MA 261 EXAM 1 Spring 2001 Page 1/6NAMESTUDENT IDRECITATION INSTRUCTORRECITATION TIMEPage 1 /12Page 2 /7Page 3 /18Page 4 /18Page 5 /27Page 6 /18TOTAL /100DIRECTIONS1. Write your name, student ID number, recitation instructor’s name and recitation timein the space provided above. Also write your name at the top of pages 2–6.2. The exam has six (6) pages, including this one.3. Circle the correct answer for problems 1–3. Write your answer in the box providedfor problems 4–12.4. You must show sufficient work to justify your answers.5. Credit for each problem is given in parentheses in the left hand margin.6. No books, notes or calculators may be used on this exam.(5) 1. Let ~a =~i − 2~j +3~k and~b =3~i +4~j +7~k.Then~a ·~bk~ak=A. 8B.3314C.33√14D.16√14E.87(7) 2. Symmetric equations for the tangent line to the curve ~r(t)=et~i+(2t+3)~j+(5−sin t)~kat the point (1, 3, 0) are:A.x − 11=y − 32=z−1B.x − 11=y − 33=z5C.x − 1et=y − 32=z−cos tD. x =1+t, y =3+2t, z = −tE. x =1+t, y =3+3t, z =5tMA 261 Exam 1 Spring 2001 Name Page 2/6(7) 3. Which of the following surfaces represents the graph of f (x, y)=4x2+ y2− 4?MA 261 Exam 1 Spring 2001 Name Page 3/6(9) 4. Find an equation of the plane through the points (1, 2, −3), (4, 1, 1), and (5, 0, 2).(9) 5. If a particle has velocity ~v(t)=2~i +3t2~j + et~k and initial position ~r(0) =~i +2~k, findthe position ~r(t) of the partial at time t.~r(t)=MA 261 Exam 1 Spring 2001 Name Page 4/6(9) 6. If w = f(t2, 2t3), where f(x, y) is differentiable, fx(1, 2) = 5 and fy(1, 2) = 8, computedwdtat t =1.dwdtt=1=(9) 7. Find the directional derivative of f(x, y)=13x3+ x ln y at the point (2, 1) in thedirection from (2, 1) to (5, 5).D~uf(2, 1) =MA 261 Exam 1 Spring 2001 Name Page 5/6(9) 8. Find the length, L, of the curve ~r(t)=13(1+t)3/2~i +13(1−t)3/2~j +12t~k for −1 ≤ t ≤ 1.L =(9) 9. Find an equation of the plane tangent to the graph of f (x, y)=x +1y − 1at the point(3, 2, 4).tangent plane:(9) 10. Find the critical point(s) of f (x, y) = (sin x)(cos y) in the square, 0 ≤ x ≤ π,0 ≤ y ≤ π.critical point(s):MA 261 Exam 1 Spring 2001 Name Page 6/6(9) 11. Apply the second partial derivative test to determine whetherf(x, y)=x3+ y3− xy − 2x −2yhas a relative maximum, a relative minimum, or a saddle point at its critical point(1, 1). Circle the correct answer. (Give reasons for your answer.)Relative MaximumRelative MinimumSaddle Point(9) 12. Find the extreme value(s) of f (x, y)=x2− 6y on the circle x2+ y2= 25.Extreme
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