we can conclude that ϕ(x, y, z) = sin (xz) + e yz + const. If ϕ is written as a vector but the above three calculations are right, you lose 3pts.
|
|
- Hollie Reed
- 6 years ago
- Views:
Transcription
1 5 微甲 6- 班期末考解答和評分標準. (%) Let F = z cos(z)i + ze yz j + ( cos(z) + ye yz )k. (a) (8%) Find a scalar function ϕ(, y, z) such that ϕ = F. (b) (%) Evaluate C F dr, where C is the curve r(t) = (cos(t ), ln(t + ), tan (t)), t. (a) Because ϕ (, y, z) = z cos (z) ϕ(, y, z) = sin (z) + g (y, z) ϕ y (, y, z) = ze yz ϕ(, y, z) = e yz + g (, z) ϕ z (, y, z) = cos (z) + ye yz ϕ(, y, z) = sin (z) + e yz + g (, y), we can conclude that ϕ(, y, z) = sin (z) + e yz + const. If ϕ is written as a vector but the above three calculations are right, you lose pts. (b) Because F is conservative, C F dr = ϕ(r()) ϕ(r()) = ϕ(, ln, /) ϕ(,, ) = + /.. (%) Let C be the polar curve defined by r = cos θ in the first quadrant. Evaluate C y ds. Let = r cos θ, y = r sin θ (pt) C yds = r sin θ r + ( dr dθ ) dθ (pt) = r sin θ sin θ cos θ + ( cos θ ) dθ = r sin θ cos θ dθ (pt) = sin θdθ = cos θ = (pt). (%) Sketch the region of integration and evaluate the integral 8 cos ( + y + ) dyd. Page of 7
2 ( points) 8 cos( + y = = = ) dy d y + cos( ) d dy y + [(y + ) sin( y + )] (y + ) [sin(y )] dy y dy ( points) Let u = y +, dv = sin(y )dy, and by applying Integration by parts, (y + ) [sin(y )] dy = [ (y + )cos(y )] + [sin(y )] ( points) = cos() + + sin() ( points) = + sin() cos() Note:. If you sketch the wrong part of the region, you will gain ( points) for sketching part.. If you write ( points). 8 y instead of the correct answer, you will gain. If you write ( point). 8 y instead of the correct answer, you will gain. If you miss the minus in Integration by parts, you will gain ( point).. (%) Evaluate R e +y y da, where R is the region in the y-plane bounded by the four lines y =, y =, Page of 7
3 y = and y =. Method : (, y) Let u = + y ; v = y, then we can get J = (u, v) = 5 Therefore, e +y y da = R v v e u v J dudv = v v Method : (, y) Let u = y ; v = y, then we can get J = (u, v) = 5 Therefore, e +y y da = R Method : u e u v u J dvdu = u Let u = y ; v = y), then we can get J = (, y (u, v) = Therefore, e +y y da = R e v+ v J dudv = e u v 5 dudv = 5 ve u v u=v u=v = 5 e v u e 5 dvdu = (e e). u (v ) e v+ v u (v ) dudv = (e e). 給分標準 : J 算錯扣 分積分上下界找錯扣 分前面兩項找對答案算錯扣 分. v(e e) = (e e). 5. (6%) Let E be the space region bounded by the surfaces S = {(, y, z) z = + + y, z }, S = {(, y, z) z = y, z }, and V(, y, z) = yi + j + zk. z S (a) (6%) Evaluate divv dv, where Ω is the solid region enclosed by S S. Ω S y (b) (%) State the Divergence Theorem and evaluate the total outward flu S S V ds. (c) (6%) Compute the upward flu of V across S. Solution 5(a). Ω divv dv = Ω dv = = 5 6 r +r rdzdrdθ Note:. The shape of region is not hemisphere.. divv =, points. Volume of region =, points Page of 7
4 . Volume of region =, points Solution 5(b). Divergence Theorem: Ω: simple solid region S:boudary surface of Ω with positive outward orientation. V = (P, Q, R): vector field with P, Q y, R z continuous on an open region that contains Ω. Then we have Ω V dv = S V ds V ds = 5 S S 6 Note:. Ω, point. P, Q y, R z, point. Ω V dv = S V ds, point. S S V ds = 5 6, point Solution 5(c). Method : Direct calculation r(u, v) = (u cos v, u sin v, u ), r u = (cos v, sin v, u) r v = ( u sin v, u cos v, ) r u r v = (u cos v, u sin v, u) V r u r v = u u u, v S V ds = u u dudv = Method Let S = {(, y, z) + y, z = }. Then V n = on S where n = k which implies This leads to S V ds = S V ds = Ω V dv S V ds = where volume of region has been calculated in part (a). Note: S V ds = Ω V dv is WRONG. Method S V ds = Ω V dv S V ds = 5 6 where the surface integral of S is as follows: = r(u, v) = (u cos v, u sin v, + u), r u = (cos v, sin v, ) r v = ( u sin v, u cos v, ) r v r u = (u cos v, u sin v, u) V r u r v = u u u, v Page of 7
5 Note: Be careful of the sign of the normal S V ds = u u dudv = 6. (%) Evaluate C ( y + y)d (y )dy, where C is described by the polar equation r = cos θ oriented counterclockwise. Let D = { r cos θ, θ } be the region bound by the curve C. Apply Green s Theorem, C ( y + y) d (y ) dy = D = D ( y + ) ( y + ) da ( y y ) da. (6 points) Use polar coordinate = r cos θ, y = r sin θ, D ( y y ) da = = = ( r sin θ) r dr dθ (5 points) (r r= cos θ 5 r5 sin θ ) dθ r= ( cos θ) 5 sin θ( cos θ)5 dθ = + + [ ( cos θ)6 ] θ= θ= =. ( points) 7. (%) Find the area of the sphere + y + z = lying inside the cylinder ( ) + y =. The area = + z ( ) +y + zyda [ points] (z = y, z = y, z y y = y ) [ points] = ( ) +y y da [ points] (Use the polar coordinate to calculate) = cos θ rdrdθ r [ points] = = 8 = 8 cos θ ( r ) dr dθ [ points] ( r ) r= cos θ r= dθ [ points] sin θdθ [ points] = 8( ) [ points] 8. (%) Compute S curlf ds, where F = e z i+( +z )j+(y+cos z)k and where S = {(, y, z) + y 9 + z = and + z } oriented so that the boundary is counterclockwise when viewed from above. Page 5 of 7
6 y y z 9 S z z Method : curlf = F = z, e z, ( pts) Let T be the intersection of the ellipsoid with the plane + z =. Note that the boundary of S and T are the same. On the plane + z = which is z =. Parametrize T by γ(, y) = (, y, γ = (,, ), γ y = (,, ) γ γ y = ( Let D be the projection of T onto y plane. Now, by Stoke s Theorem 5 By symmetry, D ddy =. So ) with the condition + y 9 ( + y,, ) ( pts) S curlf ds = T curlf ds + y 9 Thus S curlf ds =. 9 + z ). = +, e, (,, )ddy D = + y ddy (6 pts) + 5 ddy = Area(D) = = Method : C is the intersection of + y 9 + z = and + z =. We have z = and + y =. So we can parametrize C as 9 ( pts) On C, we have γ(θ) = ( cos(θ) cos(θ), sin(θ), ), θ ( pts) F = e cos (θ), 5 cos(θ) cos (θ), sin(θ) + cos( ) ( pt) γ (θ) = sin(θ) sin(θ), cos(θ), ( pt) F γ (θ) = sin(θ)e cos (θ) + 5 cos (θ) + sin (θ) + sin(θ) cos( cos(θ) ) Then F dr = ( sin(θ)e cos (θ) + 5 cos (θ) + sin (θ) + sin(θ) cos( cos(θ) ) )dθ ( pts) Page 6 of 7
7 Note that f(θ) = sin(θ)e cos (θ) cos(θ) sin(θ) cos( ) + is an odd function. Thus f(θ)dθ = f(θ)dθ =. We also have cos (θ)dθ =. So C F dr = sin (θ) dθ = ( cos(θ)) = = ( pts) Thus, by Stoke s Theorem S curlf ds = ( pts). Page 7 of 7
2( 2 r 2 2r) rdrdθ. 4. Your result fits the correct answer: get 2 pts, if you make a slight mistake, get 1 pt. 0 r 1
Page 1 of 1 112 微甲 7-11 班期末考解答和評分標準 1. (1%) Find the volume of the solid bounded below by the cone z 2 4(x 2 + y 2 ) and above by the ellipsoid 4(x 2 + y 2 ) + z 2 8. Method 1 Use cylindrical coordinates:
More informationon an open connected region D, then F is conservative on D. (c) If curl F=curl G on R 3, then C F dr = C G dr for all closed path C.
. (5%) Determine the statement is true ( ) or false ( ). 微甲 -4 班期末考解答和評分標準 (a) If f(x, y) is continuous on the rectangle R = {(x, y) a x b, c y d} except for finitely many points, then f(x, y) is integrable
More information1. (13%) Find the orthogonal trajectories of the family of curves y = tan 1 (kx), where k is an arbitrary constant. Solution: For the original curves:
5 微甲 6- 班期末考解答和評分標準. (%) Find the orthogonal trajectories of the family of curves y = tan (kx), where k is an arbitrary constant. For the original curves: dy dx = tan y k = +k x x sin y cos y = +tan y
More informationdzdydx. [Hint: switch dzdydx into dxdydz.] z(2 z) dzdydx. z(2 z) 2 )dz z(2 z) 2 sin(πz)dz = 1 x = u + 3v (2 points) y = v (2 points) (u, v) = 1 2
Page of 6. (%) Evaluate x z( z) 微甲 8- 班期末考解答和評分標準 dzddx. [Hint: switch dzddx into dxddz.] We need to compute x z( z) dzddx. The region of integration is bounded b [, ] [, ] [, ] {(x,, z) 3 x +, z }. (4%
More information1 dx (5%) andˆ x dx converges. x2 +1 a
微甲 - 班期末考解答和評分標準. (%) (a) (7%) Find the indefinite integrals of secθ dθ.) d (5%) and + d (%). (You may use the integral formula + (b) (%) Find the value of the constant a for which the improper integral
More informationtan θ(t) = 5 [3 points] And, we are given that d [1 points] Therefore, the velocity of the plane is dx [4 points] (km/min.) [2 points] (The other way)
1051 微甲 06-10 班期中考解答和評分標準 1. (10%) A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is π/3, this angle is decreasing
More informationSolutions to the Final Exam, Math 53, Summer 2012
olutions to the Final Exam, Math 5, ummer. (a) ( points) Let be the boundary of the region enclosedby the parabola y = x and the line y = with counterclockwise orientation. alculate (y + e x )dx + xdy.
More informationPractice 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 informationx + 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 - Problem Drill 24: Stokes and Divergence Theorem
alculus III - Problem Drill 4: tokes and Divergence Theorem Question No. 1 of 1 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as needed () Pick the 1. Use
More informationMath 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 informationMLC 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 informationMAC2313 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 informationName: 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 informationPractice 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 informationMATH 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 informationName: 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 informationMTH101A (2016), Tentative Marking Scheme - End sem. exam
MTH11A (16), Tentative Marking Scheme - End sem. eam 1. (a) Let f(, y, z) = yz and S be + y + z = 6. Using Lagrange multipliers method, find the maimum and minimum values of f on S. [7] Lag. Eqns.: yz
More information7a3 2. (c) πa 3 (d) πa 3 (e) πa3
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
More informationMath 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 informationMTH 234 Solutions to Exam 2 April 13, Multiple Choice. Circle the best answer. No work needed. No partial credit available.
MTH 234 Solutions to Exam 2 April 3, 25 Multiple Choice. Circle the best answer. No work needed. No partial credit available.. (5 points) Parametrize of the part of the plane 3x+2y +z = that lies above
More information1 + 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 informationStokes s Theorem 17.2
Stokes s Theorem 17.2 6 December 213 Stokes s Theorem is the generalization of Green s Theorem to surfaces not just flat surfaces (regions in R 2 ). Relate a double integral over a surface with a line
More informationln e 2s+2t σ(m) = 1 + h 2 x + h 2 yda = dA = 90 da R
olution to et 5, Friday ay 7th ection 5.6: 15, 17. ection 5.7:, 5, 7, 16. (1) (ection 5.5, Problem ) Find a parametrization of the suface + y 9 between z and z. olution: cost, y sint and z s with t π and
More informationMATH 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 informationSOLUTIONS 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 informationAPJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE (SUPPLEMENTARY) EXAMINATION, FEBRUARY 2017 (2015 ADMISSION)
B116S (015 dmission) Pages: RegNo Name PJ BDUL KLM TECHNOLOGICL UNIVERSITY FIRST SEMESTER BTECH DEGREE (SUPPLEMENTRY) EXMINTION, FEBRURY 017 (015 DMISSION) MaMarks : 100 Course Code: M 101 Course Name:
More informationReview 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 informationAPJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 2017 MA101: CALCULUS PART A
A B1A003 Pages:3 (016 ADMISSIONS) Reg. No:... Name:... APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 017 MA101: CALCULUS Ma. Marks: 100 Duration: 3 Hours PART
More informationMath 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 informationPractice 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 informationPractice 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 informationFinal Exam Review Sheet : Comments and Selected Solutions
MATH 55 Applied Honors alculus III Winter Final xam Review heet : omments and elected olutions Note: The final exam will cover % among topics in chain rule, linear approximation, maximum and minimum values,
More informationPage 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 informationOne 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 informatione x2 dxdy, e x2 da, e x2 x 3 dx = e
STS26-4 Calculus II: The fourth exam Dec 15, 214 Please show all your work! Answers without supporting work will be not given credit. Write answers in spaces provided. You have 1 hour and 2minutes to complete
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)
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 informationMath 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 informationDivergence Theorem December 2013
Divergence Theorem 17.3 11 December 2013 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:
More informationJim 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 informationDivergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem
Divergence Theorem 17.3 11 December 213 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:
More informationName (please print) π cos(θ) + sin(θ)dθ
Mathematics 2443-3 Final Eamination Form B December 2, 27 Instructions: Give brief, clear answers. I. Evaluate by changing to polar coordinates: 2 + y 2 3 and above the -ais. + y d 23 3 )/3. π 3 Name please
More information1 4 (1 cos(4θ))dθ = θ 4 sin(4θ)
M48M Final Exam Solutions, December 9, 5 ) A polar curve Let C be the portion of the cloverleaf curve r = sin(θ) that lies in the first quadrant a) Draw a rough sketch of C This looks like one quarter
More informationCalculus 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 informationName (please print) π cos(θ) + sin(θ)dθ
Mathematics 2443-3 Final Eamination Form A December 2, 27 Instructions: Give brief, clear answers. I. Evaluate by changing to polar coordinates: 2 + y 2 2 and above the -ais. + y d 2(2 2 )/3. π 2 (r cos(θ)
More informationContents. 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 informationMATHEMATICS 317 December 2010 Final Exam Solutions
MATHEMATI 317 December 1 Final Eam olutions 1. Let r(t) = ( 3 cos t, 3 sin t, 4t ) be the position vector of a particle as a function of time t. (a) Find the velocity of the particle as a function of time
More informationIn 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 informationPage 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 informationMATH 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 informationn n 2 + 4i = lim 2 n lim 1 + 4x 2 dx = 1 2 tan ( 2i 2 x x dx = 1 2 tan 1 2 = 2 n, x i = a + i x = 2i
. ( poits) Fid the limits. (a) (6 poits) lim ( + + + 3 (6 poits) lim h h h 6 微甲 - 班期末考解答和評分標準 +h + + + t3 dt. + 3 +... + 5 ) = lim + i= + i. Solutio: (a) lim i= + i = lim i= + ( i ) = lim x i= + x i =
More informationMath 212-Lecture Integration in cylindrical and spherical coordinates
Math 22-Lecture 6 4.7 Integration in cylindrical and spherical coordinates Cylindrical he Jacobian is J = (x, y, z) (r, θ, z) = cos θ r sin θ sin θ r cos θ = r. Hence, d rdrdθdz. If we draw a picture,
More informationName: 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(a) 0 (b) 1/4 (c) 1/3 (d) 1/2 (e) 2/3 (f) 3/4 (g) 1 (h) 4/3
Math 114 Practice Problems for Test 3 omments: 0. urface integrals, tokes Theorem and Gauss Theorem used to be in the Math40 syllabus until last year, so we will look at some of the questions from those
More informationSept , 17, 23, 29, 37, 41, 45, 47, , 5, 13, 17, 19, 29, 33. Exam Sept 26. Covers Sept 30-Oct 4.
MATH 23, FALL 2013 Text: Calculus, Early Transcendentals or Multivariable Calculus, 7th edition, Stewart, Brooks/Cole. We will cover chapters 12 through 16, so the multivariable volume will be fine. WebAssign
More informationHOMEWORK 8 SOLUTIONS
HOMEWOK 8 OLUTION. Let and φ = xdy dz + ydz dx + zdx dy. let be the disk at height given by: : x + y, z =, let X be the region in 3 bounded by the cone and the disk. We orient X via dx dy dz, then by definition
More informationGreen s, Divergence, Stokes: Statements and First Applications
Math 425 Notes 12: Green s, Divergence, tokes: tatements and First Applications The Theorems Theorem 1 (Divergence (planar version)). Let F be a vector field in the plane. Let be a nice region of the plane
More informationMath 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 information1. 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 informationSections 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 informationMULTIVARIABLE INTEGRATION
MULTIVARIABLE INTEGRATION (PLANE & CYLINDRICAL POLAR COORDINATES) PLANE POLAR COORDINATES Question 1 The finite region on the x-y plane satisfies 1 x + y 4, y 0. Find, in terms of π, the value of I. I
More informationCOMPLETE Chapter 15 Multiple Integrals. Section 15.1 Double Integrals Over Rectangles. Section 15.2 Iterated Integrals
Mat 7 Calculus III Updated on /3/7 Dr. Firoz COMPLT Chapter 5 Multiple Integrals Section 5. Double Integrals Over ectangles amples:. valuate the iterated integral a) (5 ) da, {(, ), } and b) (4 ) da, [,]
More informationFaculty of Engineering, Mathematics and Science School of Mathematics
Faculty of Engineering, Mathematics and Science School of Mathematics SF Engineers SF MSISS SF MEMS MAE: Engineering Mathematics IV Trinity Term 18 May,??? Sports Centre??? 9.3 11.3??? Prof. Sergey Frolov
More informationMath 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 informationDouble Integrals. Advanced Calculus. Lecture 2 Dr. Lahcen Laayouni. Department of Mathematics and Statistics McGill University.
Lecture Department of Mathematics and Statistics McGill University January 9, 7 Polar coordinates Change of variables formula Polar coordinates In polar coordinates, we have x = r cosθ, r = x + y y = r
More informationReview 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 informationMath 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 informationCreated by T. Madas SURFACE INTEGRALS. Created by T. Madas
SURFACE INTEGRALS Question 1 Find the area of the plane with equation x + 3y + 6z = 60, 0 x 4, 0 y 6. 8 Question A surface has Cartesian equation y z x + + = 1. 4 5 Determine the area of the surface which
More informationMATH 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 informationSolutions 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 informationMath 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 informationMath 234 Review Problems for the Final Exam
Math 234 eview Problems for the Final Eam Marc Conrad ecember 13, 2007 irections: Answer each of the following questions. Pages 1 and 2 contain the problems. The solutions are on pages 3 through 7. Problem
More informationReview for the Final Test
Math 7 Review for the Final Test () Decide if the limit exists and if it exists, evaluate it. lim (x,y,z) (0,0,0) xz. x +y +z () Use implicit differentiation to find z if x + y z = 9 () Find the unit tangent
More informationPractice 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 informationSolutions 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 informationMath 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 informationMA 441 Advanced Engineering Mathematics I Assignments - Spring 2014
MA 441 Advanced Engineering Mathematics I Assignments - Spring 2014 Dr. E. Jacobs The main texts for this course are Calculus by James Stewart and Fundamentals of Differential Equations by Nagle, Saff
More informationTufts University Math 13 Department of Mathematics April 2, 2012 Exam 2 12:00 pm to 1:20 pm
Tufts University Math Department of Mathematics April, Eam : pm to : pm Instructions: No calculators, notes or books are allowed. Unless otherwise stated, you must show all work to receive full credit.
More informationMath 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 informationARNOLD PIZER rochester problib from CVS Summer 2003
ARNOLD PIZER rochester problib from VS Summer 003 WeBWorK assignment Vectoralculus due 5/3/08 at :00 AM.( pt) setvectoralculus/ur V.pg onsider the transformation T : x 8 53 u 45 45 53v y 53 u 8 53 v A.
More informationMultiple 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 informationMATH 1080 Test 2 -Version A-SOLUTIONS Fall a. (8 pts) Find the exact length of the curve on the given interval.
MATH 8 Test -Version A-SOLUTIONS Fall 4. Consider the curve defined by y = ln( sec x), x. a. (8 pts) Find the exact length of the curve on the given interval. sec x tan x = = tan x sec x L = + tan x =
More information2. Below are four algebraic vector fields and four sketches of vector fields. Match them.
Math 511: alc III - Practice Eam 3 1. State the meaning or definitions of the following terms: a) vector field, conservative vector field, potential function of a vector field, volume, length of a curve,
More informationNote: 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 informationCalculus III 2004 Summer Practice Final 8/3/2004
.. Calculus III 4 ummer Practice Final 8/3/4. Compute the following limits if they exist: (a) lim (x,y) (,) e xy x+. cos x (b) lim x. (x,y) (,) x 4 +y 4 (a) ince lim (x,y) (,) exy and lim x + 6 in a (x,y)
More information3, 1, 3 3, 1, 1 3, 1, 1. x(t) = t cos(πt) y(t) = t sin(πt), z(t) = t 2 t 0
Math 5 Final Eam olutions ecember 5, Problem. ( pts.) (a 5 pts.) Find the distance from the point P (,, 7) to the plane z +. olution. We can easil find a point P on the plane b choosing some values for
More informationVector Calculus Gateway Exam
Vector alculus Gateway Exam 3 Minutes; No alculators; No Notes Work Justifying Answers equired (see below) core (out of 6) Deduction Grade 5 6 No 1% trong Effort. 4 No core Please invest more time and
More informationMATHS 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(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 informationMA 351 Fall 2008 Exam #3 Review Solutions 1. (2) = λ = x 2y OR x = y = 0. = y = x 2y (2x + 2) = 2x2 + 2x 2y = 2y 2 = 2x 2 + 2x = y 2 = x 2 + x
MA 5 Fall 8 Eam # Review Solutions. Find the maimum of f, y y restricted to the curve + + y. Give both the coordinates of the point and the value of f. f, y y g, y + + y f < y, > g < +, y > solve y λ +
More information51. 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 informationMA FINAL EXAM Form 01 May 1, 2017
MA 26100 FINAL EXAM Form 01 May 1, 2017 NAME STUDENT ID # YOUR TA S NAME RECITATION TIME 1. You must use a #2 pencil on the scantron 2. a. Write 01 in the TEST/QUIZ NUMBER boxes and darken the appropriate
More informationMath 11 Fall 2007 Practice Problem Solutions
Math 11 Fall 27 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 informationAP Calculus (BC) Chapter 10 Test No Calculator Section. Name: Date: Period:
AP Calculus (BC) Chapter 10 Test No Calculator Section Name: Date: Period: Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The graph in the xy-plane represented
More informationPRACTICE 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= lim(x + 1) lim x 1 x 1 (x 2 + 1) 2 (for the latter let y = x2 + 1) lim
1061 微乙 01-05 班期中考解答和評分標準 1. (10%) (x + 1)( (a) 求 x+1 9). x 1 x 1 tan (π(x )) (b) 求. x (x ) x (a) (5 points) Method without L Hospital rule: (x + 1)( x+1 9) = (x + 1) x+1 x 1 x 1 x 1 x 1 (x + 1) (for the
More informationF x = 2x λy 2 z 3 = 0 (1) F y = 2y λ2xyz 3 = 0 (2) F z = 2z λ3xy 2 z 2 = 0 (3) F λ = (xy 2 z 3 2) = 0. (4) 2z 3xy 2 z 2. 2x y 2 z 3 = 2y 2xyz 3 = ) 2
0 微甲 07- 班期中考解答和評分標準 5%) Fid the poits o the surfce xy z = tht re closest to the origi d lso the shortest distce betwee the surfce d the origi Solutio Cosider the Lgrge fuctio F x, y, z, λ) = x + y + z
More informationMath 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 informationCalculus II Practice Test 1 Problems: , 6.5, Page 1 of 10
Calculus II Practice Test Problems: 6.-6.3, 6.5, 7.-7.3 Page of This is in no way an inclusive set of problems there can be other types of problems on the actual test. To prepare for the test: review homework,
More informationAnswers and Solutions to Section 13.7 Homework Problems 1 19 (odd) S. F. Ellermeyer April 23, 2004
Answers and olutions to ection 1.7 Homework Problems 1 19 (odd). F. Ellermeyer April 2, 24 1. The hemisphere and the paraboloid both have the same boundary curve, the circle x 2 y 2 4. Therefore, by tokes
More information