Math 241 Midterm exam 1 Spring 2014 Name Section and TA name READ ALL INSTRUCTIONS CAREFULLY Write legibly and use the boxes and or insert check marks for your nal answer where provided Be sure to use correct notation in particular distinguish vectors from scalars by arrow notation use explicit clearly visible dots for dot products etc An answer alone without justi cation will not earn full credit with the exception of the two multiple choice problems 3 and 4 If you make a mistake cross out all of your incorrect work We will take points off for incorrect work that is not crossed out even if the correct answer is given elsewhere on the page Problem Point Value Test Score 1 6 2 4 3 4 4 6 5 4 6 6 7 6 8 3 9 6 10 5 Total 50 Math 241 Midterm exam 1 Spring 2014 1 4 2 points Show your work a Find a scalar equation of the plane through the three points P 2 4 3 Q 5 6 3 and R 3 3 2 x y z 0 b Is the line x 4 t y 1 6t z 4t parallel to the plane in part a If not what is the point of intersection If you could not solve part a use the vector n 4 3 2 for part b Point of intersection enter PARALLEL if the line is parallel to the plane Math 241 Midterm exam 1 Spring 2014 2 4 points Are the two planes x 2y 2z 2 and 6x 3y 2z 0 parallel If not nd the cosine of the angle between the two planes If yes nd the distance d between the two planes Write your answer in the appropriate box below Enter N A in the other answer box cos Distance d 3 4 points 1 point each Which of the following expressions are meaningful Which are meaningless For each expression determine if the expression is A a scalar or B a vector or C meaningless and pencil your answers into the corresponding lines in your Scantron bubble sheet Options D and E are not valid answers in this problem and if marked will automatically result in a score of zero in that line Also check the corresponding box for each expression in case the scanner is unable to read your answers In each expression a b and c denote three dimensional vectors The notation a is used in your textbook in some lectures this was denoted a You should know what either one means a a b c is a scalar A a vector B meaningless C Line 1 b a b c is a scalar A a vector B meaningless C Line 2 c a b c is a scalar A a vector B meaningless C Line 3 d a b c is a scalar A a vector B meaningless C Line 4 Math 241 Midterm exam 1 Spring 2014 4 6 points Match the functions with their graphs and with their level curves Each level curves diagram consists of level curves where the function is constant drawn for evenly spaced constants k a 1 1 point f x y y2 x2 b 1 1 point g x y cos x y c 1 1 point h x y x2 e y 2 Each graph and level curves diagram below is labeled with the corresponding line number in your Scantron bubble sheet Pencil in your answers using the following key 2 A f x y y2 x2 B g x y cos x y C h x y x2 e y or D none of the above Option E is not a valid answer in this problem and if marked will automatically result in a score of zero in that line Also label the graphs diagrams in case the scanner is unable to read your answers 5 6 7 8 9 10 11 12 13 14 15 16 Math 241 Midterm exam 1 Spring 2014 5 4 points Does x y 2 x y 0 0 x2 y2 lim exist If yes compute it If not show why it does not exist Enter the limit in the box below or write DNE if the limit does not exist You must show your work to receive credit for this problem x y 2 x y 0 0 x2 y2 lim Math 241 Midterm exam 1 Spring 2014 6 6 points 2 points each Find the rst partial derivatives of the function f x y z x2 z e x cos y Then evaluate them at the point 0 2 1 Show your work fx x y z fx 0 2 1 fy x y z fy 0 2 1 fz x y z fz 0 2 1 Math 241 Midterm exam 1 Spring 2014 7 6 points Suppose that f x y is a differentiable function of the two variables x and y and has the table of values shown below Further suppose that x s2 t and y 2st in turn are differentiable functions of the two variables s and t x y f x y fx x y fy x y 4 2 1 1 7 2 1 2 5 1 3 4 3 2 3 1 4 1 3 2 Let F s t f x s t y s t Use the table of values to calculate the partial derivative You must show your work to receive credit for this problem F 2 1 t F 2 1 t Math 241 Midterm exam 1 Spring 2014 8 3 points Find a scalar equation of the tangent plane to the hyperboloid x2 z2 11 3y2 4 6 2 at the point 4 1 3 of your Scantron bubble sheet Pencil in your answer from the ve choices below into line 17 A 2 x 4 6 y 1 1 z 3 0 B 4 x 4 1 y 3 3 z 32 0 C 4 x 4 3 y 1 6 z 3 0 D 2 x 4 6 y 1 1 z 3 0 E 4 x 4 3 y 1 6 z 3 0 Also mark the correct answer among the ve choices above in case the scanner is unable to read your answer Show your work You may use the space below for your computations Math 241 Midterm exam 1 Spring 2014 9 6 points Suppose that a function z f x y has the table of values and partial derivatives shown below and has continuous rst and second partial derivatives Identify all critical points of f shown in the table and classify them as local minimum maximum or saddle point Also nd the linear approximation of f at the point 2 3 and use it to estimate the values of the function f at the two points 3 4 and 3 2 x y f x y fx x y fy x y fxx x y fxy x y fyy x y Line 18 4 3 9 3 0 2 1 2 Line 19 1 0 4 0 0 1 1 2 Line 20 2 3 …
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