Calculus 3 Exam 2 31 October 2017 Name: Instructions: Be sure to read each problem s directions. Write clearly during the exam and fully erase or mark out anything you do not want graded. You may use your calculator or Mathematica for any calculation or derivation unless otherwise stated so long as you show your work leading up to that point and then state that that is what you are doing. You must show all your work to receive full credit unless otherwise stated. Don t cheat. See below. Prohibited: Class Notes, Handouts Homework Notebooks Study Guides and Materials Previous Work of any Form The Book Any Electronics (including phones) besides an approved TI Calculator or Mathematica The Internet Other People s Exams Page: 3 4 5 6 7 8 9 10 Total Points: 21 30 20 15 25 10 19 10 150 Score:
MATH 225 Exam 2, Page 2 of 10 Fall 2017 Scratch Work out of a possible 0 points
MATH 225 Exam 2, Page 3 of 10 Fall 2017 True or False 1. Circle either T for True or F for False. You do not have to show any work or explanation for this section. (a) (3 points) T F The equation of the plane passing through the points: P, Q, and R will have normal vector P Q P R (b) (3 points) T F On a contour diagram, contours represented equally spaced out values but occurring with greater frequency (getting closer together) represent an increase in steepness of the surface. (c) (3 points) T F To calculate the limit of a multivariable function as (x, y) (a, b), one ever only has to check two approach paths to (a, b). (d) (3 points) T F Clairut s Theorem states that if all of the appropriate partial derivatives of a function f(x, y) exist and are continuous on an open disk D, then f xy = f yx on D. (e) (3 points) T F Given a function differentiable at P, the tangent line to the level curve through P and the gradient of the function at P are parallel. (f) (3 points) T F This question is a scale to see if you re still reading. Circle both T and F for credit. (g) (3 points) T F It is possible for a saddle point to also be a local maximum. out of a possible 21 points
MATH 225 Exam 2, Page 4 of 10 Fall 2017 Computation 2. Calculate the following quantities by hand (ie. No Mathematica). (a) (10 points) g x (x, y) if g(x, y) = y ln (2x 5) (b) (10 points) f xy (x, y) if f(x, y) = yx cos(πx) (c) (10 points) 8 h x 2 y 2 z 4 if h(x, y, z) = x7 y tan(x 9 z 12 ) out of a possible 30 points
MATH 225 Exam 2, Page 5 of 10 Fall 2017 3. (10 points) Given the function defined below, determine the local linearization of f(x, y) at the point P (2, 1). f(x, y) = e 2y x 4. (10 points) Determine the equation of the line through the point P (2, 3, 4) and parallel to the intersection of the planes defined below. Q : 3x y + z 2 = 0 & R : x + y 2z + 1 = 0 out of a possible 20 points
MATH 225 Exam 2, Page 6 of 10 Fall 2017 5. Consider the function below. f(x, y) = x 3 + 6xy 2y 2 + 1. (a) (10 points) Find and classify all local maxima, minima, and saddle points of the function f(x, y). You may use Mathematica/TI-89 to graph the surface to get a sense of which type of points you are looking at and to compute any partial derivatives but you must find and justify the points explicitly by hand using the Second Derivative Test. (b) (5 points) How fast is the surface z = f(x, y) increasing or decreasing at (1, 1) along the direction < 2, 1 >? out of a possible 15 points
MATH 225 Exam 2, Page 7 of 10 Fall 2017 Concepts 6. Consider the function f(x, y) = 36 x 2 y 2 = 36 (x 2 + y 2 ). (a) (5 points) What is the domain of f(x, y)? (b) (5 points) What is the range of f(x, y)? 7. (15 points) Match each standard form equation on the left with the appropriate quadratic surface on the right by writing the corresponding letter in the blank column denoted Match. Assume all constants are positive. You do not have to show any work for this problem. Match Equation Label Quadratic Surface x 2 a + y2 2 b + z2 2 c = 1 A Elliptical Paraboloid 2 x 2 a + y2 2 b = z2 2 c B Hyperbolic Paraboloid 2 x 2 a y2 2 b z2 2 c = 1 C Elliptical Cone 2 x 2 a + z2 2 c = y 2 b D Hyperboloid of Two Sheets x 2 a y2 2 b = z 2 c E Hyperboloid of One Sheet F Ellipsoid G Cylinder out of a possible 25 points
MATH 225 Exam 2, Page 8 of 10 Fall 2017 8. (10 points) Considering the contour diagram below of a multivariable function z = f(x, y), sketch a cross section of the surface showing height with y = 4 held constant on the xz-plane below. 6.4 10 5.6 8 6 4.8 4 2 4 0 3.2-2 -4 2.4 1.6 0.8 0 0.5 1 1.5 2 2.5 3 3.5 4 11 9 z 7 5 3 1 1 3 5 1 2 3 4 x out of a possible 10 points
MATH 225 Exam 2, Page 9 of 10 Fall 2017 9. Assume that w = f(x, y, z) where x = g(t, s), y = h(t, s) and z = j(t, s). (a) (15 points) Draw a single branch diagram showing how to find all of the Chain Rules for each derivative. (b) (4 points) Then write down the specific chain rule to determine w t. out of a possible 19 points
MATH 225 Exam 2, Page 10 of 10 Fall 2017 Applications 10. (10 points) Using Lagrange multipliers, determine the maximum and minimum values of f(x, y) = e 2xy subject to the constraint x 2 + y 2 = 16. Give your answers as ordered triples (x 0, y 0, z 0 ) being sure to label them as max or min values. 11. (3 points (bonus)) What do you think my almost three year-old daughter is dressing up as for Halloween tonight? out of a possible 10 points