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Multivariable Calculus FAMAT State Convention 2009 The answer choice E. NOTA denotes that “None of These Answers” are correct. DNE stands for “Does Not Exist”. The domain and range of functions are assumed to be either the real numbers or the appropriate subset of the real numbers. 1. Evaluate the following limit: lim lim x 1 h 0 A) 2 B) 1 4y 2. If x3 y 2 z 6 xyz 1 , what is A) 3x 2 6 yz 2 y 1 6 xy B) 2 x 2 ln( y h) 2 x 2 ln( y) h 2 C) D) 4 ln y y E) NOTA z z ? x y 3x 2 2 xz 1 6 xy C) 3x 2 6 yz 6 xz 1 6 xz D) 1 E) NOTA 4 f 3. Consider the function f ( x, y, z ) and u ( x, y, z ) a b c , where a b c 4 (a,b,c are all whole x y z numbers). What is the maximum number of distinct values of u (1, 2,3) ? A) 64 B) 20 C) 32 D) 15 E) NOTA For questions 4 and 5, consider the volume V bounded in the 3-dimensional coordinate system by the graphs x 2 y 2 9, x z 5 , and z 0 . 4. Find the z coordinate of the center of gravity, given that the volume bound is V 45 units 3 . A) 113 45 B) 119 45 C) 13 5 D) 3 E) NOTA 5. Find the radius of gyration k z with respect to the z-axis I z A) 2 2 B) 3 2 C) 3 2 2 D) 3 ( x 2 V 2 y 2 )dV k z V . E) NOTA 6. Find the distance between the point (1, 2, 0) and the tangent plane to the surface 2 x 2 4 y 2 z 2 9 at the point (0, 2, 5) . A) 0 B) 17 89 89 C) 7 89 89 D) 5 87 E) NOTA 3 1 For questions 7 and 8, consider the production function P( K , L) 2 K 4 L 4 , where P is production, L is labor, and K is capital. 7. If K remains constant, find C such that PL ( K , C) 2 PL ( K , 2) , where PL ( K , L) is the partial derivative of the production function with respect to L. 1 2 1 A) 3 B) 3 C) e D) e 2 2 E) NOTA 8. Find the marginal increase in production with respect to capital when K = 16 and L = 81. A) 6 4 B) 5 6 8 C) 3 6 2 D) 3 E) NOTA Multivariable Calculus FAMAT State Convention 2009 9. The electric field vector at a distance d away from a charged particle has magnitude of E k q directed away from positive charges (where k is a constant, q is magnitude of d2 the charge). Find the magnitude of the vertical component of the electric field at a point 2 meters above the center of a charged solid disk that is 2 meters in radius given E dE . Take the charge density of the disk to be 0.1 C A) 30 k B) 40 k C) 2 20 k D) 2 400 m2 k . E) NOTA For questions 10 and 11, consider f ( x, y) x 2 y 3 4 y . 10. Find the directional derivative of f ( x, y ) at the point (2, 1) in the direction of the vector 2,1 . A) 1 B) 16 5 5 C) 16 5 5 D) 0 E) NOTA 11. If the directional derivative of f ( x, y ) at the point (2, 1) in the direction of the unit vector a, b is a minimum, find a . b A) 0 B) 1 2 C) 2 D) 1 E) NOTA 12. Using the linearization method of approximation at the point (2,1,8) , estimate f (2.05,.95) for the function z f ( x, y) x 2 3xy 2 y 2 with x .05, y .05 . A) 7.75 B) 7.90 C) 8.05 D) 7.80 E) NOTA 13. Given that f F and f 2 xy y 2 z 2 2, x 2 2 z 2 xyz 2 3, 2 y 2 xy 2 z z , find F (1,1,1) if F (0,1, 0) 4 . A) 9 2 B) 1 2 C) 4 D) 3 E) NOTA 14. Let A be the number of saddle points, B the number of local minimums, and C the number of local maximums for the function f ( x, y) x 4 y 4 4 xy 1 . Find A B C . A) 2 B) 1 C) 1 D) 0 E) NOTA 15. Find the equation of the best fit line for the points A(2,1), B(0,3), C (1, 4) . The best fit line Ŷ 0 1 X is determined by the values of 0 , 1 that minimize 3 Y i 1 i 1 X i . 2 0 Multivariable Calculus 15 A) Yˆ X 4 FAMAT State Convention 2009 B) Yˆ X 3 C) Yˆ X 7 2 D) Yˆ X 11 3 E) NOTA y (1) 0 16. Consider the third order differential equation y xy y y and the initial conditions y ( 1) 1 . y(1) 1 Using Euler’s method with x 0.5 , find y(0) . A) 1.5 B) 2.5 C) 3.5 D) 2 E) NOTA 17. Given that f ( x, y ) x 2 y 2 , which of the following critical points occurs at ( x, y ) (0, 0) ? A) Local minimum C) Local maximum E) NOTA B) Impossible to determine D) Saddle point 42 x 18. Which of the following integrals represents a reversal of the order of integration of 2 4 y 2 x 4 A) xdxdy x 2 B) xdxdy 2 2 y 4 42 x C) 2 xdydx ? x 2 x 2 ydydx x D) xdxdy x E) NOTA 4 19. The density of a certain Martian sphere 1 meter in radius is given by the r kg where r is the distance to the center of the sphere. Find the mass of m3 2 the sphere (in kilograms). 4 3 A) B) C) D) E) NOTA 3 32 2 equation (r ) 20. Given that the surface area of a rectangular box with no lid is 48 in 2 , find the maximum volume of the box (answers are in in3 ). A) 16 2 B) 20 C) 32 D) 24 E) NOTA 21. Use differentials to estimate the volume of steel used to construct a uniform cylindrical steel container which can hold a column of water 3 inches in radius and 10 inches in height. The thickness of the steel is 1 inch (all answers are in in3 ). 2 A) 46 B) 39 C) 42 D) 143 4 E) NOTA Multivariable Calculus FAMAT State Convention 2009 xy . ( x , y ) (0,0) x y 2 B) 0 C) 22. Evaluate the following limit A) Does not exist lim 2 E) NOTA 1 2 4 2 32 . Find the length of the arc of the function t 2t , t 2 3 23. Consider the vector function r (t ) from t 0 to t 4 . A) 16 B) 12 D) Cannot be determined C) 10 D) 8 E) NOTA x1 x2 1 24. Find the maximum value of the sum x1 2 x2 subject to the constraints: x2 2 and x1 , x2 0 . x1 x2 4 A) 4 B) 6 C) 8 D) 10 E) NOTA 25. Find the curvature of the parametric vector function r (t ) 3sin t ,3cos t . A) 3 3 B) 1 2 26. Evaluate the integral C) 1 D) 3 E) NOTA xyds , where C is the upper right quarter of the ellipse defined by the C equation A) 3 x2 y 2 1. 16 9 B) 2+ 3 C) 148 7 D) 2 3 7 E) NOTA For questions 27-29, consider the vector field defined by F ( x, y, z ) ( xyz )iˆ x 2 z 3 ˆj y 2 z kˆ . 27. Find curl F at the point ( x, y, z ) (1, 2,3) . A) 15iˆ 2 ˆj 57kˆ C) 39iˆ 2 ˆj 57kˆ B) 15iˆ 2 ˆj 51kˆ D) 39iˆ 2 ˆj 57kˆ 28. Find div F at the point ( x, y, z ) (1, 2,3) . A) 12 B) 6 C) 10 29. Find div curl F at the point ( x, y, z ) (1, 2,3) . A) 0 B) Undefined C) 12 E) NOTA D) 2 E) NOTA D) 1 E) NOTA 30. Evaluate the discriminant of the curve f ( x, y) x 2 y xy 3 at the point (1, 2) . A) 52 B) 148 C) 100 D) 0 E) NOTA