MAT 224 Final, Fall 2015

NDU MAT 224 Calculus IV Final Exam Thursday, June 16, 2016 Duration: 2 hours Name Section Instructor Cell phones are forbidden You have®exercises an@ $ pages. 1/8 1) 65 points) For each of the multiple choice questions below, circle the letter of the correct answer. If more than one letter is circled in the same problem, you will receive no credit for that problem. Question A 3 4.4 Let flxy)= {es = a } Then: x+y a) f(xy) can be extended to become continuous at (0,0) by defining f (0.0) =0 b) f(x,3) can be extended to become continuous at (0,0) by defining f(0,0) =e. c) f(x¥) can be extended to become continuous at (0,0) by defining f(0,0) =] d) f' (x, y) cannot be extended in any way to make it continuous at (0,0). Question B Let w= f(x, y,Z) be a differentiable function withy =sinu+Inv,y=sinv+Inu, z= a , then v ow hm. at (u,v) =(—,—) 18: (ay) “Cap Vv an Ane2y Neg vi 2 Fi pln lly +e a 2 a bp Baty a) 2 pte Ay 2 ca a Question-C The absolute minimum mm and absolute maximum Af of f(x, y)= 2x? + y? —4x on the disk x* + y? <9 have the values a) m= 72, M55 b) m=-2; ME 30 c) m=5; M =30 d) m=5; M=6 2/8 Question D Consider the following two surfaces S, and S, defined by S,:-2x?-2? =2y—4 and S,: y? =x? =-z". The equation of the line tangent to the curve of 2 intersection of the two surfaces at the poin 8 Q ai ——=t+-——, tis areal parameter 3 V3 ——, tis areal parameter a) a(e?-l) b) a(e? +1) a 2) Fed Evaluate i ty —* sin » The Debate Club d) “sin4 8 3/8 Question G Consider the solid $ bounded laterally by x’ + y’ = 1, below by the xy-plane and above by 2 = 4x? +4y*. Then [[fx"d¥’ in cylindri 5 Find the work do: / =yi tay+3 (0, 1, 0) to (1, 2, 1) w (C,): The curve y= x° +1in the xy-plane , 0S x <1 (C,): The line segment from (1, 2, 0) to (1, 2, 1) _* 2 a or ala Mwliwnr oO The Debate Club 4/8 Question I Apply Green’s theorem to evaluate fe — yjde+(2Qxy+ y’)dy, where C is the boundary of the CG region in the first quadrant enclosed by the x-axis, the line x =1, and the curve y=x’. 8 15 Question J Consider the integral ff +y)cos(3y—x)dydx, where R is the triangular region in R the xj-plane bounded by the lines y = 0, =—x, and 3y—x=4. Let G be the region in the uy- plane which is the image of R under the transformation u=3y—x and. v=x+y. Then Part 1 ffie+ cosBy—x)dydx = a) — [[ 4vcosududv G b) {f-rcosudtuay G fe) Way cos ududy d) ff 4y cos ududy G Part 2: The region G in the uv -plane is bounded by the lines: a)u=4,u>—-y, v=0 b) 3v-#34° 4 Sy = 0 c)u=4,u=9, p33 d)u=3, u=3v, y=0 5/8 2) (24 points) a)(12points) Let D be the region that is bounded from below by z = 1 and from above by x? + jy +z’ =4. Sketch D and set up triple integrals in spherical coordinates representing its volume according to the order of integrationdp dé dé. b) (12 points) Set up triple integrals in — its volume ac to the order of integrationdd dp d@ The Debate Club 3) (8 points) The derivative of a function f(x, y,Z)at a point P is greatest in the direction of vai ~2} +2k In this direction, the value of the derivative is 2. Find Vf “at P. 4) (5 points) Find th t y+ zover the path #(t)= 207 +9 +(2-20k, The Debate Club 78 5)(10 points) Consider the field : F (x.y. z)=(2xcos y 22°) +6 +2ye? — x" sin yy +(e? -6x2")k a) (7 points) Show that F’ is conservative and find a potential function for F The Debate Club 8/8 6) (8 points) Use Green’ s theorem to find the outward flux for the field B= 2Qxy? b+x 2 2g across the curve (C): the boundary of the triangular region bounded by the lines y=x, y=—x and p=1 —s =— The Debate Club