MATH 152 CALCULUS II MIDTERM 1 EXAMINATION Solution Set

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1 MATH 152 CALCULUS II MIDTERM 1 EXAMINATION Solution Set Department of Mathematics Spring Term November 06, 2012 Fair Use Disclaimer, This document contains copyrighted material. We are making such a material available in our efforts to advance understanding in the education of mathematics. We believe this constitutes a fair use of any such copyrighted material as provided by the TRNC or EU Copyright Law. This document is distributed without profit to those who have expressed a prior interest in receiving the included information for research and educational purposes. If you wish to use this copyrighted document for purposes of your own that go beyond fair use, you must obtain permission from the copyright owner, Department of Mathematics, Eastern Mediterranean University. The Department of Mathematics at Eastern Mediterranean University accepts no liability for the content, use or reproduction of such materials. Permission to reproduce this document in digital or printed form must be obtained from the Department of Mathematics Chair s office at EMU. Permission will be voided unless all copyrights and credits are displayed with the information reproduced.

2 Student s Name Surname: Signature: MATH 152 CALCULUS II MIDTERM 1 EXAMINATION November 06, 2012 Number: Group: Question No TOTAL Weight Received Instructions 1. There are 6 questions in this examination. 2. No books, notes, calculators, cell phones or other electronic devices are allowed. 3. Students has to shut down their cell phones before the exam starts. 4. Duration is 90 minutes. 5. Results of this examination will be announced on Nov., 13, 2012 on 6. You may check your paper. To do so, visit your instructor s office. 7. You may find the solutions of the examination questions on Some Useful Information 0,, cos,,,, cos cos, sin cos sin sin!, lim cos, sin,, tan

3 QUESTION 1. a) Find the parametric equations of the line passing through point 3,5,4 and is parallel to another line given by 1 3, 1 2, 3,. Direction vectors of the two lines are also parallel 3, 2,1, thus the parametric equations of line are: b) Show that the planes 1 and 22 are neither orthogonal nor parallel. Then find the parametric equations for the line of intersection of these planes. Plane 1: 1,1,1 Plane 2: 1, 1, planes are not orthogonal 32 0 planes are not parallel Hence they are intersecting. To find the line of intersection, a point on the line and a direction vector is required. To find a point on the line let 0, then 1 2 Solving for and, we obtain, 3 2and 1 2and a point on the line of intersection is: 3 2,1 2,0 Direction vector is: 32 Hence a set of parametric equations for the line of intersection is, QUESTION 2. a) Use second order Taylor polynomial to approximate the value of 6., ! b) Find the center, radius and interval of convergence of the power series, 2 Center is at 0 Radius is, lim 2 lim So the interval of convergence is 1, 1 But analysis at the two end points is required At the left end point 1, and the series is 21 convergent alternating series At the right end point 1, and the series is 2 convergent p series Thus the interval of convergence is 1, 1

4 QUESTION 3. ( pts) a) Show that the vectors 23 and 46 are parallel. 2 a scalar multiple Then the two vectors are parallel. (Also you may find that their cross product is zero vector, implying that the vectors are parallel) QUESTION 4. Let, a) Find the domain of, Domain is:, :, 0,0 ( pts) b) Show that the vectors 34 and 43 are orthogonal imlying that the two vectors are orthogonal b) Find lim,,, lim, lim,,,, c) Let 23 and 26 Find c) Find lim,,, Let for any lim,, lim lim lim 1 1 Hence for different values of (i.e. on different paths) limit has different values. By the two path test limit does not exist.

5 QUESTION 5. a) Let,6,4. Compute the derivative of the function.. QUESTION 6. (4 pts each) a) Consider the parametric equations 2, 34, 22. Sketch the curve for the values 2, 1, 0, 1,2. On the graph show the positive orientation of the curve. 2, 6, 0 2,6,0 2,6, b) Evaluate the integral b) Eliminate, and obtain an equation of the curve of part (a) in terms of and ,44 c) Graph the polar point 4, 3 2. Write one alternative set of polar pair for this point. 4, 3 2 4, d) Express the polar point 2, 2 in Cartesian coordinates. 2cos 2 0,2sin 2 2 The point is 0, 2 in Cartesian coordinates e) Express the point 2, 2 in polar coordinates , tan thus the point is 2 2, 5 4