# exam1 - 1 1(1-1 1 and(2 3 0(b(5 points Find the area of the...

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Name: Math 211A Fall 2005 Exam # 1 Honor Code Write and sign the honor code here after you have finished the quiz. There will be no calculators allowed during the exam. You may use a 3x5 inch note card. Use full sentences on all explanations and put a box around your final answer. Good Luck!! 1. True or False (2 points each) a. The intersection of two surfaces is always a line. b. The area of a parallelogram with sides a and b is given by | a · b | . c. For any vectors a and b , we have a · ( a × b ) = 0. d. The graph of f ( x, y ) = x + y sin x is a space curve. e. Suppose r ( s ) defines a vector function which is parametrized by arclength. Then 1 0 | r ( s ) | ds = 2 1 | r ( s ) | ds .

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2. (a) (5 points) Find an equation of the line through the point (7 , 2 , - 5), which is normal to the plane x - 2 3 = y + 4 2 ; z = 3 . (b) (5 points) Find an equation of the line tangent to the space curve r ( t ) = < sin t, cos 2 t, t 3 > at t = π .
3. (a) (5 points) Find an equation of the plane through the points (1

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Unformatted text preview: , 1 , 1) , (1 ,-1 , 1) , and (2 , 3 , 0). (b) (5 points) Find the area of the triangle with vertices (1 , 1 , 1) , (1 ,-1 , 1) , and (2 , 3 , 0). 4. (a) (6 points) Draw (and clearly label) four traces of the function f ( x, y ) = x sin( y ) (b) (2 points) Which of the following is the graph of f ( x, y ) = x sin( y )? (a) (b) (c) (d) 5. (5 points) Set up, but do not evaluate the integral which gives the arclength of the space curve r ( t ) = < t 3 , e 4 t , 2 cos( πt ) > between the points (0 , 1 , 2) and (8 , e 8 , 2). Bonus: (5 points) What is the curvature of the space curve r ( t ) = < (sin √ 3 t 13 ) t , 4(sin √ 3 t 13 ) t , (sin √ 3 t 13 ) t > at t = 1?...
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