MIDTERM April 24, 2002 Instructions. Please show our work. You will receive little or no credit for an answer not accompanied b appropriate explanations, even if the answer is correct. If ou have a question about a particular problem, please raise our hand and one of the proctors will come and talk to ou. Calculators or computers of an kind are not allowed. You are not allowed to consult an other materials of an kind, including books, notes and our neighbors. You will find a list of some useful formulas on page 2 of the exam. At the end of the exam, please hand the exam paper to our TA. Please be prepared to show our universit ID upon request. If ou have a question about the grading of a particular problem, please come and see me or one of the TAs within 4 das of the exam. Name: Student ID: # #2 #3 #4 #5 Total
C! #"%$'&#( ) * +-,$'&#( ). / 0 23 465 98-: 4#7 8-< 8-; = >? 8-; @BA C 4#7 4#5 GF H/I>? 4 ED 4 )D @ F SUTW SX ZY SUT Y[Y SX Y. ) Y SUT\ SX YZY KJMLNP 6R SUT Y[Y SX Y[]6 ] 2
Y s d SMT Problem. Let be the vector ^ _G`_Ga[b SX and be the vector ^ _c _Gd b. (a) Find the unit vector in the same direction SeT S T as SX. (b) Find the angle between the vectors and. (c) Let f be the line through d`_gd_gd N SUT in the direction of and f be the line through g _Gd_0a N SX in the direction of. Write down the parametric equations of f and f. Then determine if these lines are parallel, skew, or intersecting. SeT Solution. (a) The length of is h i ja h k lm onp h cl q. Thus the desired unit vector is " r 't. (b) We have that S T/ SX k u k ja a a ) Y SUT Y[Y SX h cl h v jd h *w h xz So W ) * " { s }. (c) f : ~, Z, Zaƒ, m ; f : ~ ˆŠ/, ˆ, a, Š Wˆ v S T SX. Since the angle between the vectors and is not d, the vectors are not parallel, and so the lines f and f are not parallel. Let us check if the are intersecting. At the point of intersection we must have ~ IˆŒi /ˆ aƒ Wa From the last equation ; thus î. The first equation is _Ž`_0a satisfied, so that the lines do intersect at N. 3
SMT Problem 2. Let be the vector ^ _) _cg b SX and S T be the vector SX ^ _) _c b. Find the equation of the plane parallel to both and, and passing through the point g _cg _cg N. SMTi\ SX ^ `_)Š`_0d[b. This gives Solution. We compute the cross product us a normal vector to the plane. Thus the equation is or N ~ W j N W od ~ d N W Wd_ 4
Problem 3. Find an equation in polar coordinates that describes the line a ~ v. In our answer, please indicate the range of the parameter. Solution. Let s sa that LNP. Then we will have ~ ) * LNP L2N-. Thus LNP /a ~ všwa ) * W LNP Solving this for LNP gives so that LN-*)N 3a ) * 5 and LN-* 3a ) For LNP to make sense, we must have a ), i.e., + 0 * " a. Thus the range of is from d to, excluding all values of 0 * " a.
š š N a š F F Problem 4. Consider the polar curve Š3 Œa (a) Sketch the curve. (b) Find the area between the ~ -axis and the part of the curve which lies below the ~ axis. Solution.(a) The graph is shown below: 6 3 (2-sin(3*t))*cos(t), (2-sin(3*t))*sin(t) 2.5 2.5 0.5 0-0.5 - -.5-2 -3-2 - 0 2 3 (b) From the picture, we see that the part of the curve that lies below the W. Thus the area is ~ axis corresponds to š H m3 Œa F N š lœ3l Œa o a * l* u ža š š Ÿ Ÿ š Ÿ nƒ l l a 3 ž
A N A A Problem 5. Sketch the following planar curve, indicating foci, intersections with the coordinate axes and asmptotes, as approprite: ~ d Solution.Rewriting the equation we get ~ E h N h From this we find that the foci are located on the -axis, at I #. The hperbola intersects the axis at I ~. The hperbola looks like this: j 7. Asmptoticall, 2 sqrt(x*x + 5)/5.5 0.5 0-0.5 - -.5-2 -2 -.5 - -0.5 0 0.5.5 2