math318.midterm1 Spring 2013 - Math 318 Krantz First...

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Math 318 Krantz February 20, 2013 First Midterm General Instructions: Read each problem carefully. Do only what is requested—nothing more nor less. Always show your work on each problem —unless the problem explicitly tells you not to show work. You will not get full credit for just writing down the answer. The point value of each problem is shown. The total points on this exam is 100. Ask questions if anything is unclear. 1. (10 points) Give an explicit verification that V = { ( x 1 , x 2 , x 3 ) R 3 : 3 x 1 - 2 x 2 + x 3 = 0 } is a subspace of R 3 . 1
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2. (10 points) Give the formal definition of a linear transformation. 3. (10 points) Find the equation of the plane that passes through (2 , 5 , 3) and is per- pendicular to the vector v = h 1 , 1 , 2 i . 2
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4. (10 points) Let A = 2 1 3 4 0 2 1 1 - 1 and B = - 1 1 0 0 0 2 1 0 - 1 . Calculate AB and BA . 5. (10 points) Find a nonzero vector in R 3 which is perpendicular to both u = h 2 , 4 , 1 i and v = h- 1 , 0 , 1 i . 3
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6. (10 points) Sketch the trace of the parametrized curve r ( t ) = h t, cos t, sin t i in R 3 . 7.
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Unformatted text preview: 3 . 7. (8 points) Sketch the graph (which is a 2-dimensional surface in R 3 ) of the function f ( x,y ) = x-y + 2. 4 8. (10 points) Sketch the vector field in the plane given by f ( x,y ) = h x,-2 y i . [ Hint: Your answer should be a set of axes with a bunch of arrows exhibiting the flow of the vector field.] 5 9. Which of the following sets is open and which is closed? Which is nei-ther? Give brief reasons for your answers. (a) (3 points) S = { ( x,y ) ∈ R 2 : x 2 + y 2 < 1 } (b) (3 points) T = { ( x,y ) ∈ R 2 : 1 ≤ x < 2 } (c) (3 points) U = { ( x,y,z ) ∈ R 3 :-1 ≤ x ≤ 1 ,-1 ≤ y ≤ 1 ,-1 ≤ z ≤ 1 } (d) (3 points) V = { ( x,y,z ) ∈ R 3 : 1 ≤ x 2 + y 2 + z 2 < 2 } 6 10. (10 points) Explain why this function f has no limit at the origin: f ( x,y ) = xy x 2 + y 2 . 7...
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