Math 2321 Review for Test 2 Fall 11

Transcription

1 Math 2321 Review for Test 2 Fall 11 The test will cover chapter 15 and sections of chapter 16. These review sheets consist of problems similar to ones that could appear on the test. Some problems may be a mix of ideas. There will also be problems that test your understanding of the ideas of the class by getting you to write about them. Some sample questions appear below. Critical Points and Max-Min problems 1A.) Find the critical points of f(x, y) = 6x 2x 3 + 3y 3 9y 2. Use the second derivative test to tell if the critical points are local maximums, minimums or saddle points. (Answer: Points are (1, 0), (1, 2), ( 1, 0), ( 1, 2), types in order are: max,saddle,saddle, min.) 1B) Find and classify the critical points of f(x, y) = 1 3 x3 +xy 2 9x (For the detailed answer, see the solution to problem 1 test 2 Sp 08.) The following problems should be solved by Lagrange multipliers. For each problem ask yourself, What is the objective function?, What are the decision variables?, What are the relations between the decision variables? What is a formula for the objective function in terms of the decision variables?, even if the problem doesn t call for this. if you get the right answers to these questions, and set up the Lagrange multipier equations you will get at least 2/3 credit on most problems. 2.) (F03) A company operates two plants which manufacture the same item and whose joint cost functions (in thousands of dollars) is C(x, y) = x + y 2, where x and y are the quantities (in thousands of units) produced by each plant. The company s objective is to maximize the total production f(x, y) = x + y subject to the budget constraint C(x, y) 4000 a) Sketch the points in the x, y plane that satisfy all the constraints of the problem, and sketch some of the level curves f(x, y) = c of the production function f. Then observe what happens as c increases. Based on your sketch and observation explain why the maximum value of f occurs at a point where C(x, y) is equal to b) Now use Lagrange multipliers to determine how many units should be supplied by each plant, to maximize production. Answer: x = 13, y = 50 3.) (F02 exam) Light intensity varies inversely with the square of the distance. The greater the intensity, the warmer the point. What point on the plane z x 2y = 12 is warmed the most by a light placed at (9, 12, 3)? Answer: (6, 6, 6) 1

2 4.) (Su99) A contractor is building a house and the crossection of the house is drawn below. (The crossection is a rectangle topped by an isosceles triangle.) The cost of materials to build the house is proportional to the perimeter. Since homebuyers like a lot of room, the builder wants to make the crossectional area A as large as possible for a given fixed perimeter P. a) Find a formula for the area A in terms of x, y and θ. b) Find a formula for the perimeter P in terms of x, y and θ. c) Find the angle θ that will maximize the area A for fixed perimeter P. Useful facts: sec (θ) = sec(θ) tan(θ), tan (θ) = sec 2 (θ). Answer: θ = π/6 Integration problems 5.) (a) Draw the graph of the function f(x, y) = 3x 2 + y (b) Use an iterated integral to evaluate the volume of the solid under the graph of f(x, y) and above the triangle defined by the lines y = 1, y = 2x + 1, and x = 1 in the xy plane. (ans. 32) (c) Suppose the base was the triangle defined by the lines y = 1, y = 2x + 1 and y = 2x + 1, what are the limits? (d) Use a Riemann sum with m = n = 2 to get an upper bound for the volume under the graph of f(x, y) = 3x 2 +y 2 +6, over the unit square. 6) Suppose the density of a tetrahedron at (x, y, z) is given by the height of the point above the xy plane. Find the mass of the tetrahedron if the base is the plane defined by z = 2, and walls defined by y = 0, x = 0, z + 2x + 3y = 14. A) Draw the solid, and label each smooth boundary with its equation. B) Draw the shadow, and label each smooth boundary with its equation. C) Setup the integral. 7) Suppose the density of a solid is given by the distance to the xz plane. Find the mass if the solid has as floor the plane defined by z = y, and walls defined by y = 0, y = 4 x 2, and ceiling the graph of the equation z + y = 14. A) Draw the solid, and label each smooth boundary with its equation. 2

3 B) Draw the shadow, and label each smooth boundary with its equation. C) Setup the integral. 8) Find the center of mass of the part of the disk of radius 10 centered at the origin in the first quadrant, assuming the density is constant. ( Hint: Use symmetry to minimize your work.) (answer < x, ȳ >=< 40/3π, 40/3π >.) Integrals in Cylindrical Coordinates. 9) Calculate the volume of the solid bounded above by the sphere of radius R, centered at (0, 0, 0), and below by the plane z = a where R > a > 0. (Answer 2π(1/3R 3 ar 2 /2 + a 3 /6).) 10) Calculate the volume of the sphere of radius R centered at the origin, from which a cylinder of radius a, symmetric around the z axis has been drilled out, a < R.(Answer: (4π/3)(R 2 a 2 ) 3/2.) Integrals in Spherical coordinates 12) Use integration in spherical coordinates to find the center of mass of a hemisphere of radius R with center at the origin, of density proportional to the distance to the xy plane, bounded below by the xy plane. (Answer: < 0, 0, 8R/15 >.) 13) Use integration in spherical coordinates to find the center of mass of an orange slice of radius R of constant density with center at at the origin formed from the planes x = 0 and the plane which makes a 30 angle with the plane y = 0. (Volume of a sphere of radius R is (4/3)πR 3. Answer: < 9R/32, 9R 3/32, 0 >.) 14) Use integration in spherical coordinates to find the center of mass of the hemispherical shell of constant density, if the outer radius is 10 meters and the inner radius is 9 meters, both spheres centered at the origin, and the hemispheres are the part of the spheres above the xy plane. (Answer: < 0, 0, 7395/1736 > meters) 15) Setup the integral for #9 in spherical coordinates. 3

4 Interpreting Plots 11.) Below is a plot of the level curves and gradient field of a function f(x, y). (To make it easy to see the gradient vectors we have made them unit vectors.) The circle is the level curve of a constraint equation g(x, y) = c. a) Circle the critical points of f and say what kind of critical point they are; use M for max, m for min and S for saddle. b) Suppose you start at (4, 1), and move tangent to the gradient field of f. Where do you end up? c) Decide where the restriction of f to the constraint curve has its maximum and minimum values; label these with M for max, m for min as well. Verbalization Problems 12 Using Riemann sums, explain why the mass of a plate is the integral of the density of the plate over the plate. 4

5 13 Using Riemann sums, explain why da in polar coordinates becomes rdrdθ 14 Using Riemann sums, explain why the the iterated integral x2 y 2 dxdy gives the volume under the graph of z = 4 x 2 y 2 over the square 0 x, y 1. 5