practicefinal7

# practicefinal7 - MA 261 PRACTICE PROBLEMS 1. If the line...

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MA 261 PRACTICE PROBLEMS 1. If the line has symmetric equations x - 1 2 = y - 3 = z +2 7 , ﬁnd a vector equation for the line 0 that contains the point (2 , 1 , - 3) and is parallel to . A. ~ r =(1+2 t ) ~ i - 3 t ~ j +( - 2+7 t ) ~ k B. ~ r =(2+ t ) ~ i - 3 ~ j +(7 - 2 t ) ~ k C. ~ r =(2+2 t ) ~ i +(1 - 3 t ) ~ j - 3+7 t ) ~ k D. ~ r t ) ~ i - 3+ t ) ~ j - 3 t ) ~ k E. ~ r t ) ~ i + ~ j - 3 t ) ~ k 2. Find parametric equations of the line containing the points (1 , - 1 , 0) and ( - 2 , 3 , 5). A. x =1 - 3 t,y = - 1+4 t,z =5 t B. x = = - =0 C. x - 2 = - 1+3 t D. x = - 2 =3 t E. x = - 1+ =2 - 3. Find an equation of the plane that contains the point (1 , - 1 , - 1) and has normal vector 1 2 ~ i +2 ~ j +3 ~ k . A. x - y - z + 9 2 B. x +4 y +6 z +9=0 C. x - 1 1 2 = y +1 2 = z +1 3 D. x - y - z E. 1 2 x y z 4. Find an equation of the plane that contains the points (1 , 0 , - 1), ( - 5 , 3 , 2), and (2 , - 1 , 4). A. 6 x - 11 y + z B . 6 x +11 y + z C . 11 x - 6 y + z D. ~ r =18 ~ i - 33 ~ j ~ k E. x - 6 y - 11 z =12 5. Find parametric equations of the line tangent to the curve ~ r ( t )= t ~ i + t 2 ~ j + t 3 ~ k at the point (2 , 4 , 8) A. x =2+ =4+4 =8+12 t B. x =1+2 =12+8 t C. x =4 =8 t D. x = t E. x =4+2 =8+3 t 6. The position function of an object is ~ r ( t )=cos t ~ i +3sin t ~ j - t 2 ~ k Find the velocity, acceleration, and speed of the object when t = π . Velocity Acceleration Speed A. - ~ i - π 2 ~ k - 3 ~ j - 2 π ~ k π 4 B. ~ i - 3 ~ j π ~ k - ~ i - 2 ~ k 10 + 4 π 2 C. 3 ~ j - 2 π ~ k - ~ i - 2 ~ k 9+4 π 2 D. - 3 ~ j - 2 π ~ k ~ i - 2 ~ k π 2 E. ~ i - 2 ~ k - 3 ~ j - 2 π ~ k 5 1

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7. A smooth parametrization of the semicircle which passes through the points (1 , 0 , 5), (0 , 1 , 5) and ( - 1 , 0 , 5) is A. ~ r ( t )=s in t ~ i +cos t ~ j +5 ~ k, 0 t π B. ~ r ( t )=cos t ~ i +sin t ~ j ~ k, 0 t π C. ~ r ( t t ~ i t ~ j ~ k, π 2 t 3 π 2 D. ~ r ( t t ~ i t ~ j ~ k, 0 t π 2 E. ~ r ( t )=sin t t ~ j ~ k, π 2 t 3 π 2 8. The length of the curve ~ r ( t )= 2 3 (1 + t ) 3 2 ~ i + 2 3 (1 - t ) 3 2 ~ j + t ~ k , - 1 t 1is A. 3B . 2C . 1 2 3D . 2 3E . 2 9. The level curves of the function f ( x,y p 1 - x 2 - 2 y 2 are A. circles B. lines C. parabolas D. hyperbolas E. ellipses 10. The level surface of the function f ( x,y,z z - x 2 - y 2 that passes through the point (1 , 2 , - 3) intersects the ( x,z )-plane ( y = 0) along the curve A. z = x 2 +8 B. z = x 2 - 8C . z = x 2 D. z = - x 2 - 8 E. does not intersect the ( )-plane 11. Match the graphs of the equations with their names: (1) x 2 + y 2 + z 2 = 4 (a) paraboloid (2) x 2 + z 2 = 4 (b) sphere (3) x 2 + y 2 = z 2 (c) cylinder (4) x 2 + y 2 = z (d) double cone (5) x 2 +2 y 2 +3 z 2 = 1 (e) ellipsoid A. 1b, 2c, 3d, 4a, 5e B. 1b, 2c, 3a, 4d, 5e C. 1e, 2c, 3d, 4a, 5b D. 1b, 2d, 3a, 4c, 5e E. 1d, 2a, 3b, 4e, 5c 12. Suppose that w = u 2 /v where u = g 1 ( t )and v = g 2 ( t ) are diﬀerentiable functions of t .I f g 1 (1) = 3, g 2 (1) = 2, g 0 1 (1) = 5 and g 0 2 (1) = - 4, ﬁnd dw dt when t =1. A. 6 B. 33 / . - 24 D. 33 E. 24 13. If w = e uv and u = r + s , v = rs , ﬁnd ∂w ∂r . A. e ( r + s ) rs (2 rs + r 2 )B . e ( r + s ) (2 rs + s 2 )C . e ( r + s ) (2 rs + r 2 ) D. e ( r + s ) (1 + s )E . e ( r + s ) ( r + s 2 ). 2
14. If f ( x,y )=cos( xy ), 2 f ∂x∂y = A. - xy cos( xy )B . - xy cos( xy ) - sin( xy )C . - sin( xy ) D. xy cos( xy )+sin( xy )E . - cos( xy ) 15. Assuming that the equation xy 2 +3 z =cos( z 2 ) deﬁnes z implicitly as a function of x and y , ﬁnd ∂z ∂x .

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## This note was uploaded on 01/26/2012 for the course MA 261 taught by Professor Stefanov during the Spring '08 term at Purdue.

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practicefinal7 - MA 261 PRACTICE PROBLEMS 1. If the line...

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