7a3 2. (c) πa 3 (d) πa 3 (e) πa3

Size: px
Start display at page:

Download "7a3 2. (c) πa 3 (d) πa 3 (e) πa3"

Transcription

1 1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin φ a cos θ sin φ, a sin θ sin φ, a cos φ da. As 3 2 part of the problem, you need to decide which sign you need. (a) 7a3 2 (b) 7a3 2 (c) πa 3 (d) πa 3 (e) πa3 2 πa2 2 2.(6pts) Let z = f(x, y) and suppose f(2, 1) = 3. Suppose x = g(u, t) and y = h(u, t) with g( 1, 3) = 2 and h( 1, 3) = 1. hen z is a function of u and t. Find z at the point u ( 1, 3) if gradf = 4, 5 at (2, 1), gradf = 1, 1 at ( 1, 3) and g u (u, t) = 3 at ( 1, 3), h u (u, t) = 2 at ( 1, 3). Which number below is z u (u, t) at ( 1, 3)? (a) (b) 2 (c) 1 (d) 3 (e) 4

2 3.(6pts) Compute origin. D y da where D is the upper half of the disk of radius a centered at the (a) 2a3 3 (b) a3 3 (c) 2a2 3 (d) 4a2 3 (e) 4a3 3 4.(6pts) he two level surfaces f(x, y, z) = x 2 y xyz+z 2 = 7 and g(x, y, z) = x 2 +y 2 +z 2 = 14 intersect at the point (3, 1, 2). Which equation below is an equation for the tangent line to the curve of intersection at the point (3, 1, 2)? (a) t 1, 1, + 1, 1, 2 (b) (1 t) 3, 1, 2 + t 1, 1, 1 (c) 3, 1, 2 + t 1, 1, 1 (d) (1 t) 3, 1, 2 + t 2, 1, 2 (e) 3, 1, 2 + t 2, 1, 2

3 5.(6pts) Which function below is a potential function for the field F = yz 2 + 2xyz, xz 2 + x 2 z, 2xyz + x 2 y 2z (a) f(x, y, z) = xyz 2 + x 2 yz 1 2 z4 (b) f(x, y, z) = x 3 y 2 z + x 2 xyz 2 (c) f(x, y, z) = 2xyz xy2 z y 2 (d) f(x, y, z) = xyz x3 y 2 z + x 2 (e) f(x, y, z) = xyz 2 + x 2 yz z 2 6.(6pts) Find the area of the parallelogram with vertices (1, 2), (3, 5), (4, 3) and (6, 6). (a) 8 (b) 7 (c) 11 (d) 1 (e) 9

4 7.(6pts) Find the directional derivative of f(x, y) = x 2 xy + y 3 at the point (2, 1) in the same direction as the vector u = 1, 1. (a) 1 1 3, 1 (b) (c) 2 (d) 1 1 1, 3 (e) 2 8.(6pts) Maximize the function x 2y + z subject to the requirement that the points lie on the surface x 2 + 4y 2 + z 2 = 3. (a) 3 (b) he function has no maximum value on the surface. (c) 4 (d) 3 2 (e) 2

5 9.(6pts) Which iterated integral below gives the same number as the iterated integral y x5 + 1 dx dy (a) 2 x 4 /8 x5 + 1 dy dx (b) 2 8x 4 x5 + 1 dy dx (c) 16 8x 4 x5 + 1 dy dx (d) x 4 /8 2 x5 + 1 dy dx (e) 16 x 4 /8 x5 + 1 dy dx 1.(6pts) Find the integral F dr where C is the square in the xy-plane starting at the C origin, going to (, 2), then to (2, 2), then to (2, ) and finally back to the origin, and F = 8 x2 xy, y (a) 3 (b) 2 (c) 3 (d) 4 (e)

6 11.(6pts) Let F = x, 2y, x 2 + z. Compute F ds over the part of the surface x 2 + y 2 + S zx 2 + zy 2 + z 3 = 9 which lies above the xy plane. Note that S together with the disk D x 2 + y 2 9, z = bounds a solid E. ry integrating over a different surface. (a) 49π2 2 (b) 81π 4 (c) (d) π 3 (e) 7π 4 12.(6pts) Let S be the surface r(u, v) = uv 2, uv, u 2 v and note that r(2, 1) = 2, 2, 4. Which equation below is an equation of the tangent plane to S at 2, 2, 4? (a) 2x 6y + z = 4 (b) x + y + 4z = 2 (c) 2x + y + 2z = 14 (d) x y + 4z = 16 (e) 2x 2y + z =

7 13.(6pts) Find the area of the piece of the cylinder over y = x 3, x 1 above the plane z = and below the graph of the cylinder z = 36x 3. (a) π 3 36 (b) 18π (c) 2 3( 1 1 ) (d) 12π ( 1 ) (e) (6pts) If C is the curve r(t) = (sin 2t) ( 4 t 2, t π ), 2 + sin(t), t π 2 2 find f dr where f(x, y, z) = x 2 y 3 + zy + xy z. C (a) 1 (b) (c) π (d) 16 5 (e) 2

8 15.(6pts) Let r(u, v) = u 2 + uv, v 2 + uv. Compute the Jacobian of this transformation. (a) u 2 + v 2 4uv (b) 4uv + 2u 2 + 2v 2 (c) 4uv 2u 2 + 2v 2 (d) 4uv + 2u 2 2v 2 (e) (u 2 + v 2 )uv 16.(6pts) Find the moment about the yz plane of a thin sheet of unit density bent in the shape of a surface r(u, v) = u 3, v 4, uv for 1 u 1 and 1 v 1. Suppose is the square with vertices (, ), (, 1), (1, ) and (1, 1). (a) u 3 u 6 + v 8 + u 2 v 2 da (b) u 3 16v 8 + 9u u 4 v 6 da (c) (e) v 4 16v 8 9u u 4 v 6 da v 4 16v 8 + 9u u 4 v 6 da (d) u 3 16v 8 9u u 4 v 6 da

9 17.(6pts) Find the area of the region enclosed by the curve x = sin(2t), y = sin(t), t π. Fact: sin(2t) = 2 sin(t) cos(t), cos(2t) = cos 2 (t) sin 2 (t). Hint: Use Green s heorem. (a) 8 2π 5 (b) 2π 3 (c) 2π 1 5 (d) 4 3 (e) 4π (6pts) Which number below is S xyz, xyz, xyz ds where S consists of the six faces of the cube with sides of length 2 in the first octant and with one vertex at the origin and with normal vector pointing out of the cube? (a) 4 (b) 6 (c) 12 (d) 24 (e) 1

10 19.(6pts) Let C be the intersection of the cylinder over the triangle with vertices (, ), (1, ) and (, 1) with the sphere x 2 +y 2 +z 2 = 9. Orient counterclockwise. Which integral below is equal to y, z, x dr? (a) (d) C x y 9 x2 y 1 da(b) 2 x y 9 x2 y + 2 da(e) 2 x + y 9 x2 y 1 da(c) 2 x y 9 x2 y da x y 9 x2 y 2 1 da 2.(6pts) Let r(t) = t 2, (t 1) 3, t 2. Which parametrized surface below is the result of rotating this curve about the y axis? (a) t 2, (t 1) 3 cos(θ), t 2 sin(θ) ; t 2, θ 2π (b) t 2 cos(θ), (t 1) 3, t 2 sin(θ) ; t 2, θ 2π (c) t 2, (t 1) 3 cos(θ), t 2 sin(θ) ; t 2, θ π (d) t 2 cos(θ), (t 1) 3, t 2 sin(θ) ; t 2, θ π (e) t 2 cos(θ), (t 1) 3 cos(θ), t 2 sin(θ) ; t 2, θ π

11 21.(6pts) Which point below is a local maximum for the function f(x, y) = 2x 3 + 6xy 2 3y 3 15x. Note f xx = 12x, f xy = f yx = 12y and f yy = 12x 18y. he function f has exactly four critical points listed as answers (a)-(e). (a) (5, ) (b) (3, 4) (c) ( 3, 4) (d) he function had no local maxima. (e) ( 5, ) 22.(6pts) Find the length of the curve given by r(t) =,, and 72, 3, t3, 1 2 t, 1 2 t2 between the points (a) 34 (b) 5 (c) 19 (d) 75 (e) 86

12 23.(6pts) Which number below is the cosine of the angle of intersection of the two planes 2x + 3y + z = 1 and 3x + 2y z = 4. (a) 4 5 (b) 3 16 (c) (d) 2 3 (e) he two planes do not intersect. 24.(6pts) Let F = x + e sin(yz), y + sin(x 2 + z), cos(xyz). hen div F is a function, grad(div F) is a field and so curl ( grad(div F) ) is a field. Which field below is it? (a) x, 2y, z (b) 1, 1, sin(xyz) (c) 1, 2, 1 (d) x, y, z (e),,

13 25.(6pts) Compute the divergence of F, F where F = f with f(x, y, z) = x 2 z +xyz +y 3. (a) 2xz + xz (b) 3xz + yz + 3y 2 (c) (d) 3xz + yz + 3y 2 + x 2 + xy (e) 6y + 2z

14 1. Solution. In spherical coordinates, H is given by ρ = a, θ 2π, φ π 3, so H is parametrized by r(θ, φ) = a cos θ sin φ, a sin θ sin φ, a cos φ so a normal vector is given by i j k r θ r φ = det a sin θ sin φ a cos θ sin φ = a cos θ cos φ a sin θ cos φ a sin φ a 2 cos θ sin 2 φ, a 2 sin θ sin 2 φ, a 2 sin φ cos φ = a sin φ a cos θ sin φ, a sin θ sin φ, a cos φ. he region in the θ-φ plane is given by θ 2π, φ π 3 Hence x, y, z d S = H a cos θ sin φ, a sin θ sin φ, a cos φ a sin φ a cos θ sin φ, a sin θ sin φ, a cos φ da = 2π π/3 2π a(sin φ)a 2 da = a 3 sin φ da = a 3 sin φ dφ dθ = a 3 cos φ a 3 2π 1 2. π/3 dθ = 2. Solution. Let r = g, h. hen g u (u, t) = f r u at ( 1, 3) so the answer is 4, 5 3, 2 = Solution. In polar coordinates, D is r a, θ π so y da = ( D a ) ( π ) r 2 dr sin(θ) dθ = a3 3 ( cos(θ) π ) = a3 2a3 ( 1 (1)) = 3 3 a π r sin(θ)r dθ dr =

15 4. Solution. f = 2xy yz, x 2 xz, xy + 2z and g = 2x, 2y, 2z so at the point in question f = 4, 3, 1 and g = 6, 2, 4 so a vector on the tangent line is f g = i j k det = 12 2, (16 6), 8 18 = 1, 1, 1 A parallel vector is 1, 1, Hence 3, 1, 2 + t 1, 1, Solution. f x = yz 2 + 2xyz so f = xyz 2 + x 2 yz + g(y, z). f y = xz 2 + x 2 z + g y = xz 2 + x 2 z so g y = and g(y, z) = h(z). f z = 2xyz + x 2 y + h = 2xyz + x 2 y 2z so h = 2z and h = z 2. f(x, y, z) = xyz 2 + x 2 yz z 2 i j k 6. Solution. (1, 2) + 2, 3 = (3, 5), (1, 2) + 3, 1 = (4, 3). Area is ± det 2 3 = 3 1,, 2 9 = 7 so 7.

16 7. Solution. First f = 2x y, 3y 2 x and f(2, 1) = 3, 1. he unit vector is the 1 1 direction of u is 1, 1 so D u f(2, 1) = 3, 1 1, 1 = 2 = Solution. If f = x 2y + z and g = x 2 + 4y 2 + z 2 then Lagrange says the maxima and minima occur at points where f = λ g and g = 3. Hence 1, 2, 1 = λ 2x, 8y, 2z or 1 x, y, z = 2λ, 1 ( ) 2 ( ) 2 ( ) 2 4λ, Hence g(x, y, z) = +4 + = 3 or 3 1 2λ 2λ 4λ 2λ 4λ = 3 2 or λ = ± 1 ( 2. Hence there are eight points ±1, ± 1 ). 2, ±1 he maximum occurs when all three terms are positive so f = 1 2( 1/2) + 1 = = Solution. Write the iterated integral as a double integral over the region D given by 4 8y x 2; y 2. Setting up the other way, 2 x 4 /8 x5 + 1 dy dx.

17 ( N 1. Solution. By Green s heorem Mdx + Ndy = S S x M ) da where S is the y square and F = M, N. he curve S is C so if we compute the double integral we will get the negative of the answer. We get () ( y) da = y dx dy = 2 y dy = y 2 2 = 4. S 11. Solution. S x dx 2y dy + (x 2 + z) dz + D x dx 2y dy + (x 2 + z) dz = where the normal vector to D points down. F = x, 2y, x 2 + z. o parametrize D use r(x, y) = x, y, with normal vector,, 1. x dx 2y dy + (x 2 + z) dz = D 3 2π ( 3 ) ( 2π ) x 2 da = r 2 cos 2 (θ)r dθ dr = r 3 dr cos 2 (t) dt = 81π 4. D E dv 12. Solution. r u = v 2, v, 2uv and r v = 2uv, u, u 2 so at u = 2, v = 1, i j k i j k r u r v = det v 2 v 2uv = det = (4 8), (4 16), (2 4) = 4, 12, 2 2uv u u So 4, 4, 2 x 1, y 2, z 4 = or 4x + 4y 2z = 4.

18 13. Solution. he surface is parametrized by r(x, z) = x, x 3, z over the region given by i j k x 1 and z 36x 3. he normal vector is det 1 3x 2 = 3x 2, 1, x 3 1 so 1dS = 1 + 9x4 da = 1 + 9x4 dz dx = 36 (x 3 ) 1 + 9x 4 dx. Let S 1 u = 1 + 9x 4 so du = 36x 3 dx so 36 x ( ) 1 + 9x 4 3 dx = u du = 1 1 = ( ) Solution. By the Fundamental heorem, the answer is f ( r(b) ) f ( r(a) ) ( π ) and r ( 2 π )) (, π ) 2 2, 2,, 3, r() =, π 2, 2. Now f so the answer is ( π) = π. ( r = f (,, 3) = ; f (r()) = f = = π 15. Solution. he Jacobian is det 2u + v u = (2u + v)(2v + u) (u)(v) = (4uv + 2u2 + 2v 2 + uv) (uv) = 4uv + 2u 2 + 2v 2.

19 i j k 16. Solution. he short answer is x ds. A normal vector is det 3u 2 v = 4v 4, 3u 3, 12u 2 v 3 S 4v 3 u and its length is 16v 8 + 9u u 4 v 6 so x ds = u 3 16v 8 + 9u u 4 v 6 da. S 17. Solution. r = 2 cos(2t), cos(t) he area is da =, x 2 cos(2t), cos(t) dt = A π π sin(2t) cos(t) dt = 2 sin(t) cos 2 (t) dt = 2 cos3 (y) π = ( ( 2/3) (2/3) ) = π 18. Solution. By the Divergence heorem, (yz + xz + xy) dx dy dz = (2z + 1) dz = 4 ( z 2 + z ) 2 = 24 S 2 2 xyz, xyz, xyz ds = (2yz + 2z + 2y) dy dz = U 2 F dv = (4z + 4z + 4) dz =

20 19. Solution. By Stoke s heorem, the answer is S curl F ds where S is the part of the sphere lying over the triangle. he part of the sphere we have is the graph of z = x 9 x2 y 2 over the triangle. So, d S = 9 x2 y, y 2 9 x2 y, 1 da 2 i j k det = 1, (1), 1 = 1, 1, 1 x y z y z x so we need to integrate x y 9 x2 y 1 da 2 over the triangle. 2. Solution. R(t, θ) = t 2 cos(θ), (t 1) 3, t 2 sin(θ) ; t 2, θ 2π

21 21. Solution. he critical points are located at the zeros of the gradient: f x = 6x 2 + 6y 2 15, f y = 12xy 9y 2. Since 12xy 9y 2 = 3y(4x 3y either y = or 4x 3y = so y = 4 3 x. Note f x = is the same as x 2 + y 2 = 25. If y =, x = ±5. If y = 4 3 x x x2 = 25 or 25 9 x2 = 25 so x = ±3 and the critical points are (5, ), ( 5, [ ) (3, 4) ] and [ ( 3, 4). ] [ ] fxx f he Hessian is yx 12x 12y 2x 2y = = 6 whose determinant f xy f yy 12y 12x 18y 2y 2x 3y is 4x 2 6xy 4y 2 = 2(2x 2 3xy 2y 2 ) so the type of critical point is determined by h(x, y) = 2x 2 3xy 2y 2. h(5, ) = 25 = h( 5, ) >, h(3, 4) = h( 3, 4) = = 24 <. Negative Hessian means saddle point so (pm5, ) are the only possibilities. f xx (5, ) = 6 > so (5, ) is local minimum; f xx ( 5, ) = 6 < so ( 5, ) is local maximum. 22. Solution. he parametrized curve is r, t 6. Hence the length is r (t) dt. Now r (t) = t 2, 12, t so r (t) = t t2 = t so r (t) dt = 1 3 t t 6 = Solution. he angle is the angle between the two normal vectors 2, 3, 1 and 3, 2, 1 and so cos θ =

22 24. Solution. he curl of a gradient is,, and we have a gradient. 25. Solution. F = f = 2xz + yz, xz + 3y 2, x 2 + xy so F = (2z) + (6y) + () = 6y + 2z.

SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)

SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours) SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please

More information

MATH 52 FINAL EXAM SOLUTIONS

MATH 52 FINAL EXAM SOLUTIONS MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an x-simple region. R = {(x, y) y, x y }

More information

x + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the

x + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the 1.(8pts) Find F ds where F = x + ye z + ze y, y + xe z + ze x, z and where T is the T surface in the pictures. (The two pictures are two views of the same surface.) The boundary of T is the unit circle

More information

Calculus III. Math 233 Spring Final exam May 3rd. Suggested solutions

Calculus III. Math 233 Spring Final exam May 3rd. Suggested solutions alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.

More information

Multiple Choice. Compute the Jacobian, (u, v), of the coordinate transformation x = u2 v 4, y = uv. (a) 2u 2 + 4v 4 (b) xu yv (c) 3u 2 + 7v 6

Multiple Choice. Compute the Jacobian, (u, v), of the coordinate transformation x = u2 v 4, y = uv. (a) 2u 2 + 4v 4 (b) xu yv (c) 3u 2 + 7v 6 .(5pts) y = uv. ompute the Jacobian, Multiple hoice (x, y) (u, v), of the coordinate transformation x = u v 4, (a) u + 4v 4 (b) xu yv (c) u + 7v 6 (d) u (e) u v uv 4 Solution. u v 4v u = u + 4v 4..(5pts)

More information

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant

More information

MATHS 267 Answers to Stokes Practice Dr. Jones

MATHS 267 Answers to Stokes Practice Dr. Jones MATH 267 Answers to tokes Practice Dr. Jones 1. Calculate the flux F d where is the hemisphere x2 + y 2 + z 2 1, z > and F (xz + e y2, yz, z 2 + 1). Note: the surface is open (doesn t include any of the

More information

MAT 211 Final Exam. Spring Jennings. Show your work!

MAT 211 Final Exam. Spring Jennings. Show your work! MAT 211 Final Exam. pring 215. Jennings. how your work! Hessian D = f xx f yy (f xy ) 2 (for optimization). Polar coordinates x = r cos(θ), y = r sin(θ), da = r dr dθ. ylindrical coordinates x = r cos(θ),

More information

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt Jim Lambers MAT 28 ummer emester 212-1 Practice Final Exam olution 1. Evaluate the line integral xy dx + e y dy + xz dz, where is given by r(t) t 4, t 2, t, t 1. olution From r (t) 4t, 2t, t 2, we obtain

More information

1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r (t) = 3 cos t, 0, 3 sin t, r ( 3π

1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r (t) = 3 cos t, 0, 3 sin t, r ( 3π 1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P 3, 3π, r t) 3 cos t, 4t, 3 sin t 3 ). b) 5 points) Find curvature of the curve at the point P. olution:

More information

McGill University April Calculus 3. Tuesday April 29, 2014 Solutions

McGill University April Calculus 3. Tuesday April 29, 2014 Solutions McGill University April 4 Faculty of Science Final Examination Calculus 3 Math Tuesday April 9, 4 Solutions Problem (6 points) Let r(t) = (t, cos t, sin t). i. Find the velocity r (t) and the acceleration

More information

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere. MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.

More information

Review Sheet for the Final

Review Sheet for the Final Review Sheet for the Final Math 6-4 4 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the absence

More information

e x3 dx dy. 0 y x 2, 0 x 1.

e x3 dx dy. 0 y x 2, 0 x 1. Problem 1. Evaluate by changing the order of integration y e x3 dx dy. Solution:We change the order of integration over the region y x 1. We find and x e x3 dy dx = y x, x 1. x e x3 dx = 1 x=1 3 ex3 x=

More information

Sections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed.

Sections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. MTH 34 Review for Exam 4 ections 16.1-16.8. 5 minutes. 5 to 1 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. Review for Exam 4 (16.1) Line

More information

1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l.

1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l. . If the line l has symmetric equations MA 6 PRACTICE PROBLEMS x = y = z+ 7, find a vector equation for the line l that contains the point (,, ) and is parallel to l. r = ( + t) i t j + ( + 7t) k B. r

More information

In general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute

In general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute alculus III Test 3 ample Problem Answers/olutions 1. Express the area of the surface Φ(u, v) u cosv, u sinv, 2v, with domain u 1, v 2π, as a double integral in u and v. o not evaluate the integral. In

More information

One side of each sheet is blank and may be used as scratch paper.

One side of each sheet is blank and may be used as scratch paper. Math 244 Spring 2017 (Practice) Final 5/11/2017 Time Limit: 2 hours Name: No calculators or notes are allowed. One side of each sheet is blank and may be used as scratch paper. heck your answers whenever

More information

1. For each function, find all of its critical points and then classify each point as a local extremum or saddle point.

1. For each function, find all of its critical points and then classify each point as a local extremum or saddle point. Solutions Review for Exam # Math 6. For each function, find all of its critical points and then classify each point as a local extremum or saddle point. a fx, y x + 6xy + y Solution.The gradient of f is

More information

MATH 332: Vector Analysis Summer 2005 Homework

MATH 332: Vector Analysis Summer 2005 Homework MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,

More information

Solutions to Sample Questions for Final Exam

Solutions to Sample Questions for Final Exam olutions to ample Questions for Final Exam Find the points on the surface xy z 3 that are closest to the origin. We use the method of Lagrange Multipliers, with f(x, y, z) x + y + z for the square of the

More information

1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is

1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order

More information

Math 23b Practice Final Summer 2011

Math 23b Practice Final Summer 2011 Math 2b Practice Final Summer 211 1. (1 points) Sketch or describe the region of integration for 1 x y and interchange the order to dy dx dz. f(x, y, z) dz dy dx Solution. 1 1 x z z f(x, y, z) dy dx dz

More information

DO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START

DO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START Math 265 Student name: KEY Final Exam Fall 23 Instructor & Section: This test is closed book and closed notes. A (graphing) calculator is allowed for this test but cannot also be a communication device

More information

Review problems for the final exam Calculus III Fall 2003

Review problems for the final exam Calculus III Fall 2003 Review problems for the final exam alculus III Fall 2003 1. Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t)

More information

Jim Lambers MAT 280 Fall Semester Practice Final Exam Solution

Jim Lambers MAT 280 Fall Semester Practice Final Exam Solution Jim Lambers MAT 8 Fall emester 6-7 Practice Final Exam olution. Use Lagrange multipliers to find the point on the circle x + 4 closest to the point (, 5). olution We have f(x, ) (x ) + ( 5), the square

More information

Practice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.

Practice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr. 1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line

More information

MATH 0350 PRACTICE FINAL FALL 2017 SAMUEL S. WATSON. a c. b c.

MATH 0350 PRACTICE FINAL FALL 2017 SAMUEL S. WATSON. a c. b c. MATH 35 PRACTICE FINAL FALL 17 SAMUEL S. WATSON Problem 1 Verify that if a and b are nonzero vectors, the vector c = a b + b a bisects the angle between a and b. The cosine of the angle between a and c

More information

Solutions to old Exam 3 problems

Solutions to old Exam 3 problems Solutions to old Exam 3 problems Hi students! I am putting this version of my review for the Final exam review here on the web site, place and time to be announced. Enjoy!! Best, Bill Meeks PS. There are

More information

Note: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2

Note: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2 Math Prelim II Solutions Spring Note: Each problem is worth points except numbers 5 and 6 which are 5 points. x. Compute x da where is the region in the second quadrant between the + y circles x + y and

More information

Math 234 Exam 3 Review Sheet

Math 234 Exam 3 Review Sheet Math 234 Exam 3 Review Sheet Jim Brunner LIST OF TOPIS TO KNOW Vector Fields lairaut s Theorem & onservative Vector Fields url Divergence Area & Volume Integrals Using oordinate Transforms hanging the

More information

Name: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8

Name: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8 Name: SOLUTIONS Date: /9/7 M55 alculus III Tutorial Worksheet 8. ompute R da where R is the region bounded by x + xy + y 8 using the change of variables given by x u + v and y v. Solution: We know R is

More information

MATH H53 : Final exam

MATH H53 : Final exam MATH H53 : Final exam 11 May, 18 Name: You have 18 minutes to answer the questions. Use of calculators or any electronic items is not permitted. Answer the questions in the space provided. If you run out

More information

Name: Date: 12/06/2018. M20550 Calculus III Tutorial Worksheet 11

Name: Date: 12/06/2018. M20550 Calculus III Tutorial Worksheet 11 1. ompute the surface integral M255 alculus III Tutorial Worksheet 11 x + y + z) d, where is a surface given by ru, v) u + v, u v, 1 + 2u + v and u 2, v 1. olution: First, we know x + y + z) d [ ] u +

More information

Practice problems **********************************************************

Practice problems ********************************************************** Practice problems I will not test spherical and cylindrical coordinates explicitly but these two coordinates can be used in the problems when you evaluate triple integrals. 1. Set up the integral without

More information

f(p i )Area(T i ) F ( r(u, w) ) (r u r w ) da

f(p i )Area(T i ) F ( r(u, w) ) (r u r w ) da MAH 55 Flux integrals Fall 16 1. Review 1.1. Surface integrals. Let be a surface in R. Let f : R be a function defined on. efine f ds = f(p i Area( i lim mesh(p as a limit of Riemann sums over sampled-partitions.

More information

Math 233. Practice Problems Chapter 15. i j k

Math 233. Practice Problems Chapter 15. i j k Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed

More information

Final exam (practice 1) UCLA: Math 32B, Spring 2018

Final exam (practice 1) UCLA: Math 32B, Spring 2018 Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 218 This exam has 7 questions, for a total of 8 points. Please print your working and answers neatly. Write your solutions in

More information

(b) Find the range of h(x, y) (5) Use the definition of continuity to explain whether or not the function f(x, y) is continuous at (0, 0)

(b) Find the range of h(x, y) (5) Use the definition of continuity to explain whether or not the function f(x, y) is continuous at (0, 0) eview Exam Math 43 Name Id ead each question carefully. Avoid simple mistakes. Put a box around the final answer to a question (use the back of the page if necessary). For full credit you must show your

More information

Page Points Score Total: 210. No more than 200 points may be earned on the exam.

Page Points Score Total: 210. No more than 200 points may be earned on the exam. Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Points Score 3 18 4 18 5 18 6 18 7 18 8 18 9 18 10 21 11 21 12 21 13 21 Total: 210 No more than 200

More information

Math 212. Practice Problems for the Midterm 3

Math 212. Practice Problems for the Midterm 3 Math 1 Practice Problems for the Midterm 3 Ivan Matic 1. Evaluate the surface integral x + y + z)ds, where is the part of the paraboloid z 7 x y that lies above the xy-plane.. Let γ be the curve in the

More information

McGill University April 16, Advanced Calculus for Engineers

McGill University April 16, Advanced Calculus for Engineers McGill University April 16, 2014 Faculty of cience Final examination Advanced Calculus for Engineers Math 264 April 16, 2014 Time: 6PM-9PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer

More information

Math 265H: Calculus III Practice Midterm II: Fall 2014

Math 265H: Calculus III Practice Midterm II: Fall 2014 Name: Section #: Math 65H: alculus III Practice Midterm II: Fall 14 Instructions: This exam has 7 problems. The number of points awarded for each question is indicated in the problem. Answer each question

More information

Page Problem Score Max Score a 8 12b a b 10 14c 6 6

Page Problem Score Max Score a 8 12b a b 10 14c 6 6 Fall 14 MTH 34 FINAL EXAM December 8, 14 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 5 1 3 5 4 5 5 5 6 5 7 5 8 5 9 5 1 5 11 1 3 1a

More information

MAT 211 Final Exam. Fall Jennings.

MAT 211 Final Exam. Fall Jennings. MAT 211 Final Exam. Fall 218. Jennings. Useful formulas polar coordinates spherical coordinates: SHOW YOUR WORK! x = rcos(θ) y = rsin(θ) da = r dr dθ x = ρcos(θ)cos(φ) y = ρsin(θ)cos(φ) z = ρsin(φ) dv

More information

Math 31CH - Spring Final Exam

Math 31CH - Spring Final Exam Math 3H - Spring 24 - Final Exam Problem. The parabolic cylinder y = x 2 (aligned along the z-axis) is cut by the planes y =, z = and z = y. Find the volume of the solid thus obtained. Solution:We calculate

More information

Review for the First Midterm Exam

Review for the First Midterm Exam Review for the First Midterm Exam Thomas Morrell 5 pm, Sunday, 4 April 9 B9 Van Vleck Hall For the purpose of creating questions for this review session, I did not make an effort to make any of the numbers

More information

PRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.

PRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives. PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x

More information

Math Review for Exam 3

Math Review for Exam 3 1. ompute oln: (8x + 36xy)ds = Math 235 - Review for Exam 3 (8x + 36xy)ds, where c(t) = (t, t 2, t 3 ) on the interval t 1. 1 (8t + 36t 3 ) 1 + 4t 2 + 9t 4 dt = 2 3 (1 + 4t2 + 9t 4 ) 3 2 1 = 2 3 ((14)

More information

Assignment 11 Solutions

Assignment 11 Solutions . Evaluate Math 9 Assignment olutions F n d, where F bxy,bx y,(x + y z and is the closed surface bounding the region consisting of the solid cylinder x + y a and z b. olution This is a problem for which

More information

1. (30 points) In the x-y plane, find and classify all local maxima, local minima, and saddle points of the function. f(x, y) = 3y 2 2y 3 3x 2 + 6xy.

1. (30 points) In the x-y plane, find and classify all local maxima, local minima, and saddle points of the function. f(x, y) = 3y 2 2y 3 3x 2 + 6xy. APPM 35 FINAL EXAM FALL 13 INSTUTIONS: Electronic devices, books, and crib sheets are not permitted. Write your name and your instructor s name on the front of your bluebook. Work all problems. Show your

More information

Math 263 Final. (b) The cross product is. i j k c. =< c 1, 1, 1 >

Math 263 Final. (b) The cross product is. i j k c. =< c 1, 1, 1 > Math 63 Final Problem 1: [ points, 5 points to each part] Given the points P : (1, 1, 1), Q : (1,, ), R : (,, c 1), where c is a parameter, find (a) the vector equation of the line through P and Q. (b)

More information

MAY THE FORCE BE WITH YOU, YOUNG JEDIS!!!

MAY THE FORCE BE WITH YOU, YOUNG JEDIS!!! Final Exam Math 222 Spring 2011 May 11, 2011 Name: Recitation Instructor s Initials: You may not use any type of calculator whatsoever. (Cell phones off and away!) You are not allowed to have any other

More information

MATH 2400 Final Exam Review Solutions

MATH 2400 Final Exam Review Solutions MATH Final Eam eview olutions. Find an equation for the collection of points that are equidistant to A, 5, ) and B6,, ). AP BP + ) + y 5) + z ) 6) y ) + z + ) + + + y y + 5 + z 6z + 9 + 6 + y y + + z +

More information

G G. G. x = u cos v, y = f(u), z = u sin v. H. x = u + v, y = v, z = u v. 1 + g 2 x + g 2 y du dv

G G. G. x = u cos v, y = f(u), z = u sin v. H. x = u + v, y = v, z = u v. 1 + g 2 x + g 2 y du dv 1. Matching. Fill in the appropriate letter. 1. ds for a surface z = g(x, y) A. r u r v du dv 2. ds for a surface r(u, v) B. r u r v du dv 3. ds for any surface C. G x G z, G y G z, 1 4. Unit normal N

More information

Mathematical Analysis II, 2018/19 First semester

Mathematical Analysis II, 2018/19 First semester Mathematical Analysis II, 208/9 First semester Yoh Tanimoto Dipartimento di Matematica, Università di Roma Tor Vergata Via della Ricerca Scientifica, I-0033 Roma, Italy email: hoyt@mat.uniroma2.it We basically

More information

Math 234 Final Exam (with answers) Spring 2017

Math 234 Final Exam (with answers) Spring 2017 Math 234 Final Exam (with answers) pring 217 1. onsider the points A = (1, 2, 3), B = (1, 2, 2), and = (2, 1, 4). (a) [6 points] Find the area of the triangle formed by A, B, and. olution: One way to solve

More information

SOME PROBLEMS YOU SHOULD BE ABLE TO DO

SOME PROBLEMS YOU SHOULD BE ABLE TO DO OME PROBLEM YOU HOULD BE ABLE TO DO I ve attempted to make a list of the main calculations you should be ready for on the exam, and included a handful of the more important formulas. There are no examples

More information

Practice problems. m zδdv. In our case, we can cancel δ and have z =

Practice problems. m zδdv. In our case, we can cancel δ and have z = Practice problems 1. Consider a right circular cone of uniform density. The height is H. Let s say the distance of the centroid to the base is d. What is the value d/h? We can create a coordinate system

More information

Math 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin

Math 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin Math 45 Homework et olutions Points. ( pts) The integral is, x + z y d = x + + z da 8 6 6 where is = x + z 8 x + z = 4 o, is the disk of radius centered on the origin. onverting to polar coordinates then

More information

Final Review Worksheet

Final Review Worksheet Score: Name: Final Review Worksheet Math 2110Q Fall 2014 Professor Hohn Answers (in no particular order): f(x, y) = e y + xe xy + C; 2; 3; e y cos z, e z cos x, e x cos y, e x sin y e y sin z e z sin x;

More information

Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.

Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work. Exam 3 Math 850-007 Fall 04 Odenthal Name: Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.. Evaluate the iterated integral

More information

Practice Problems for the Final Exam

Practice Problems for the Final Exam Math 114 Spring 2017 Practice Problems for the Final Exam 1. The planes 3x + 2y + z = 6 and x + y = 2 intersect in a line l. Find the distance from the origin to l. (Answer: 24 3 ) 2. Find the area of

More information

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9 MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)

More information

M273Q Multivariable Calculus Spring 2017 Review Problems for Exam 3

M273Q Multivariable Calculus Spring 2017 Review Problems for Exam 3 M7Q Multivariable alculus Spring 7 Review Problems for Exam Exam covers material from Sections 5.-5.4 and 6.-6. and 7.. As you prepare, note well that the Fall 6 Exam posted online did not cover exactly

More information

No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers.

No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers. Name: Section: Recitation Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices can be used on this exam. Clear

More information

Math Exam IV - Fall 2011

Math Exam IV - Fall 2011 Math 233 - Exam IV - Fall 2011 December 15, 2011 - Renato Feres NAME: STUDENT ID NUMBER: General instructions: This exam has 16 questions, each worth the same amount. Check that no pages are missing and

More information

MULTIVARIABLE INTEGRATION

MULTIVARIABLE INTEGRATION MULTIVARIABLE INTEGRATION (SPHERICAL POLAR COORDINATES) Question 1 a) Determine with the aid of a diagram an expression for the volume element in r, θ, ϕ. spherical polar coordinates, ( ) [You may not

More information

MLC Practice Final Exam

MLC Practice Final Exam Name: Section: Recitation/Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 13. Show all your work on the standard

More information

Without fully opening the exam, check that you have pages 1 through 12.

Without fully opening the exam, check that you have pages 1 through 12. Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 12. Show all your work on the standard

More information

Problem Points S C O R E

Problem Points S C O R E MATH 34F Final Exam March 19, 13 Name Student I # Your exam should consist of this cover sheet, followed by 7 problems. Check that you have a complete exam. Unless otherwise indicated, show all your work

More information

Vector Calculus handout

Vector Calculus handout Vector Calculus handout The Fundamental Theorem of Line Integrals Theorem 1 (The Fundamental Theorem of Line Integrals). Let C be a smooth curve given by a vector function r(t), where a t b, and let f

More information

Math 11 Fall 2018 Practice Final Exam

Math 11 Fall 2018 Practice Final Exam Math 11 Fall 218 Practice Final Exam Disclaimer: This practice exam should give you an idea of the sort of questions we may ask on the actual exam. Since the practice exam (like the real exam) is not long

More information

Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008

Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008 Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008 Code: C-031 Date and time: 17 Nov, 2008, 9:30 A.M. - 12:30 P.M. Maximum Marks: 45 Important Instructions: 1. The question

More information

Without fully opening the exam, check that you have pages 1 through 12.

Without fully opening the exam, check that you have pages 1 through 12. MTH 34 Solutions to Exam April 9th, 8 Name: Section: Recitation Instructor: INSTRUTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show

More information

MATH 52 FINAL EXAM DECEMBER 7, 2009

MATH 52 FINAL EXAM DECEMBER 7, 2009 MATH 52 FINAL EXAM DECEMBER 7, 2009 THIS IS A CLOSED BOOK, CLOSED NOTES EXAM. NO CALCULATORS OR OTHER ELECTRONIC DEVICES ARE PERMITTED. IF YOU NEED EXTRA SPACE, PLEASE USE THE BACK OF THE PREVIOUS PROB-

More information

Practice problems ********************************************************** 1. Divergence, curl

Practice problems ********************************************************** 1. Divergence, curl Practice problems 1. Set up the integral without evaluation. The volume inside (x 1) 2 + y 2 + z 2 = 1, below z = 3r but above z = r. This problem is very tricky in cylindrical or Cartesian since we must

More information

1. Find and classify the extrema of h(x, y) = sin(x) sin(y) sin(x + y) on the square[0, π] [0, π]. (Keep in mind there is a boundary to check out).

1. Find and classify the extrema of h(x, y) = sin(x) sin(y) sin(x + y) on the square[0, π] [0, π]. (Keep in mind there is a boundary to check out). . Find and classify the extrema of hx, y sinx siny sinx + y on the square[, π] [, π]. Keep in mind there is a boundary to check out. Solution: h x cos x sin y sinx + y + sin x sin y cosx + y h y sin x

More information

SOLUTIONS TO PRACTICE EXAM FOR FINAL: 1/13/2002

SOLUTIONS TO PRACTICE EXAM FOR FINAL: 1/13/2002 SOLUIONS O PRACICE EXAM OR INAL: 1/13/22 Math 21a Name: MW9 Sasha Braverman MW1 Ken Chung MW1 Jake Rasmussen MW1 WeiYang Qui MW1 Spiro Karigiannis MW11 Vivek Mohta MW11 Jake Rasmussen MW12 Ken Chung H1

More information

Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015

Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015 Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction

More information

WORKSHEET #13 MATH 1260 FALL 2014

WORKSHEET #13 MATH 1260 FALL 2014 WORKSHEET #3 MATH 26 FALL 24 NOT DUE. Short answer: (a) Find the equation of the tangent plane to z = x 2 + y 2 at the point,, 2. z x (, ) = 2x = 2, z y (, ) = 2y = 2. So then the tangent plane equation

More information

Math 20C Homework 2 Partial Solutions

Math 20C Homework 2 Partial Solutions Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we

More information

Math 350 Solutions for Final Exam Page 1. Problem 1. (10 points) (a) Compute the line integral. F ds C. z dx + y dy + x dz C

Math 350 Solutions for Final Exam Page 1. Problem 1. (10 points) (a) Compute the line integral. F ds C. z dx + y dy + x dz C Math 35 Solutions for Final Exam Page Problem. ( points) (a) ompute the line integral F ds for the path c(t) = (t 2, t 3, t) with t and the vector field F (x, y, z) = xi + zj + xk. (b) ompute the line

More information

Multivariable Calculus

Multivariable Calculus Multivariable alculus Jaron Kent-Dobias May 17, 2011 1 Lines in Space By space, we mean R 3. First, conventions. Always draw right-handed axes. You can define a L line precisely in 3-space with 2 points,

More information

Solutions for the Practice Final - Math 23B, 2016

Solutions for the Practice Final - Math 23B, 2016 olutions for the Practice Final - Math B, 6 a. True. The area of a surface is given by the expression d, and since we have a parametrization φ x, y x, y, f x, y with φ, this expands as d T x T y da xy

More information

S12.1 SOLUTIONS TO PROBLEMS 12 (ODD NUMBERS)

S12.1 SOLUTIONS TO PROBLEMS 12 (ODD NUMBERS) OLUTION TO PROBLEM 2 (ODD NUMBER) 2. The electric field is E = φ = 2xi + 2y j and at (2, ) E = 4i + 2j. Thus E = 2 5 and its direction is 2i + j. At ( 3, 2), φ = 6i + 4 j. Thus the direction of most rapid

More information

Peter Alfeld Math , Fall 2005

Peter Alfeld Math , Fall 2005 WeBWorK assignment due 9/2/05 at :59 PM..( pt) Consider the parametric equation x = 2(cosθ + θsinθ) y = 2(sinθ θcosθ) What is the length of the curve for θ = 0 to θ = 7 6 π? 2.( pt) Let a = (-2 4 2) and

More information

Math 210, Final Exam, Fall 2010 Problem 1 Solution. v cosθ = u. v Since the magnitudes of the vectors are positive, the sign of the dot product will

Math 210, Final Exam, Fall 2010 Problem 1 Solution. v cosθ = u. v Since the magnitudes of the vectors are positive, the sign of the dot product will Math, Final Exam, Fall Problem Solution. Let u,, and v,,3. (a) Is the angle between u and v acute, obtuse, or right? (b) Find an equation for the plane through (,,) containing u and v. Solution: (a) The

More information

Math 11 Fall 2016 Final Practice Problem Solutions

Math 11 Fall 2016 Final Practice Problem Solutions Math 11 Fall 216 Final Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,

More information

Direction of maximum decrease = P

Direction of maximum decrease = P APPM 35 FINAL EXAM PING 15 INTUTION: Electronic devices, books, and crib sheets are not permitted. Write your name and your instructor s name on the front of your bluebook. Work all problems. how your

More information

The Divergence Theorem

The Divergence Theorem Math 1a The Divergence Theorem 1. Parameterize the boundary of each of the following with positive orientation. (a) The solid x + 4y + 9z 36. (b) The solid x + y z 9. (c) The solid consisting of all points

More information

Multiple Integrals and Vector Calculus: Synopsis

Multiple Integrals and Vector Calculus: Synopsis Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things which are (and are not) vectors. Differentiation and integration

More information

Without fully opening the exam, check that you have pages 1 through 10.

Without fully opening the exam, check that you have pages 1 through 10. MTH 234 Solutions to Exam 2 April 11th 216 Name: Section: Recitation Instructor: INSTRUTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through

More information

Final exam (practice 1) UCLA: Math 32B, Spring 2018

Final exam (practice 1) UCLA: Math 32B, Spring 2018 Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 2018 This exam has 7 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions

More information

Name: Instructor: Lecture time: TA: Section time:

Name: Instructor: Lecture time: TA: Section time: Math 222 Final May 11, 29 Name: Instructor: Lecture time: TA: Section time: INSTRUCTIONS READ THIS NOW This test has 1 problems on 16 pages worth a total of 2 points. Look over your test package right

More information

McGill University December Intermediate Calculus. Tuesday December 17, 2014 Time: 14:00-17:00

McGill University December Intermediate Calculus. Tuesday December 17, 2014 Time: 14:00-17:00 McGill University December 214 Faculty of Science Final Examination Intermediate Calculus Math 262 Tuesday December 17, 214 Time: 14: - 17: Examiner: Dmitry Jakobson Associate Examiner: Neville Sancho

More information

McGill University April 20, Advanced Calculus for Engineers

McGill University April 20, Advanced Calculus for Engineers McGill University April 0, 016 Faculty of Science Final examination Advanced Calculus for Engineers Math 64 April 0, 016 Time: PM-5PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer Student

More information

Practice problems. 1. Evaluate the double or iterated integrals: First: change the order of integration; Second: polar.

Practice problems. 1. Evaluate the double or iterated integrals: First: change the order of integration; Second: polar. Practice problems 1. Evaluate the double or iterated integrals: R x 3 + 1dA where R = {(x, y) : 0 y 1, y x 1}. 1/ 1 y 0 3y sin(x + y )dxdy First: change the order of integration; Second: polar.. Consider

More information

51. General Surface Integrals

51. General Surface Integrals 51. General urface Integrals The area of a surface in defined parametrically by r(u, v) = x(u, v), y(u, v), z(u, v) over a region of integration in the input-variable plane is given by d = r u r v da.

More information

MATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS

MATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,

More information