MATH 2065 Midterm: MATH 2065 LSU s05exf

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turquoisegnu128

15 Feb 2019

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Answer each of the questions on your own paper. Put your name on each page of your paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without supporting work. As usual, a copy of the table of laplace transforms from the text will be provided. In exercises 1 7, solve the given di erential equation. If initial values are given, solve the initial value problem. Some problems may be solvable by more than one technique. Dirac delta function: [8 points] find a particular solution of the di erential equation y . 2 t2 y = t ln t, (t > 0) given the fact that the general solution of the associated homogeneous equation is. The integral formula yh = c1t + c2t2. 3 if 0 t < 2, if t 2. (b) g(t) = e2t(t 1)2 + e 2t cos 5t.

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INSTRUCTIONS: There are 15 problems on this exam, and each problem is worth 13 points, plus 5 points for writing down an approximate volume of the planet Earth in cubic miles at the bottomleft corner of page 3, for a total possible 200 points. For this exam, you are allowed to use one 8.5"X11" sheet of notes (both sides, in your own handwriting), pencil, eraser, ruler, scratch paper and calculator. Even though the questions are multiple-choice, you need to show enough work to justify your answer in order to receive full credit. After you have determined the correct solution to each problem, write the letter corresponding to your answer in the appropriate space on the right side of the page AND circle your answer. If your answer does not agree with any of the choices, after double-checking your work, choose option E) and prominently mark your answer (box, circle, underline, etc.). Time limit: Two hours. Attach (i.e. staple) any notecard and scratch paper to the rear of this exam paper. Good luck! MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. The northern third of Indiana is a rectangle measuring 96 miles by 132 miles. Thus, let D = [0, 96] times [0,132]. Assuming that the total annual snowfall (in inches), S(x,y), at (x,y) euro D is given by the function S(x,y) = 60e-0.001 (2x + y) with (x,y) euro D, find the average snowfall on D. 51.14 inches 52.06 inches 51.78 inches 52.44 inches Evaluate the work done between point 1 and point 2 for the conservative field F. Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. Evaluate the surface integral of G over the surface S. S is the portion of the cone Find the flux of the vector field F across the surface S in the indicated direction. Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. Integrate the function f over the given region. f(x,y) = 1/ln x over the region bounded by the x-axis, line x = 9, and curvey y = ln x Reverse the order of integration and then evaluate the integral Use a double integral in polar coordinates to find the area of the region specified in polar coordinates. the region inside both r = 7 sin Theta and r = 7 cos Theta Use spherical coordinates to find the volume of the indicated region. the region bounded above by the sphere x2 + y2 + z2 = 25 and below by the cone z = Find the center of mass of the rectangular solid of density (x,y,z) = xyz defined by 0 LE x LE 8, Use the given transformation to evaluate the integral. where R is the parallelogram bounded by the lines y = 8x + 8, y = 8x +9, y = -6x + 2, y = -6x +7 Evaluate the line integral along the curve C. Find the work done by F over the curve in the direction of increasing t.

1. (45 marks) Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answer. Unless otherwise stated, it is not necessary to simplify your answers in this question. a) Find the equation of the tangent plane to the graph of the function z = f(x,y) = sell at (0,1). Answer b) Evaluate (T, 0) if f(x,y) = sin(ry). Answer c) Graph of a function z = f(x,y) is given by: Which of the following level curves correspondes to this function. Answer d) If f(x,y) = e(3x – 2y), find lim - Math)-f(x,y) e) Find % if f(x, y) = x2 - y3 and 2 = r sin(t), y = r cos(t). Answer f) Evaluate the directional derivative of f(x, y) = !at (1, -1) in the direction of u =(2, 1). Answer 8) Does Simono converge? h) Sketch the direction field of the differential equation = 1-2y. i) Consider the differential equation ay = 15 - 31, y(0) = 0, and use Euler's method with step size 0.1 to estimate y(0.1). Answer j) Knowing that (-3,-3) is a critical point of f(x,y) = 2:3 + y2 – 4.cy -3.0, use the second derivative test to find out whether it is a local minimum, local maximum, or a saddle point. Answer k) Solve the initial value problem, dy dt 642 2y + cos(u) y(1) = . Answer 1) Is ū=(2,2, -1) perpendicular to v = (5,-4, 2)? Answer m) Find the area between the graph of y = 1 + sin(x) and y=1-sin(x) from 0 to 7. Answer n) Determine the lentgh of the curve y = 8/2,0 SIS 3. Evaluate and simplify your answer. Answer o) Express the volume of the solid obtained by rotating the bounded region that lies between curves y = (x-2)2 and y=1,150 $4 about the vertical line 2 = 5. Do not evaluate this integral. Answer

Math 1113 Fall 2017 . Fulk Project 2 Solve the following problems below individually or in groups of up to three. Each group will submit only one project. Each group member will also submit an explanation their contribution to the group project. Show all work leading to your solutions. To receive credit, you must clearly support you answers. Turn in on Wednesday, November 29 Points may be taken off for sloppy your project at the beginning of class disorganized, or poorly written work 1. (20 points) As viewed from the earth, the angle subtended by the full moon is s 0518Â°. Use this information and the fact that the distance AB from the earth to the moon is 236,900 mi to find the radius of the moon. (20 points) Variable stars are ones whose brightness varies periodically. One of the most visible is R Leonis; its brightness is modeled by the function b) -7.9-2.1 measured in days. a) Find the period of R Leonis. b) Find the maximum and minimum brightness c) Graph the function b. 2. where t is 3. (20 points) A movie theater has a 25-foot-high screen that is 8 feet above eye level. If you sit too close, your viewing angle is too small. If you sit too far back, the image is too small. If you sit x feet back from the screen, your viewing angle, Î¸, is given by Find the viewing angle, in radians, at distances of 5 feet, 10 feet, 15 feet, 20 feet, and 25 feet. Then graph the function in a graphing calculator with a viewing window of [0, 50, 10] by 10, 1, 0.1]. Use the calculator to find the distance at which you have a maximum viewing angle. [Hint: Viewing windows are written as [minmax Xscl by Dmin.Jmax Yscil. These values can be adjusted by pressing the WINDOW button on the calculator. To find a maximum, press 2ND TRACE, choose maximum, move to the left of the maximum and press ENTER, move to the right of the maximum and press ENTER, and then press ENTER again. (x-79j+12, (20 points) The number of hours of daylight in Boston is given by y-3sin where : s he number of days after lanuary 1. Within a year,when does Boston have 13.5 hours of daylight? Give your answer in days after January 1 and round to the nearest day. 4

MATH 2065 Midterm: MATH 2065 LSU s05exf

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