B C A NAME FINAL EXAM for MATH 233 Fall 2017 UNC EMAIL PID INSTRUCTOR SECTION INSTRUCTIONS This exam consists of 12 exercises each worth 10 points Write clear solutions so that you can get partial points for your reasoning Write your nal answer in the BOXES provided Calculators and other materials are NOT allowed The duration of the exam is 3 hours I certify that no unauthorized assistance has been received or given in the completion of this work HONOR PLEDGE SIGNATURE This page WILL NOT be graded Use it for scrap work Problem 1 1 Find an equation of the tangent plane to the graph of f x y yex at the point 0 2 Express the plane in ax by cz d form 2 Find a linear approximation of the function f x y yex at the point 0 2 3 Find an approximation of the value 2 03e0 1 This page WILL NOT be graded Use it for scrap work Problem 2 a Sketch the vector 2u 0 5v b Sketch the vector projvu v u v u c Let u h1 1 0i and v h1 0 1i Find the angle between u and v d Let u h1 1 0i and v h1 0 1i Find the area of the parallelogram of sides u and v This page WILL NOT be graded Use it for scrap work Problem 3 Find the absolute minimum value of the function f x y x2 2y2 x on the region x2 y2 4 This page WILL NOT be graded Use it for scrap work Problem 4 Consider the curve in space parametrized by r t ht2 t4 t3i 0 t 2 1 Find parametric equations for the tangent line to the curve at the point 1 1 1 2 Find the work done by the force F x y z hy 0 zi on a particle that moves along the curve described above This page WILL NOT be graded Use it for scrap work Problem 5 Let where f is an unknown function Suppose you know that g s t f 3s est s2 sin t2 2t g 1 0 3 g 4 1 0 f 1 0 5 f 4 1 8 fx 1 0 1 fx 4 1 1 fy 1 0 4 fy 4 1 2 Find 1 0 g s g s 1 0 This page WILL NOT be graded Use it for scrap work Problem 6 Consider the vector eld F x y D2xy2 5 2x2y 3y2E 1 Is F conservative If the answer is no please justify If the answer is yes nd a potential that is a function f so that rf F 2 Evaluate C1 F dr where C1 is given by r t t2i ptj between 0 0 and 1 1 3 Evaluate C2 F dr where C2 is the circle x2 y2 4 travelled in the counter clockwise direction starting at 2 0 This page WILL NOT be graded Use it for scrap work Problem 7 A bug is crawling on a sheet of metal modelled by the xy plane The temperature of the sheet is given by the function T x y 2x2y 3xy 10 1 If the bug is currently at the point 3 1 nd the direction in which the bug should go to decrease temperature most rapidly 2 Find the rate of change of the temperature at the point 3 1 in the direction of h 1 1i 3 Suppose the bug is now at the point 3 1 Find an equation for a curve on which the bug should walk if it wants to stay at exactly the same temperature as it is now This page WILL NOT be graded Use it for scrap work Problem 8 A lamina is shaped as the region in R2 bounded by the curves x 0 y x 2 and y 1 Sketch the region that the lamina occupies shade it on the coordinate plane below and nd the mass of the lamina if its density is given by x y x cos y3 1 2 This page WILL NOT be graded Use it for scrap work Problem 9 Find the volume of the solid that lies in the rst octant and is enclosed by the paraboloid z 1 x2 y2 and the plane x y 2 This page WILL NOT be graded Use it for scrap work Problem 10 Consider the line integral D x2y dx y2x dy where D is the region in the rst quadrant enclosed between the coordinate axes and the circle x2 y2 4 and where D is the boundary curve of the region D traversed in counterclockwise direction Use Green s Theorem to compute this integral This page WILL NOT be graded Use it for scrap work Problem 11 Find the volume of the solid that lies within the sphere x2 y2 z2 1 above the xy plane and below the cone z px2 y2 This page WILL NOT be graded Use it for scrap work Problem 12 Find the surface area of the part of the paraboloid x y2 z2 between the planes x 0 and x 4 This page WILL NOT be graded Use it for scrap work
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