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Unformatted text preview: Math 215 — Second Midterm March 25, 2010 Name: EXAM SOLUTIONS Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 9 pages including this cover. There are 7 problems. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 3. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor when you hand in the exam. 4. Please read the instructions for each individual problem carefully. One of the skills being tested on this exam is your ability to interpret mathematical questions, so instructors will not answer questions about exam problems during the exam. 5. Show an appropriate amount of work (including appropriate explanation) for each problem, so that graders can see not only your answer but how you obtained it. Include units in your answer where that is appropriate. 6. You may use no aids (e.g., calculators or notecards) on this exam. 7. If you use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 8. Turn off all cell phones and pagers , and remove all headphones. 9. Because no calculators or notecards are allowed, you are not required to evaluate expressions you may obtain in the solution of these problems. In some cases it may be to your advantage to simplify expressions, but it will not be required for full credit. 10. Note that problems 1–4 will be graded giving very little partial credit. 11. There is a list of possibly useful formulas included as the second page of this exam. Problem Points Score 1 14 2 12 3 12 4 16 5 16 6 14 7 16 Total 100 Math 215 / Exam 2 (March 25, 2010) page 2 Some possibly useful formulas Areas and Volumes ◦ Area of a triangle: A = 1 2 bh , where b is the length of one side and h the length of the perpendicular from that side to the opposite angle. ◦ Area of a sector of a circle: A = 1 2 r 2 θ , where r is the radius of the circle and θ the angle of the sector. ◦ Volume and surface area of a sphere: V = 4 3 π r 3 ; S = 4 π r 2 . ◦ Volume of right circular cylinder: V = π r 2 h , where r is the radius and h the height of the cylinder. ◦ Volume of a cone: V = 1 3 π r 2 h , where r is the radius of the base of the cone and h its height. Trigonometry ◦ cos 2 x + sin 2 x = 1 ◦ cos( x + y ) = cos( x )cos( y ) − sin( x )sin( y ); sin( x + y ) = cos( x )sin( y ) + cos( y )sin( x ) ◦ cos(2 x ) = cos 2 x − sin 2 x ; sin(2 x ) = 2 sin x cos x ◦ cos 2 x = 1 2 (1 + cos(2 x )); sin 2 x = 1 2 (1 − cos(2 x )) Vector Formulas ◦ a × b = − b × a ; c a × b = c ( a × b ) = a × ( c b ) ◦ a × ( b + c ) = a × b + a × c ; ( a + b ) × c = a × c + a × b ◦ The unit tangent T , normal N and binormal B vectors for a space curve r ( t ) are T ( t ) = r ′ ( t ) /  r ′ ( t )  , N ( t ) = T ′ ( t ) /  T ′...
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This note was uploaded on 01/28/2012 for the course MATH 215 taught by Professor Fish during the Fall '08 term at University of Michigan.
 Fall '08
 Fish
 Math, Calculus

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