S01_First_Midterm_Makeup-G.Bergman

S01_First_Midterm_Makeup-G.Bergman - function on the...

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l0/04/2O0I THU 16:14 FAX 6434990 George M. Bergman 959 Evans MOFFITT LIBRARY Spring 2001, Math 53M First Midterm - Make-up Exam @ oor 23 February,2O0l l0:10-11:00AM 1. (a) (54 points, 9 points apiece) Find the following. If an expression is undefined, say so. dy,/dx, where ,r= 2sin ("t), y = 5cos(et). Express your answer as a function of t. (b) The length of the space curve given by the parametric equations x = 2 "r, y = "2r, z=t (-1 <l(+1). (") tifl,,-'(o,o) (lxl+2)/( lrl+7). (d) Thc equation of the plane tangent to the surface ,= (*2 +y)% at the point where x=3, !=7. a .\L t") #En f?yz) where / is a clifferentiable function. @xpress your answer in terms of / and its derivatives.) * "-"j+ (tan t)k))dt (where i, j and k are the standard basis vectors 2. (34 points) (a) (20 points) Let f be a positive continuous real-valued
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Unformatted text preview: function on the interval f-n/4, n/4'1. L'et A denote the area between the curve whose expression in polar coordinates is r = f(o) (-n/4 &lt; o &lt; n/4) andthe two lines o = -n/4 and 0 = n/4. Let B denote the area between the curve whose expression in polar coordinates is r=f(0/2) (-n/2&lt;o&lt;n/2) andtheverticalaxis a=!n/2. showthat B=zA. You may assume area formulas given in Stewart. (b) (14 points) Find the area between the y-axis and the curve whose expression in polar coordinates is r = sec O/2. You may use the result of part (a) whether or not you have proved it; or you may use any other method that gives the correct answer. 3. (12 points) Find equations in Canesian (i.e., (x,y, z)) and spherical coordinates for the surface described in cylindrical coordinates by the equation y2 = 72 + r. {D [i&lt;tz x 1t2i in IR')....
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