1 a Consider the points A 0 1 2 B 2 2 3 and C 1 3 4 Compute the vectors v AB and w AC 2 points v b Consider the points D 0 2 1 E 1 4 1 and F 3 8 2 and the vectors a 1 2 0 b 3 6 1 and c 2 4 1 as shown at right Find a normal vector n to the plane containing the points D E and F 3 points w D a E b c F n c Let P 2 1 1 and u 3 2 4 Find a linear equation for the plane that contains P and has normal vector u 2 points Equation x y z d For two vectors v and w in R3 which of the following does v w measure Circle your answer 1 point The length of v w The area of the parallelogram determined by v and w The volume of the parallelepiped determined by v w and v w 2 a Find the midpoint M of the straight line segment between the points P 1 3 2 and Q 3 3 4 2 points M b Let L be the line parametrized by r t a t b for a 1 0 2 and b 2 1 1 Find the point Q of intersection of the line L with the plane whose equation is 3x 2y z 14 3 points Q x y x2 y 3 Consider the function f x y 2 for x y 0 0 Evaluate lim f x y or explain why it does x y 0 0 x y2 not exist 5 points 4 For each function a x 2 y 2 b cos x y c y 2 e x 2 label its graph and its level set diagram from among the options below Here each level set diagram consists of level sets f x y c i drawn for evenly spaced c i 9 points 5 a Compute the partial derivatives of f x y x 3 2x y y 2 points f x f y b Consider the graph of f R2 R shown at right If P is the point a b f a b find the sign of each of the quantities below Circle your answers 1 point each y z P f x a b positive negative 0 f y a b positive negative 0 f xx a b positive negative 0 Graph of f x y x c Use the contour plot of f x y shown at right to estimate f x 2 1 and f y 2 1 For each circle the number below that is closest to your estimate 4 points f 3 f 2 f 1 2 f 0 f 1 1 f 2 f 3 0 1 2 3 f x 2 1 3 2 5 2 1 5 1 0 5 0 0 5 1 1 5 2 2 5 3 f y 2 1 3 2 5 2 1 5 1 0 5 0 0 5 1 1 5 2 2 5 3 6 Suppose a function f R2 R has f 1 2 5 tion to estimate f 1 1 1 9 3 points f 1 2 2 x and f 1 2 3 Use linear approxima y f 1 1 1 9 7 Suppose f x y R2 R x s t R2 R and are functions Let F s t f x s t y s t be their composition a Write the formula for y s t R2 R F using the Chain Rule 1 points s F s b Suppose x s t 2s t and y s t s 2 t 1 and that f x y has the table of values and partial derivatives shown at F right Compute 2 1 4 points s x y f x y 2 3 2 1 2 4 5 3 0 2 3 1 f x 3 2 4 3 F 2 1 s f y 6 1 6 5 Extra credit problem Let E R2 R be given by E x y 2x y 2 Find a 0 so that E h 0 01 for all h x y with h Carefully justify why the you provide is good enough 2 points Scratch space below and on back
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