MTH 234 Exam 1 February 19th, 2018Name:Section: Recitation Instructor:INSTRUCTIONS• Fill in your name, etc. on this first 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 answersclearly! Include enough steps for the grader to be able to follow your work. Don’tskip limits or equal signs, etc. Include words to clarify your reasoning.• Do first all of the problems you know how to do immediately. Do not spend toomuch time on any particular problem. Return to difficult 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 thisexam. If you have questions, consult only the proctor(s).• Books, notes, calculators, phones, or any other electronic devices are not allowed onthe 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 academicdishonesty. Penalties for academic dishonesty can be very severe. All cases ofacademic dishonesty will be reported immediately to the Dean of UndergraduateStudies and added to the student’s academic record.I have read and understand theabove instructions and statementsregarding academic honesty:. SIGNATURE? Page 1 of 11MTH 234 Exam 1 February 19th, 2018Standard Response Questions. Show all work to receive credit. Please BOX your final 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) = hsin(2t), t, cos(2t)i from t = 0 to t = π?Page 2 of 11MTH 234 Exam 1 February 19th, 20183. (7 points) Evaluate the limit or show that it does not exist: limx→1y→−1√x −√2 + yx − y −24. (7 points) Evaluate the limit or show that it does not exist: limx→0+y→0+4x2+ 9y22xyPage 3 of 11MTH 234 Exam 1 February 19th, 20185. Consider the function f(x, y) =px2+ 6xy + 9y2in the first quadrant.(a) (5 points) Calculate the partial derivatives, fxand fy.(b) (5 points) Use your answer in part (a) to find 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 11MTH 234 Exam 1 February 19th, 20186. Consider the function f(x, y) = 2 −py2+ x − 1(a) (5 points) Sketch the domain of f.xy(b) (6 points) Sketch the level curves of f for levels k = 0, k = 1, and k = 2.xy(c) (3 points) What is the range of f ?Page 5 of 11MTH 234 Exam 1 February 19th, 2018Multiple Choice. Circle the best answer. No work needed. No partial credit available.7. (4 points) The intersection of the quadric surfaces defined by x2+ y2+ z2= 4 and x2+ z2= 4 is:A. one pointB. two pointsC. a circleD. a straight lineE. a parabola8. (4 points) Find the equation for the sphere in standard form centered at (1, 2, 3) with radius 3.A. (x + 1)2+ (y + 2)2+ (z + 3)2= 9B. (x − 1)2+ (y − 2)2+ (z − 3)2= 8C. (x − 1)2+ (y − 2)2+ (z − 3)2= 9D. (x − 1) + (y −2) + (z − 3) = 3E. x + 2y + 3z = 39. (4 points) What is the positive value of parameter t corresponding to a point wherethe line defined by h4t, 3t, 1i intersects the cone z2= x2+ y2?A. 5/2B. 3/10C. 8/5D. 1/5E. 15Page 6 of 11MTH 234 Exam 1 February 19th, 201810. (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. 4B.p15/2C.√11D.√10E. 311. (4 points) Which of the following is a parametrization for the full curve of intersection betweenx2+ y2− z2= 0 and z = x + 1A. r(t) =t,√2t + 1, t + 1with t ∈ [−1/2, ∞)B. r(t) =Dt2−12, t,t2+12Ewith t ∈ [0, ∞)C. r(t) =Dt2−12, t,t2+12Ewith t ∈ (−∞, ∞)D. r(t) = hcos(t), sin(t), −2i with t ∈ [0, 2π]E. r(t) = hcos(t), sin(t), 0i with t ∈ [0, 2π]12. (4 points) Which of the following is the unit tangent vector to the curve: r(t) =2t, t2,23t3at r(1) = h2, 1, 2/3iA. r(t) = h2, 1, 2/3iB. r(t) = h2, 2, 2iC. r(t) =1√3h1, 1, 1iD. r(t) = h2, 2t, 2t2iE. r(t) = h2, 0, −cos(t)iPage 7 of 11MTH 234 Exam 1 February 19th, 201813. (4 points) Consider the function f (x, y) = ln(y)x. Calculate fxx+ fxy.A. fxx+ fxy=1y+ 1B. fxx+ fxy=1yC. fxx+ fxy= 0D. fxx+ fxy= 1E. fxx+ fxy=1x2y2−2x3y14. (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 todfdtat t = 0?A. 6B. 24C. −14D. −15E. −315. (4 points) Find the vector a with |a| = 6 orthogonal to the plane given by x + 2y + 2z + 5 = 0.A. h0, 3, −1iB.D0,3√10,−1√10EC. h2, 0, t2iD. h1, 2, 2iE. h2, 4, 4iPage 8 of 11MTH 234 Exam 1 February 19th, 2018More Challenging Problem(s). Show all work to receive credit.16. (7 points) Find the distance between the two skew lines given by vector equationsh1 + 2t, 0, ti and h−1 + t, 2 + t, 1 + ti17. (7 points) An object moves along the surface x2+ z2= 9 in R3via the parametric equations:x(t) = 3 cos(πt) and y(t) = 4πtWhat is the speed and of the object at time t = 2?Page 9 of 11MTH 234 Exam 1 February 19th, 2018Please have your MSU student ID ready so that is can be checked.When you are completely happy with your work please bring your exam to the front to be handed in.Go back and check your solutions for accuracy and clarity. Make sure your final answers are BOXED .Congratulations you are now done with the exam!DO NOT WRITE BELOW THIS LINE.Page Points Score2 143 144 145 146 127 128 129 14Total: 106No more than 100 points may be earned on the exam.Page 10 of 11MTH 234 Exam 1 February 19th, 2018FORMULA SHEETVectors in SpaceSuppose u = hu1, u2, u3i and v = hv1, v2, v3i:• Unit Vectors:i = h1, 0, 0ij = h0, 1, 0ik = h0, 0, 1i• Length of vector u|u| =pu12+ u22+ u32• Dot Product:u · v = u1v1+ u2v2+ u3v3= |u||v|cos θ• Cross Product:u × v =i j ku1u2u3v1v2v3• Vector Projection: projuv =u · v|u|2uPartial Derivatives• Chain Rule: Suppose z = f(x, y) andx = g(t) and y = h(t) are all differentiablethendzdt=∂f∂xdxdt+∂f∂ydydtCurves and Planes in Space• Line parallel to v: r(t) = r0+ tv• Plane normal to n = ha, b, ci:a(x − x0) + b(y − y0) + c(z − z0) = 0• Arc Length of curve r(t) for