2 to the surface z = xy. (125) (29) Find the work done by the force F(x, y, z) = 2xi + 3xyj + 4zk acting on a particle that moves along the curve r(t) = ti + 2t^2 j + 3t^3 k from the origin to (1, 2 , 3). ( 1195 ) (30) Find the mass of the thin wire shaped in the form of the circular arc√ y = 4 − x^2 , 0 ≤ x ≤ 2, if the density function is δ(x, y) = kxy^3 /^2 , k > 0. ( 16
√ 2 5 k) (31) Determine whether the vector field F = (1 +
y)i + 2 √xy j is conservative. If it is, find a potential function for it. (φ(x, y) = x + x
y + C) (32) Find the work done by the conservative force F(x, y) = − (^) xy 2 sinh yx i + 1 x sinh^
y x j^ as it acts on a particle moving from^ P^ (1,^ 1) to^ Q(2,^ 2). (0) (33) Use Green’s theorem to evaluate
C
(3x^2 + y) dx + 4xy dy where C is the
triangle with vertices (0, 0), (2, 0), and (0, 4) traversed in a counterclockwise direction. ( 523 )
