Practice Exam for Math 241 Instructions Calculators books notes and suchlike aids to gracious living are not permitted Show all your work as credit will not be given for correct answers without proper justification except for Problems 2 and 6 Note Problem 6 b and the very last part of 6 a are based on material from Friday s lecture so you won t be able to do those yet 1 Consider the points A 0 0 2 B 1 0 3 and C 0 1 3 in R3 a Compute the vectors v AB and w AC 2 points b Find a normal vector n to the plane P containing the points A B C c Find the area of the triangle spanned by A B C 3 points 2 points d Find an equation which describes P If you can t do b take n 1 2 1 1 point e Consider the line L given by the parameterization r t 2 2t 3 1 2t Is L parallel to the plane P Why or why not 2 points 2 Match the following functions with their graphs and level set diagrams Here each level set diagram consists of level sets f x ci drawn for evenly spaced ci 9 points q c x 2 y 2 a 1 1 x 2 y 2 b cos x 2 y 2 3 Consider the function f x y exists y2 for x y 0 0 Compute the following limit if it x2 y 2 5 points lim x y 0 0 4 Consider the composition of the f R2 R with x y R2 R that is h s t f x s t y s t f x y function h 1 2 using the chain rule and the s table of values at right 5 points Compute input x y f x s y s f x f y 0 1 1 1 4 1 2 7 3 1 1 1 2 6 1 1 6 2 1 2 0 1 5 2 3 5 1 2 3 2 3 4 0 1 4 1 5 Consider the function f R2 R given by f x y x 2 a Compute the partial derivatives fx fy and fxy b Is f differentiable at 2 1 Why or why not x y 3 points 2 points c Give the linear approximation of f at the point 2 1 f 2 x 1 y d Give the equation of the tangent plane to the graph of f at 2 1 6 2 points 6 The picture below shows some level sets of a function f R2 R y f 3 f 2 f 1 v f 0 f 1 p f 2 x a At the point p shown determine the sign of each of the below quantities 1 points each f p positive negative 0 fx p positive negative 0 fy p positive negative 0 fxx p positive negative 0 Dv f p positive negative 0 b Draw f p on the picture 1 points Extra credit problem Let E R2 R be given by E x y 3x 2 xy 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 3 points
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