Final Examination MATH 2321Fall 2010 #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Total Extra Credit Name: Instructor: Students are allowed to bring a 8 1 2 11 page of formulas. Answers must be supported by detailed calculations to get credit. Some helpful formulas: Volume of a sphere of radius r is (4/3)πr 3, surface area is 4πr 2. Volume of a cone of radius r and height h is (1/3)πr 2 h. 1 (8 points) The pressure P of a certain quantity of gas depends on its temperature T and volume V, as given by the table below: T=300 T=310 T=320 T=330 V = 2.0 11.5 11.9 12.3 12.7 V = 2.2 10.5 10.8 11.2 11.5 V = 2.4 9.6 9.9 10.3 10.6 V = 2.6 8.9 9.2 9.5 9.8 (a) Use the table to approximate P/ T at T = 310 and V = 2.4. (b) Use the table to approximate P/ V at T = 310 and V = 2.4. (c) Using your answers from the previous two questions, write the linearization of this function at T = 310 and V = 2.4. (d) Use the linearization to approximate the pressure when T = 308 and V = 2.5. 2 (8 points) Find the mass of the solid lying above the rectangle with sides 0 x 10 and 0 y 20 in the xy-plane, and below the plane z = x, with density function ρ(x, y, z) = x.
3 (11 points) The barometric pressure P in millibars is measured at various points around the Boston area. Measuring from the Blue Hills Observatory, it is discovered that the barometric pressure drops at a rate.04 millibars for each kilometer you move north, and increases at a rate of.03 millibars for each kilometer you go east. (Assume the positive y-axis points North, and the Blue Hills observatory is at position (0, 0).) (a) Use the data to find the gradient of P at (0, 0). (b) Use your answer to part a) to find the rate of change of P in the Northwest direction at (0, 0). (c) A packet of air released from rest will move in the direction in which pressure falls the fastest. If the air was released at (0, 0) in which direction would it move? (Your answer should be a unit vector.) (d) The curves along which the pressure is constant are called isobars. Geostrophic winds are the air currents created by the rotation of the earth and action of the pressure gradient P. These two forces combine to make these air currents blow parallel to the isobars in the direction such that the pressure is greater to the right. In which direction does the geostrophic wind move at (0, 0)? (Your answer should be a unit vector.) 4 (10 points) Compute S curl F d S where F = x 3 +zx+y, ln(z +y), cos(x+y +z) and S is the part of paraboloid z = 4 x 2 y 2, z 0 oriented downward.
5 (10 points) The Tao chemical company which has three factories, all of which are overloaded, receives a rush order for 6 tons of fertilizer. The cost of producing fertilizer for an overloaded factory is the product of the overload rate for the factory and the square of the amount of fertilizer produced by the factory. We are given that the overload rate for factory A is 2, for factory B is 4 and the overload rate for factory C is 5. We are wondering how much fertilizer each factory should produce. (a) ) Find a formula for the total cost of producing fertilizer in terms of the amounts produced by each factory. (b) Use Lagrange multipliers to find out how much fertilizer each factory should produce to minimize cost.
6 (10 points) Consider the following integral: (a) Sketch the region of integration. 1 e 0 e x 2y dy dx ln y (b) Switch the order of integration. (c) Evaluate your new integral. 7 (12 points) Make a solid S by starting with the solid hemisphere of radius 10 centered at the origin with z 0, and removing the part lying above the cone with equation z 2 = 3(x 2 + y 2 ). Assume S has constant density. (a) Set up an integral in spherical coordinates for the z coordinate of the center of mass. Do not evaluate the integral. (b) Using the geometry of the solid, find the the x and y coordinates of the center of mass, providing a justification of your answer.
8 (10 points) (a) Let C be the clockwise path around a triangle with vertices at (0, 0), (1, 1), and (1, 0). How much work does the force field F (x, y) = (x 2 + 1) 3/2 + e y, x 2 + sin(y 5 ) do on an object that moves along C? (b) Let C be the path along the curve x = 1/y from (1, 1) to (1/2, 2). Find (πy cos(πx)) dx + (sin(πx)) dy. C
9 (10 points) In this problem we look at the electric flux due to an electric field in different situations. (a) Suppose the electric field due to a charge distribution is given by E(x, y, z) = k < 3x, 3y, z +4 >. Find the the electric flux over the surface which is the boundary of the solid hemisphere with equation x 2 + y 2 + z 2 R 2, z 0. (b) What is the electric flux over the hemisphere with equation x 2 + y 2 + z 2 = R 2, z 0, with E(x, y, z) = k < 3x, 3y, z + 4 >? Explain how and why the answers of A) and B) are related. (Assume the orientation of the hemisphere is away from the origin.) (c) (extra credit 3 points) Suppose E = < x, y, z3 x 2 y 2 > x2 + y 2, and S is a cylinder with equation x 2 + y 2 = R 2, h/2 z h/2, ends not included, with outward orientation. What is the flux over the cylinder?
10 (11 points) Below is a plot of the level curves and the gradient field of a function f(x, y). The black ellipse is the graph of a constraint equation g(x, y) = c. (a) Circle the critical points of f and say what type they are; use M for max, m for min and S for saddle. (b) Suppose you start at (6, 10), and move tangent to the gradient field of f (ie. in the direction of f). Draw the path you follow on the plot. Where do you end up? (c) Label with M and m the points where the restriction of f to the constraint curve has its local maximum and minimum values. (d) Draw a curve along which the work done by f is negative. Justify your answer. (e) If you start at a point on the ellipse which you have labeled with an M and go once around the ellipse, what is the work done? Justify your answer.