Download MATH 251-Spring 2006 Midterm 1 Exam and more Exams Calculus in PDF only on Docsity! MATH 251-3, Spring 2006 Simon Fraser University Midterm 1 7 February 2006, 5:30โ6:20pm Instructor: Ralf Wittenberg Last Name: First Name: Student Number: Signature: Instructions 1. Please do not open this booklet un- til invited to do so. 2. Write your last name, first name(s) and student number in the box above in block letters, and sign your name in the space provided. 3. This exam contains 5 questions on 5 pages (after this title page). Once the exam be- gins please check to make sure your exam is complete. 4. The total time available is 50 minutes, and there are 50 points, so allow about a minute per point; for example, you should aim to spend about 10 minutes on a 10-point ques- tion. Attempt all problems! 5. This is a closed book exam. Only non- programmable scientific calculators are al- lowed. 6. Use the reverse side of the previous page if you need more room for your answer, and clearly indicate where the solution contin- ues. 7. Show all your work, and explain your an- swers clearly. 8. Good luck! Question Maximum Score 1 18 2 4 3 10 4 10 5 8 Total 50 Math 251 : Midterm 1 2 1. Given the four points A(1, 2, 1), B(2, 0, 1), C(โ1, 2, 0), and D(3, 3,โ1) : (a) [2 points] Compute โโ AB ร โโ AC. (b) [2 points] Find the area of the triangle ABC. (c) [4 points] Find the angle ฮธ between the vectors โโ AB and โโ AC. (d) [6 points] Find an equation of the plane that passes through A and B and is parallel to the line through C and D. Math 251 : Midterm 1 5 4. A particle moves in space with parametric equations x = t, y = t2, z = 4 3 t3/2, t > 0. (a) [4 points] Find the velocity and acceleration vectors, and the speed of the particle as a function of t. (b) [6 points] Determine the curvature ฮบ and radius of curvature of the curve traced out by the particle at time t = 1. Math 251 : Midterm 1 6 5. Consider the surface described by the equation x2 + y2 โ z2 = 4. (a) [4 points] Briefly describe the traces (cross-sections) of the surface in horizontal planes of the form z = z0, and the traces (cross-sections) of the surface in vertical planes of the form x = x0. (You may wish to include a rough sketch in your description.) (b) [4 points] Write the equation for the surface in both cylindrical and spherical coordinates.