F13 Study Guide/Practice Exam 3

Size: px

Start display at page:

Download "F13 Study Guide/Practice Exam 3"

James Lester
5 years ago
Views:

1 F13 Study Guide/Practice Exam 3 This study guide/practice exam covers only the material since exam 2. The final exam, however, is cumulative so you should be sure to thoroughly study earlier material. The distribution of content on this practice exam is not necessarily representative of the distribution of content on the actual exam. There are far more problems here than on the actual exam and some of the problems are more difficult than those that will appear on the actual exam. For reference, the sections from the text which we have covered this semester are (portions of) , , , 14.6, 14.7, , You should also study the notes on the course webpage concerning Taylor polynomials and Taylor series. Those are more representative of what we did in class than chapter 11 of Stewart. (1) Let f(x, y) = x 2 sin(x) + cos(y). Find the linear approximatin to f based at (0, 0). (2) Let f(x, y) = x 3 + x 2 y y 2. Find and classify all critical points of f. (3) Let f(x, y) = e x2 +y 2 e (x2 +y 2). Find and classify all critical points of f. (4) Find the point on the plane x 2y +z = 1 that is closest to the point (1, 1, 0) and prove that you ve found the point giving minimum distance. (Hint: minimize the square of the distance.) (5) Let g(x, y) = 1. Find the points on the graph of g which are closest xy to the origin in 3. (Hint: Let s(x, y, z) = x 2 +y 2 +z 2 be the square of the distance from (x, y, z) to the origin and set z = g(x, y).) (6) A company operates two plants which manufacture the same item and whose total cost functions are C 1 = q 2 1 C 2 = q 2 2 where q 1 and q 2 are the quantities produced by each plant. If the item costs p dollars then p = 60.04(q 1 + q 2 ). The goal is to find values for q 1 and q 2 which will maximize the company s profit. 1

2 Carefully set up and describe how you would solve this problem using multi-variable derivatives. You need not actually perform the calculations. (7) Let f(x, y) = x + y and let be the rectangle in 2 with corners (1, 1), (1, 3), (2, 1) and (2, 3). (a) Subdivide into four subrectangles of equal area and write down a sum which approximates (b) Write f da as an iterated integral and solve. (8) For the following functions f and regions set up (but do not solve) an iterated integral equal to Your answer should be something that can be plugged into Mathematica to find the answer. (a) f(x, y) = x 3 y and is a disc of radius 1 centered at the point (1, 1). (b) f(x, y) = sin(xy) and is the triangular region with corners (0, 0), (2, 0), and (1, 5). (c) f(x, y) = x 2 y 2 and is the region bounded by the graphs of y = x 5 and y = x 3. (9) Give an example of the following regions in 2 : (a) A Type I (vertically convex) but not Type II (horizontally convex) region. (b) A Type II but not Type I region. (c) A Type III (both vertically and horizontally convex) region (d) A region that is not of Type I, II, or III. (10) Explain why Fubini s theorem is true. (11) Describe two ways to define the double integral of a function f(x, y) over a non-rectangular region D in 2. (12) Suppose that is a rectangle and that for each n, has been subdivided into n subrectangles each of area n = Area()/n. Let x i be a point in the ith subrectangle. Suppose that f is a continuous function on. Prove that the limit of the average values of f(x i ) is 1 equal to Area() (13) Set up iterated triple integrals to find the volumes (or masses) of the following objects in 3. You do not need to solve the integrals. 2

The green cylinder is x 2 +y 2 = 1 and the orange cylinder is x 2 + z 2 = 1. (c) A certain snow globe is the upper hemisphere of a sphere of radius 2 centered at the origin.

3 (a) The region trapped between the graphs of y = 1, y = 1, y = x 3, z = x, and z = 5. See Figure 1. 3 FIGUE 1. The red plane is the graph of z = 5. The green plane is z = x and the two blue planes are y = 1 and y = 1. (b) The region which is trapped between the cylinders x 2 + y 2 = 1 and x 2 + z 2 = 1. See Figure 2. FIGUE 2. The green cylinder is x 2 +y 2 = 1 and the orange cylinder is x 2 + z 2 = 1. (c) A certain snow globe is the upper hemisphere of a sphere of radius 2 centered at the origin. Its density is given by the equation δ(x, y, z) = x 2 + y 2 + z 2. Write down an iterated integral in spherical coordinates which will equal the mass of the snowglobe. (d) A parking ramp is described in cylindrical coordinates by the equation z = θ for 1 r 2 and 0 θ 2π. Find the volume of the region lying between the ramp and the xy-plane.

4 (14) Carefully state the Change of Variables Theorem for double integrals. (15) Let f(x, y) = y 4 (x 2 + y 2 ). Let be the half disc defined by x 2 + y 2 1 and y 0. Set up an iterated integral in polar coordinates which is equal to (16) ecall that for a 2-dimensional region D, the centroid of D is the point (x, y) defined by: x = y = D x da D da D y da D da Let D be the region defined by the inequalities 1 x 2 + y 2 4 and x 0 and 0 y x. Find the centroid of D. (Hint: Use polar coordinates.) (17) The following problems each give a function f, a region, and a change of coordinates. For each, write down an iterated integral in the new coordinate system which equals (a) f(x, y) = x y. is the triangle with corners (0, 0), (3, 1) and (0, 2). The coordinate change is given by x = s t, y = s t. (b) f(x, y) = 4 x + y. is the region bounded by x 2 y 2 = 1, x 2 y 2 = 4, y = 0, and y = x/2. The region appears in Figure 3. The coordinate change is given by x = r cosh θ, y = r sinh θ. It may help to remember the following facts: d/dθ cosh θ = sinh θ d/dθ sinh θ = cosh θ cosh 2 θ sinh 2 θ = 1 cosh θ = (e θ + e θ )/2 sinh θ = (e θ e θ )/2 4 FIGUE 3

5 (c) Let f(x, y) = 1. Let be the elliptical region Ax 2 + Bxy + Cy 2 1, where A, B, and C are positive constants such that C > B 2 /(4A 2 ). Use the coordinate change s = ( x + B y) A 2A t = y C B2. 4A Computing f da will give the area of. 5

MATH Exam 2 Solutions November 16, 2015

MATH Exam 2 Solutions November 16, 2015 MATH 1.54 Exam Solutions November 16, 15 1. Suppose f(x, y) is a differentiable function such that it and its derivatives take on the following values: (x, y) f(x, y) f x (x, y) f y (x, y) f xx (x, y)