Exam 1 February 19th 2018 MTH 234 Name Section Recitation Instructor INSTRUCTIONS Fill in your name etc on this rst page Without fully opening the exam check that you have pages 1 through 11 Show all your work on the standard response questions Write your answers clearly Include enough steps for the grader to be able to follow your work Don t skip limits or equal signs etc Include words to clarify your reasoning Do rst all of the problems you know how to do immediately Do not spend too much time on any particular problem Return to di cult problems later If you have any questions please raise your hand and a proctor will come to you You will be given exactly 90 minutes for this exam Remove and utilize the formula sheet provided to you at the end of this exam ACADEMIC HONESTY Do not open the exam booklet until you are instructed to do so Do not seek or obtain any kind of help from anyone to answer questions on this exam If you have questions consult only the proctor s Books notes calculators phones or any other electronic devices are not allowed on the exam Students should store them in their backpacks No scratch paper is permitted If you need more room use the back of a page Anyone who violates these instructions will have committed an act of academic dishonesty Penalties for academic dishonesty can be very severe All cases of academic dishonesty will be reported immediately to the Dean of Undergraduate Studies and added to the student s academic record I have read and understand the above instructions and statements regarding academic honesty SIGNATURE cid 63 Page 1 of 11 MTH 234 Exam 1 February 19th 2018 Standard Response Questions Show all work to receive credit Please BOX your nal answer 1 7 points Find a parametrization of the line of intersection between planes 2x z 1 and y z 2 2 7 points What is the length of the curve r t sin 2t t cos 2t cid 105 cid 104 from t 0 to t Page 2 of 11 MTH 234 Exam 1 February 19th 2018 3 7 points Evaluate the limit or show that it does not exist x x 2 y y 2 lim 1 x 1 y 4 7 points Evaluate the limit or show that it does not exist 4x2 9y2 2xy lim 0 x 0 y Page 3 of 11 MTH 234 Exam 1 February 19th 2018 5 Consider the function f x y x2 6xy 9y2 in the rst quadrant a 5 points Calculate the partial derivatives fx and fy cid 112 b 5 points Use your answer in part a to nd the linearization of f at 1 2 c 4 points Use your answer in part b to approximate f 0 7 2 1 Page 4 of 11 MTH 234 Exam 1 February 19th 2018 6 Consider the function f x y 2 a 5 points Sketch the domain of f y2 x 1 cid 112 y b 6 points Sketch the level curves of f for levels k 0 k 1 and k 2 y c 3 points What is the range of f x x Page 5 of 11 MTH 234 Exam 1 February 19th 2018 Multiple Choice Circle the best answer No work needed No partial credit available 7 4 points The intersection of the quadric surfaces de ned by x2 y2 z2 4 and x2 z2 4 is A one point B two points C a circle D a straight line E a parabola A x 1 2 y 2 2 z 3 2 9 3 2 8 3 2 9 1 2 y 1 2 y 2 2 z 2 2 z C x B x D x 1 y 2 z 3 3 E x 2y 3z 3 8 4 points Find the equation for the sphere in standard form centered at 1 2 3 with radius 3 9 4 points What is the positive value of parameter t corresponding to a point where the line de ned by intersects the cone z2 x2 y2 4t 3t 1 cid 105 cid 104 A 5 2 B 3 10 C 8 5 D 1 5 E 15 Page 6 of 11 MTH 234 Exam 1 February 19th 2018 10 4 points Find the area of the triangle formed by the points O 0 0 0 P 1 0 2 and Q 2 1 1 A 4 B 15 2 C 11 cid 112 D 10 E 3 11 4 points Which of the following is a parametrization for the full curve of intersection between x2 y2 z2 0 and z x 1 A r t B r t C r t D r t E r t with t 1 2 0 cid 11 with t 1 2 t2 cid 10 t 2t 1 t 1 2 t t2 1 2 t t2 1 cid 68 cid 69 cos t sin t cid 104 cid 69 cid 68 t2 2 1 with t 0 2 with t 2 cid 105 with t cos t sin t 0 cid 105 cid 104 0 2 12 4 points Which of the following is the unit tangent vector to the curve r t 2t t2 t3 2 3 cid 28 cid 29 at r 1 2 1 2 3 cid 105 cid 104 A r t B r t C r t 2 1 2 3 cid 105 cid 104 cid 104 2 2 2 cid 105 1 1 1 1 3 cid 104 cid 105 2 2t 2t2 D r t E r t 2 0 cid 104 cid 104 cid 105 cos t cid 105 Page 7 of 11 MTH 234 Exam 1 February 19th 2018 13 4 points Consider the function f x y ln y x Calculate fxx fxy 1 A fxx fxy 1 y 1 y C fxx fxy 0 B fxx fxy D fxx fxy 1 E fxx fxy 1 x2y2 2 x3y df dt a A 6 B 24 C D E 14 15 3 A B C D E 1 10 0 0 3 cid 104 1 cid 105 3 10 cid 68 2 0 t2 cid 104 1 2 2 cid 105 cid 104 2 4 4 cid 105 cid 104 cid 105 cid 69 14 4 points Suppose f x y x2y where x t 3 cos t 2 sin t and y t 5 cos t 4 sin t Which of the following is equal to at t 0 15 4 points Find the vector a with 6 orthogonal to the plane given by x 2y 2z 5 0 Page 8 of 11 MTH 234 Exam 1 February 19th 2018 More Challenging Problem s Show all work to receive credit 16 7 points Find the distance between the two skew lines given by vector equations 1 2t 0 t cid 105 cid 104 and 1 t 2 t 1 t cid 105 cid 104 17 7 points An object moves along the surface x2 z2 9 in R3 via the parametric …