f11_math2263_exam3

MATH 2263 Name (Print): Fall 2011 Student ID: Exam 3 Section: December 1, 2011 TA: Time limit: 50 minutes Signature: This exam contains 8 pages (including this cover page) and 6 problems. Check to make sure you have all 8 pages. Enter all requested information at the top of this page, and put your initials on the top of every page, in case the pages become separated. Do not give numerical approximations to quantities such as sin 5, pi, or √ 2. However, you should simplify cos pi4 = √ 2/2, e0 = 1, and so on. The following rules apply: • Show your work, in a reasonably neat and coherent way, in the space provided. All an- swers must be justified by valid mathematical reasoning, including the evaluation of definite and indefinite integrals. To receive full credit on a problem, you must show enough work so that your solution can be followed by someone without a calculator. • Mysterious or unsupported answers will not receive full credit. Your work should be mathematically correct and carefully and legibly written. • A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit. • Full credit will be given only for work that is presented neatly and logically; work scattered all over the page without a clear ordering will receive little to no credit. In the event that you cannot fit your entire answer in the space provided, clearly indicate where the answer continues. You may find the following facts useful: sin2 α = 1 2 − cos 2α 2 cos2 α = 1 2 + cos 2α 2∫ u dv = uv − ∫ v du (Integration by Parts) 1 20 pts 2 15 pts 3 15 pts 4 15 pts 5 20 pts 6 15 pts TOTAL 100 pts MATH2263 Fall 2011 Exam 3 – Page 2 of 8 December 1, 2011 1. (20 pts) Let C be the triangular path from (−1, 1) to (0, 0), to (1, 1), and back to (−1, 1) along straight lines. Compute ∫ C F · dr where F(x, y) = (y sinx− ex)i + (xy +√y)j. MATH2263 Fall 2011 Exam 3 – Page 3 of 8 December 1, 2011 2. (15 pts) Let C be the line segment from (1, 0, 3) to (4, 2,−1). Compute ∫ C (x− yz) ds. MATH2263 Fall 2011 Exam 3 – Page 4 of 8 December 1, 2011 3. (15 pts) Compute the work done by the force field F(x, y) = (2x− y)i + (cos y − x)j in moving an object along the curve r(t) = (cos t,−pi2 sin t), pi2 ≤ t ≤ pi. MATH2263 Fall 2011 Exam 3 – Page 5 of 8 December 1, 2011 4. (15 pts) Suppose F and G are vector fields defined on all of R3 whose component functions have continuous partial derivatives. Let C1 be the straight line from (0, 0, 0) to (pi2, pi, 0) and let C2 be a different curve from (0, 0, 0) to (pi2, pi, 0), parametrized by r(t) = (t2, t, sin t), 0 ≤ t ≤ pi. (a) Suppose you know that ∫ C1 F·dr = 2pi and ∫C2 F·dr = 2pi. Is F conservative, not conservative, or do you not have enough information to decide? Why? (b) Suppose you know that ∫ C1 G · dr = 6 and ∫C2 G · dr = 4. Is G conservative, not conservative, or do you not have enough information to decide? Why? (c) Suppose you know div (F) = 3xyz + 1. Is there a vector field H with curl (H) = F? Why or why not, or do you not have enough information to decide? MATH2263 Fall 2011 Exam 3 – Page 6 of 8 December 1, 2011 5. (20 pts) Let M be the surface which is the part of the paraboloid z = x2 + y2 with 1 ≤ z ≤ 4. Give a parametrization of this surface. Use your parametrization to compute the surface area of M . MATH2263 Fall 2011 Exam 3 – Page 7 of 8 December 1, 2011 6. (15 pts) Compute ∫ C F · dr where F(x, y, z) = 2x i + z j + (xy − xz)k and C is the helix curve r(t) = (cos t, sin t, t), 0 ≤ t ≤ pi2 . MATH2263 Fall 2011 Exam 3 – Page 8 of 8 December 1, 2011 SCRATCH PAGE