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sampletest11

# sampletest11 - (b Find the straight line passing through...

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Math 137 Sample problems for Test I 1. The vector a starts from the point (1 , 1 , 1) and end at (2 , 2 , 2) . The vector b starts from the point (1 , 2 , 3) and end at (2 , 3 , 4). Find a . Are vectors a and b different? 2. Given a = (1 , 1 , 2) and b = (2 , 3 , 4), find a · b and a × b . 3. The equations for two straight lines are x 1 = y 2 = z 3 and x 1 = y 1 = z - 1 respectively. Do these two lines pass the point (0 , 0 , 0)? Are they perpendicular to each other? 4. A plane contains three points: (0 , 0 , 0), (1 , 1 , 1) and (3 , 2 , 1). (a) Find an equation for the plane. (b) Find the area of the triangle with these three points as its vertices. (c) Find a parametric equation for the line passing through the point (1 , 2 , 3). (d) Find the point P where the line in (c) passing through the plane. (e) Find the distance between the point (1 , 2 , 3) to the plane, using the following two methods: The first one is to use the result in (c) and (d). The other is to use the formula in the textbook. 5. Given a straight line L, x = (1 , 2 , 3) + t (4 , 5 , 6), and a point P = (1 , 1 , 1). (a) Find the distance between the point P to the line.
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Unformatted text preview: (b) Find the straight line passing through the point P and intersect the line L at the right angle. Hint: ﬁnd the direction vector for this line ﬁrst via Fig 11.43. 6. Do the following limit exist? If yes, ﬁnd them (a) lim ( x,y ) → (0 , 0) x 2 +2 xy + y 2 x + y . (b) lim ( x,y ) → (0 , 0) x 2 +2 xy + y 2 x 2 + y 2 . 7. Find the following for the vector function F ( t ) = ( t 2 ,e 2 t +3 , sin t ). (a) lim t → F ( t ) . (b) d dt F ( t ) . (c) R F ( t ) dt. 8. Find the tangent line of the curve x = ( t 2 ,e 2 t +3 ,sint ) at t = 0. 9. Graph the surface z 2-x 2-y 2 = 0. 10. Assume a particle is moving with a constant speed. Show that its velocity v ( t ) must be perpendicular to its acceleration. Here t is the time. Hint the assumption means || v || 2 = v · v = constant. 11. Find the partial derivatives f x ,f y ,f xx ,f xy for f ( x,y ) = x 2 y 3 + sin( x/y ). 1...
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