MATH 1151 Midterm: Math_1151_Sample_Midterm_1_SP18

Pens, pencils, erasers are allowed. Note sheets, formula sheets and öthèr sia atus alt not allowed. No additional material is allowed. must be justified, unsupported answers may receive litle to no credit. SHOW ALL WORK. Justifications such as "my calculator says so" are not acceptable. Give EXACT answers, not decimal approximations. For example, v2 is exact, but 1.414 and 1.414213562 are approximations. The total number of points is 100. The number of points for each problem is indicated in square brackets > 7 Problem 1. [18] Find all critical points of the function fx,y)-36a Classify cach critical point as a local minimum, local maximum or saddle point Problem 2. [18] Let R be the region in the first quadrant that is bounded by the curves whose equations are y and y-7x. For your convenience, a graph of R is included on page Evaluate the integral Problem 3. [14] Consider the following iterated double integral: (a) Sketeh the region of integration b Write an equivalent doable integral with the order of integration reversed. DO NOT EVALUATE THE INTEGRAL PLEASE TURN OVER FOR PROBLEMS4THROUGH 6

DUE: Wednesday, November 20 You may work together with so. How other class members on this project. In fact, I encourage you to do ever, each person must turn in their own project, written up neatly on a separate sheet of paper. Do not ask f or assistance on this project from the Math Center tutors. Show your work to receive credit for these problems. Points will be deducted for sloppy and/or unorganized work. Yo u will also lose points for poor or improper notation. As always, you are more than welcome to come ask me for help. (1) This problem will walk you through a development of the equation of the ellipse with center at e origin (0,0). Recall the definition of the ellipse is the set of points whose sum of distances from two given points (the foci) is a specified constant, say d a) (2 points) If the center of the ellipse is the origin and the ellipse is oriented horizontally, what are the coordinates for the two foci? (Recall the definition of variable c when giving your answer.) b) (6 points) Give the distance from an arbitrary point (x, y) to each of the two foci. NOTE: You will have two different answers. Variables x, y, and c will be used in your answers. c) (2 points) Write an equation for the ellipse using its distance definition. Your answer will contain variables x, y, c, and d. d) (10 points) Use algebraic manipulations to rewrite the equation in our standard form for the ellipse. e) (2 points) Using your answer from (d), find a in terms of c and d D(2 points) Using your answer from (d), find b in terms of c and d g) ( 1 point) Justify the Pythagorean relationship, a2 = b2 + c2 . * This project is continued on the back of this page*

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.