Math 250 - Answers

Math 250: Final Review, Spring 2022   13.5) Lines and Planes in Space. 1-10, 11-26, 27-30, 31-37, 43-58, 61-64, 65-68, 71-72, 73-76, 77-80.   1. (a) ⃗r(t) = (1 + 2t, −1 + 3t, 2 − t) (b) ⃗r(t) = ⟨2t, 2 − t, 3 − 2t⟩ . (c) ⃗r(t) = ⟨−1 − 2t, 2 + t, t⟩. 2. (a) 5(x − 2) − (y − 3) + 3(z − 1) = 0 or 5x − y + 3z = 10. (b) (x − 1) + (y − 2) − 2z = 0 or x + y − 2z = 3. (c) x − y + z = 2.  13.6) Cylinders and Quadric Surfaces. 1-6, 7-12, 15-20, 21-28, 29-51, 54-58, 60.  1. (a) Elliptic cylinder parallel to y=axis. (b) Parabolic cylinder parallel to x-axis. 2. (a) xy: hyperbola, xz: hyperbola, yz: no solution. Hyperboloid of two sheets. (b) xy: hyperbola, xz: hyperbola, yz: ellipse. Hyperboloid of one sheet. (c) xy : y = ±3x. xz : (0, 0), yz : y = ±3z. Cone.  15.1) Graphs and Level Curves. 25-33, 34, 35, 36-43, 74-77.  1. (a) plane. (b) elliptic paraboloid. (c) hyperbolic paraboloid. 2. (a) lines. (b) c = 0, y = ±x. c = −1, 1, 2, hyperbolas. (c) c = 2, (0, 0). c = −1, 0, 1 circles.  15.2) Limits and Continuity. 10-12, 13-27, 29-34, 35-50, 52-53, 62-67, 71.  1. (a) Function is continuous at (−1, 2) so lim f (x, y) = f (−1, 2) = 0. (x,y)→(−1,2) (b) lim(x,y)→(3,−1) (x+3y)(x−3y) y(x+3y) = lim(x,y)→(3,−1) 1 x−3y y = −6. (t,0)→(0,0) f (x, y) = 21 , lim f (x, y) = 0, 2. (a) lim (b) (t,0)→(0,0) lim f (x, y) = 2. (0,t)→(0,0) lim (t,t)→(0,0) f (x, y) = 12 .  15.3) Partial Derivatives. 1-9, 11-14, 15-30, 32-34, 38-46, 48-53, 54-59.  (3(x+h)−y)−(3x−y) h h→0 = 3, lim (x+h)y 2 −xy 2 h h→0 x(y+h)2 −xy 2 h h→0 1. (a) lim (b) lim (3x−(y+h))−(3x−y) h h→0 = y 2 , lim = −1. = 2xy. 2. (a) fxx = −y 2 sin(xy), fxy = cos(xy) − xy sin(xy), fyx = cos(xy) − xy sin(xy), fyy = −x2 sin(xy). 2(y 3 −x2 ) , fxy (x2 +y 3 )2 (b) fxx = = fyx = −6xy 2 , fyy (x2 +y 3 )2 = 6x2 y−3y 4 (x2 +y 3 )2  15.4) The Chain Rule. 9-18, 19-26, 27-28, 29-30, 35-40, 57-59, 65, 67-69, 72-73, 75.  1. dw dt = 3x2 y 2 (2t) + 2x3 y(1 + e2−t ) = 3(25)(1)(4) + 2(53 )(1)(2) = 800. 2. ∂w ∂u ∂w ∂v = x1 (2) − = x1 (3) + 3. (a) (b) ∂F (0) ∂x ∂y ∂z = + 1 v2 y u2 1 2v y u = 2 − 21 = 32 . = 3 − 2 = 1. ∂F ∂y ( ) ∂y ∂z + ∂F (1) ∂z =0⇒ ∂y ∂z = − ∂F / ∂F . ∂z ∂y −xyexz −xyeyz exz +xzeyz −2xy−z 3 ∂z , y 2 +3xz 2 ∂y −2yz . y 2 +3xz 2 4. ∂z ∂x = 5. dE dt = 12 2v dv + g sin(θ) dθ = −g sin(θ)v + g sin(θ)v = 0. dt dt =   15.5) Directional Derivatives and Gradient. 1-10, 11-12. 13-20, 21-30, 31-36, 43-44, 47-50, 59-64, 69-72, 74, 75-78, 81, 82, 85.   1. (∇f )(−1, 2, 1) = (−4, 1, 3). (D⃗u f )(−1, 2, 1) = √ 8 2. (a) 145, ⟨ √145 , √−9 ⟩. 145 8 3 + 23 + 1. (b) ⟨9, 8⟩. 3. f (x, y) = xy. (a) Graph the level set of f through the point (2, 1). (b) Include the vector ∇f (2, 1) on your graph. (c) (x − 2) + 2(y − 1) = 0 ⇒ y = − 12 x + 2. (d) The vector ∇f (2, 1) is perpendicular to the level set of f at (2, 1). 2  15.6) Tangent Planes. 3-4, 9, 11, 13-28, 29-32, 54-56.  1. Find the tangent plane to the surface at the given point. (a) −4(x + 1) − 1(x − 2) + 4(x − 2) = 0. (b) 1(x − 3) + 6(y − 0) + 0(z − 2) = 0 or x + 6y = 3. (c) 1 (x 2 − 3) − 2(y − 1) − 21 (z + 1) = 0. 2. Find the tangent plane to the surface at the given point. (a) 5 (x 3 − 5) − 43 (x − 4) − (z − 3) = 0. (b) 0(x − 2) + 2(y − 0) + 1(z − 3) = 0 or 2y − 3z = 3. (c) 3(x − 2) − (y − 1) + 4(z + 1) = 0.  15.7) Maximum/Minimum Problems. 9-12, 13-22, 23-37, 41-42, 43-46, 62-66, 71.  1. Find any critical points and classify each as relative maximum, relative minimum, or saddle. (a) Local max at (0, 0). (b) local min at (0, 1) (c) Saddle at (1, 2). (d) Local min (±1, 0). Local max (0, 0). 2. min x2 + y 2 + z 2 = 2 at (x, y) = (1, 0). 3. max V = 2 at (x, y) = (1, 1).  15.8) Lagrange Multipliers. 3-4, 5-6, 7-23, 26, 27-36.  √ √ 1. max = 3 3, min = −3 3. 2. Area = 32. √ 3. D = 30.  16.1) Double Integrals, Rectangular Regions. 1-3, 5-6, 7-24, 25-35, 36-39, 40-45, 46-50, 53-54.  1. 1 4 ∫2∫3 0 2. (a) 1 2y − x dydx = 3. ∫ ln(3) ∫ 1 0 0 xyexy dydx = 1 − 12 ln(3). 2 3 ∫3∫1 (b) 0 0 √ y dxdy 1+xy = 10 . 3   16.2) Double Integrals, General Regions. 5-8, 9-10, 11-27, 28-34, 35-42, 43-53, 57-62, 63-68, 69, 70, 71, 73-80, 85-90, 95-96, 99-102.  1. ∫ 2 ∫ 3y+1 1 y 2 +3  2y − 1dxdy = 13 . ∫ 3 ∫ 2y 2. (a) (b) ey dxdy = e9 − 1. 2 0 0 ∫2 ∫4 −2 x2 e12x−x dydx = 13 (e16 − e−16 ) 3   16.3) Double Integrals in Polar Coordinates. 7-10, 11-14, 15-18, 19-20, 21-30, 31-40, 42, 44-46,47, 49-50, 53-54, 57-60, 65-68.   1. Evaluate the integral by converting to polar. ∫4∫ π (a) 0 02 (r cos(θ))(r sin(θ))r drdθ = 32. ∫π (b) −2π (1+r1 2 )2 r drdθ = 2π . 5 2 ∫ π ∫ 2 sec(θ) (c) −4π sec(θ) r drdθ = 3. 4  16.4) Triple Integrals. 4-6, 7-14, 15-29, 30-35, 36-37, 38-46, 47-50, 51-54, 57-58, 62-63, 67-70.  1. ∫ 2 ∫ 1− 1 x2 ∫ 4−4y−x2 4 ∫01 ∫0√4−4y ∫04−4y−x2 0 1 ∫02 ∫04−x2 ∫ 1− z− 1 x2 4 4 dzdydx dzdxdy ∫04 ∫0√4−z ∫0 1− 1 z− 1 x2 4 ∫04 ∫01− 1 z ∫ 0 4 ∫01 ∫04−4y ∫ 0 2. (a) (b) 0 √ √ 4 dydzdx dydxdz 4−4y−z dxdydz 4−4y−z dxdzdy ∫ 2 ∫ √4−2z ∫ 4−2z √ − 4−2z x2 √ ∫ 4 ∫ y ∫ 2− 1 y 2 √ 0 − y 0 0 dydxdz = dzdxdy = 256 . 30 256 . 30   16.5) Cylindrical and Spherical Coordinates. 3-4, 9-10, 11-14, 15-22, 23-28, 29-34, 35-38, 41-47, 48-54, 58-61, 62-63, 64-65, 66-72, 77-79.  1. 2. ∫ 2π ∫ 2 ∫ 4−r2 0 1 0 ∫ π ∫ π ∫2 2 0 2 0 0  z 3 r2 rdzdrdθ = 73 π. ρ4 cos(θ) sin3 (ϕ) cos(ϕ)dρdϕdθ = 4 32 . 15 ∫ 2π ∫ 3 ∫ 3 3. (a) r dzdrdθ = 9π. 0 0 r ∫ 2π ∫ π ∫ 3 sec(ϕ) 2 (b) 0 04 0 ρ sin(ϕ)dρdϕdθ = 9π. ∫ 2π ∫ 5 ∫ √25−r2 4. (a) 3 0 cot−1 (4/3) ∫ 2π ∫ π2 (b) 0 0 r dzdrdθ = ∫5 3 csc(ϕ) 128π . 3 ρ2 sin(ϕ) dρdϕdθ = 128π . 3   16.7) Change of Variables. 5-11, 13-16, 17-22, 23-26, 27-30, 31-36, 37-39, 41-44, 46-47, 48, 50-52, 53, 56.  1. 2. 3.  ∫2∫3 1 1 1 2v ∫2∫3 0 1 (v − 2u)( 12 ) dudv = −6. ∫π∫1 0 0 dvdu = 12 ln(3). 10r2 sin(θ) drdθ = 20 . 3  17.1) Vector Fields. 2, 8-15, 18, 24, 25-30, 35-42, 43-45, 47-48, 49-52.  1. 2.   17.2) Line Integrals. 4-10, 12-16, 17-34, 35-36, 39-40, 41-46, 47-48, 49-56, 57-60, 62, 64-65, 68, 70-72, 73.   1. Evaluate the line integrals. ∫2 (a) 0 t3 (0) + t(2t) dt = 16 . 3 ∫1 (b) 0 (2t)(−1 + 2t)(3) dt = 1. ∫2 (c) −1 (t4 − t)(4t3 ) + 2t dt = 104.1.  17.3) Conservative Vector Fields. 7, 9-16, 17-30, 31-34, 39-42, 44, 45-50, 51-52, 54-56, 59-62, 63-64.  1. (a) ϕ(x, y) = √ x2 + y 2 . (b) ϕ(3, 4) − ϕ(1, 0) = 4. 2. F⃗ = ⟨1 + cos(y), 2y − x sin(y)⟩. (a) ∂g ∂x = ∂f ∂y = − sin(x). (b) ϕ(x, y) = x + x cos(y) + y 2 . (c) ϕ(2, π) − ϕ(1, 0) = π 2 − 2. 5 3. ∇ × F⃗ = ⃗0, ϕ(x, y, z) = x2 + xy + 2y + z 3 .  17.4) Green’s Theorem. 9-14, 15-16, 17-20, 21-25, 27-30, 31-40, 41-45, 48, 53-54.  ∫2∫1 2x − 2x dydx = 0. ∫1 ∫1 2. (a) −1 t − t2 + (t + t2 )(2t) dt + 0 (1 − 2t − 1)(−2) dt = 23 + 2 = 38 . ∫1 ∫1 (b) −1 x2 2 dydx = 83 . ∫ 1 ∫ 2x 3. 0 0 4 dydx = 4. ∫ 2π ∫ 2π ∫ 2 4. 0 16 cos4 (t)+16 sin4 (t) dt = 24π, 0 0 3r3 drdθ = 24π.(T hef irstintegralisalittletoohard/tedioust 1. 0 0  17.5) Divergence and Curl. 9-16, 17, 19, 21-22, 27-34, 41, 42, 44, 65, 67-69, 73.  1. F⃗ should be ⟨x3 , x2 y, yz 3 ⟩. ∇ · F⃗ = 4x2 + 3yz 2 , ∇ × F⃗ = ⟨z 3 , 0, 2xy⟩ . 2. ∇ × ⟨ ∂ϕ , ∂ϕ , ∂ϕ ⟩ = ⃗0. ∂x ∂y ∂z 3. ∇ × F⃗ = ⟨−xy, −zexz , yz − 2y⟩ ̸= ⃗0.  17.6) Surface Integrals. 9-14, 15, 17-18, 19-24, 25-28, 29-34∗ , 35-38∗ , 43-48∗ , 52, 70-72, 74-75.  *: For 29-34, 35-38, 43-48, you must parametrize the surface by r(u, v) and use the parametric form to do the integrals, rather than the ‘explicit’ form indicated in the book here. 1. 2. ∫ 2π ∫ 1 √ 0 0 3 1 + r2 r drdθ = π6 (5 2 − 1) ∫ 2π ∫ 2 (− cos(θ) + 2)r drdθ = 8π √ ∫ 3 ∫ 3u √ 3. 0 u2 6 dvdu = 4.5 6 0 0 √ ⃗ = ⟨−f (v)f ′ (v), f (v) cos(u), f (v) sin(u)⟩ dudv, dS = f (v) 1 + (f ′ (v))2 dudv 4. dS ∫π∫1∫ 2 5. (a) 0 0 u cos(v) sin(v) − u cos(v) sin(v) + uv dudv = π4 . √ ∫π∫1 3 (b) 0 0 u sin(v) 1 + u2 dudv = 23 (2 2 − 1).  17.7) Stokes’ Theorem. 5-10, 11-16, 17-24, 30-33, 45.  6 1. ∫ 2π 0 4 sin2 (t) + 4 cos2 (t) dt = 8π, ∫ 2π ∫ 2 0 0 2 rdrdθ = 8π. ∫ 2π 9 sin2 (t) + 9 cos2 (t) dt = 18π. ∫ 1 ∫ 1−u ∫ 3. 0 0 2 dvdu = 1 2. 0  17.8) Divergence Theorem. 9-12, 13-16, 17-24, 25-27, 30.  ∫ 2π ∫ 2 5 r drdθ + 16rdrdθ = 128π . 3 0 0 0 0 0 0 s ⃗ The surfaces have outward pointing 2. Use the divergence theorem to evaluate S F⃗ · dS. normal. ∫ 2 ∫ 3−1.5z ∫ 6−2y−3z (a) 0 0 3dxdydz = 18. 0 ∫ 2π ∫ π ∫ 3 5 3 (b) 0 0 0 3ρ sin (ϕ) cos(ϕ) dρdϕdθ = 0. 1. ∫ 2π ∫ 2 ∫ 2 r2 r dzdrdθ = 128π . 3 − ∫ 2π ∫ 2 7