MATH 201 Exam 2, Spring 2008

i : 7 Ne Lyre) An overview of the exe % S AEA USIVENSIL 4 IBRARY ‘ (the problems are repeated inside the booklet). ; . oe “ oa d OF BEERCT 1. a) (4 points) Sketch the following curves in polar coordinates and find the two inter- section points. There are no tricks involved!!! ft, Gere WW © f4alh Zo\ Cy: r= wl w Cy: r=1+cosé b) (4 points) Using polar coordinates, SET UP a-1 integral which gives the area of the region that is outside C and inside Cp. DO NOT EVALUATE THE INTEGRAL. 2. (8 points) Find the equation of the tangent plane to the surface z = x? + 3y? — 7 at the point (2,—1,0). 3. Consider the function ™ f(z,y,z)=2+ 4—a5 —y?. a) (2 points} Describe the domain of f and sketch a picture. b) (8 points) Is the domain open? Why or why not? c} (3 points) Is the domain bounded? Why or why not? 4. (8 points) Prove that the function . g- sin 2? if (x,y) # (0,0) f(y) = wry G, if (x, y) == (0, 0) is continuous at (0,0). 5. Let RB be the region in the zy-plane between the 1 1e y = 2x and the parabola y = z?. a) (8 points) Write down the iterated integrals for f fyx GA in both orders (this mets both dy dx and dz dy). : b) (4 points) Evaluate ONE of the integrals from part a). a 6. We are given a differentiable function f(x,y) suck that: GF(0,1) = -t+9 Of, -4) = 2427 of(,2) =i+3. ) a) (8 points) By about how much will f(x, y) chauge if the point P(x, y) moves from Po(0, 1) a distance of ds = 0.05 units in the direction f the vector 7 = 31+ 47? b) (8 points) Let the point P(z,y) follow the «urve (x(t), y(t)) = (t? — 8,é — 1). Compute d Sl s(a.uo)] at -=3 7. a) (4 points) Find the critical point(s) of f(x,y) = 22 +y? — 122+ 6y and classify each critical point as a local maximum, a local minimum o:; a saddle point. b) (8 points) Find the absolute maximum and minimum values of f when we restrict, (x,y) to lie in the region 2? + y” < 5. ii Ay SID aaned \ 7 La h, ‘