Math 241 Midterm exam 1 Spring 2014Name:Section and TA name:READ ALL INSTRUCTIONS CAREFULLY. Write legibly, and use the boxes and/or insert check marks ✓□ foryour final answer where provided.Be sure to use correct notation; in particular, distinguish vectors from scalars by arrow notation, use explicitclearly visible dots for dot products, etc.An answer alone, without justification, will not earn full credit (with the exception of the two multiplechoice problems 3 and 4). If you make a mistake, cross out all of your incorrect work. We will take pointsoff for incorrect work that is not crossed out, even if the correct answer is given elsewhere on the page.Problem Point Value Test Score1 62 43 44 65 46 67 68 39 610 5Total 50Math 241 Midterm exam 1 Spring 20141. (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 20142. (4 points) Are the two planes x − 2y + 2z = 2 and 6x + 3y + 2z = 0 parallel?If not, find the cosine of the angle θ between the two planes.If yes, find the distance d between the two planes.Write your answer in the appropriate box below. Enter N/A in the other answer box.cos θ =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 = meaninglessand pencil your answers into the corresponding lines in your Scantron bubble sheet. (Options D andE are not valid answers in this problem, and if marked, will automatically result in a score of zero inthat line.)Also check the corresponding box ✓□ for each expression in case the scanner is unable to read youranswers.In each expression, a, b, and c denote three-dimensional vectors. The notation |a| is used in yourtextbook; 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..4Math 241 Midterm exam 1 Spring 20144. (6 points) Match the functions with their graphs and with their level curves. Each level curves diagramconsists 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) = x2e−y2Each graph and level curves diagram below is labeled with the corresponding..line number in yourScantron bubble sheet. Pencil in your answers using the following key:..A f(x, y) = y2− x2,..B g(x, y) = cos(x + y),..C h(x, y) = x2e−y2, 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 ofzero in that line.) Also label the graphs/diagrams in case the scanner is unable to read your answers.5678910111213141516Math 241 Midterm exam 1 Spring 20145. (4 points) Doeslim(x,y)→(0,0)(x + y)2x2+ y2exist? 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 workto receive credit for this problem.lim(x,y)→(0,0)(x + y)2x2+ y2=Math 241 Midterm exam 1 Spring 20146. (6 points = 2 points each) Find the first partial derivatives of the functionf(x, y, z) = x2z + e−xcos y.Then evaluate them at the point (0,π2, 1). Show your work.fx(x, y, z) =fx(0,π2, 1) =fy(x, y, z) =fy(0,π2, 1) =fz(x, y, z) =fz(0,π2, 1) =Math 241 Midterm exam 1 Spring 20147. (6 points) Suppose that f(x, y) is a differentiable function of the two variables x and y, and has thetable of values shown below. Further suppose that x = s2− t and y = 2st in turn are differentiablefunctions 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 −2Let F(s, t) = f(x(s, t), y(s, t)). Use the table of values to calculate the partial derivative∂F∂t(2, 1).You must show your work to receive credit for this problem.∂F∂t(2, 1) =Math 241 Midterm exam 1 Spring 20148. (3 points) Find a scalar equation of the tangent plane to the hyperboloidx24+ 3y2−z26=112at the point (−4, 1, 3).Pencil in your answer from the five choices below into line..17 of your Scantron bubble sheet:□..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) = 0Also mark the correct answer ✓□ among the five choices above in case the scanner is unable to readyour answer.Show your work.You may use the space below for your computations.Math 241 Midterm exam 1 Spring 20149. (6 points) Suppose that a function z = f(x, y) has the table of values and partial derivatives shownbelow (and has continuous first and second partial derivatives).Identify all critical points of f shown in the table, and classify them as local minimum, maximum, orsaddle point.Also find the linear approximation of f at the point (2, 3), and use it to estimate the values of thefunction 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) 2 6 −1 1 5 2..Line..21 (0, 0) 0 −1 1 2 −4 0..Line..22 (5, −2) 8 0 0 2 3 4..For each of the five points above, pencil your answers into the corresponding..line in your Scantronbubble sheet using the following key:..A = local min..B = local max..C = saddle point..D = critical point but impossible to classify..E = not a critical pointAlso fill in the circle in the last column of each row with your choice..A −..E in case the scanneris unable to read your answers.You may use the space below for your computations.The linear approximation is f(x, y) ≈ near the point (2, 3).In the lines
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