MATH 201 Exam 2, Fall 2003

aN TN AEN ToMERGAN any | LIBRARY ov BERET I OT Math. 201 Quiz Il : N. Nahlus & H. Yamani . Dec. 8 , 2003 NAamMe wecsnsseneneennnecennenee LD seccssncessneeneesenenerse Circle your section number (Sec 9 at 1 W) (sec 10 at 12:30T) _ (sec 11 at 27) (sec 12 at 3:30T) 1. (10%) Find the equation of the tangent plane to the surface 2x? + y2 +4xz—7=0 at the point (J, 1, 1). 2. (10%) Find the points on the surface 2y4 + x? the xy plane -—9 =4yz where the tangent plane is perpendicular to 3. (10%) The equation Pa - xty! +2yz~-8=0 defines y implicitly as a function of x and z. Find the value of Gy/& at the point (1,1,4) (by any method) f(uyex? +~p -5xytl. 4) (10%) Investigate the critical point (1, 1) in the function . os 5a. (5%) Find |ym *Y (sin 4 cos —>) - Justify your apswer x y (x,y (0,0) 4.6 ~) (Hint: Use the curve y =—x® +mx'® ) 5b. (5%) Investigate ]jm im, xo +y? Ba. (5%) Setup but do not evaluate the integral(s) m polar coordinates that represent | the area outside the circle = 1 and inside the lemniscate r= =2c0s26. (Hint: sketch carefully) 8b. (5%) Set up but_do not evaluate the integral(s) in polar coordinates to find the area of the plane region 1n the 1" quadrant hounded by = 3x’, the y-axis, and the straight line y = 3. the parabola y doe venewocepenmmampanejsncnne ten ACTON EI VMN ARATE TS NE LT A IE RS ANY 9, (10%) Find the minimum possible directional derivative of f(x,y) at the point p(-1, 2) given that (D, f)(p) = 20 & (Dy f)(p) = 4 where v=4i+3j & w=7j. (Hint: Use 2 equations in 2 unknowns) 10. (10%) Find the absolute maximum & absolute minimum of the function T(x, y) = 3x? +2y? -—4y +30 | on the circle x? +’ =9. | | | | | | \