MA 261 Exam 1 Spring 2000 Page 1/9NAMESTUDENT ID #RECITATION INSTRUCTORRECITATION TIMEDIRECTIONS1) Fill in the above information. Also write your name at the top of each page of theexam.2) The test has 9 pages, including this one.3) Problems 1 through 6 are multiple choice; circle the correct answer.4) Problems 7 through 10 are problems to be worked out. Write your answer in thebox provided. YOU MUST SHOW SUFFICIENT WORK TO JUSTIFY YOURANSWERS. CORRECT ANSWERS WITH INCONSISTENT WORK MAY NOTRECEIVE CREDIT.5) Points for each problem are given in parenthesis in the left margin.6) No books, notes, or calculators may be used on this test.Page 2 /20Page 3 /20Page 4 /10Page 5 /10Page 6 /10Page 7 /10Page 8 /10Page 9 /10TOTAL /100MA 261 Exam 1 Spring 2000 Name: Page 2/9(10) 1) Parametric equations for the line that contains the point (1, −2, 3) and is perpendicularto the plane 3x − 4y +2z =8are:A. x =1+3t, y = −2 − 4t, z =3+2tB. x =3+t, y = −4+2t, z =2+3tC. x =8+3t, y =8− 4t, z =8+2tD. x = −1+3t, y =2− 4t, z = −3+2tE. x = −1 − 3t, y =2+4t, z = −3 − 2t(10) 2) lim(x,y)→(0,0)sin(x2+ y2)(x2+ y2)· (y +2)isequalto:A. 0B. 1C. 2D. 4E. Does not exist.MA 261 Exam 1 Spring 2000 Name: Page 3/9(10) 3) Symmetric equations for the line tangent to the curve−→r(t)=t2~i +(3t − 4)~j +(2− t2)~kat the point (4, 2, −2) are given by:A.x − 44=y − 23=z − 2−4B. x =4andy − 22=z +23C.x − 44=y +2−3=z − 2−4D.x − 44=y − 23=z +2−4E.x − 44=y − 32=z +4−2(10) 4) Let S be the level surface of f(x, y, z)=x2− y2−z24corresponding to c =1. Theintersection of S with the xy plane is:A. two linesB. a circleC. a parabolaD. an ellipseE. a hyperbolaMA 261 Exam 1 Spring 2000 Name: Page 4/95) An object has acceleration−→a(t)=et~i+2~k, initial velocity−→v(0) =~i, and initial position−→r(0) = 2~j. Find the position vector of the object at time t =1.A. (e − 1)~i − 2~j +~kB. (e − 1)~i +2~j +~kC. e~i − 2~jD. e~i +2~j +~kE. e~i +2~j −~kMA 261 Exam 1 Spring 2000 Name: Page 5/9(10) 6) Let f (x, y)=ln(x2+ y2) with x = g(t)andy = h(t). Assuming that g(0) = 1,h(0) = 3,g0(0) = 2, and h0(0) = 4, the value ofddt(f(g(t),h(t)) when t =0is:A.15B.25C.35D.75E.145MA 261 Exam 1 Spring 2000 Name: Page 6/97. Consider the plane containing the points (0, 1, 2), (1, 2, 3, and (2, 1, 0).(5) a) Find a vector ~n which is perpendicular to the plane. (Put your answer in the boxbelow.)Answer to 7.a)~n =(5) b) Find the equation for the plane.Answer to 7.b)MA 261 Exam 1 Spring 2000 Name: Page 7/98. Consider the curve given by:−→r(t)=t2~i +2t~j + (ln t)~k, 1 ≤ t ≤ e.(6) a) Write down an integral that gives the arclength L of this curve (including limitsof integration).Answer to 8.a)L =Z(4) b) Compute the integral in 8.a) to get the exact value of the arclength L.Answer to 8.b)L =MA 261 Exam 1 Spring 2000 Name: Page 8/99. Let f (x, y)=x2exy.(6) a) Find∂2f∂x∂y.Answer to 9.a)∂2f∂x∂y=(4)b) Whatis∂2f∂x∂yat the point (1, 0)?Answer to 9.b)∂2f∂x∂y(1, 0) =MA 261 Exam 1 Spring 2000 Name: Page 9/910. A function f(x, y) is positive if y>2, negative if y<2. The graph of f is a planewhich intersects the xy plane at a 45–degree angle.(3) a) Find∂f∂x(0, 2).Answer to 10.a)∂f∂x(0, 2) =(3) b) Find∂f∂y(0, 2).Answer to 10.b)∂f∂y(0, 2) =(4) c) Find the directional derivative of f at (0, 2) in the direction~i −~j.Answer to