C3-fin-ex-ans

C3-fin-ex-ans - MATH 2433 Problem 1. FINAL EXAM REVIEW...

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MATH 2433 FINAL EXAM REVIEW QUESTIONS Problem 1. (a) The points (3 , - 1 , 2) and ( - 1 , 3 , - 4) are the endpoints of a diameter of a sphere. ( i ) Determine the center and radius of the sphere. ( ii ) Find an equation for the sphere. Answer: ( i ) C :(1 , 1 , - 1); r = 17 ( ii )( x - 1) 2 +( y - 1) 2 x +1) 2 =17 (b) Given the vectors a =2 i - j +2 k , b =3 i j - k , c = i k . ( i ) Calculate 2 a · ( b - 3 c ). ( ii ) Determine the vector projection of c onto b . ( iii ) Find the cosine of the angle between a and b . ( iv ) Find a unit vector that is perpendicular to the plane determined by a and c . Answer: ( i ) - 32 ( ii ) 1 14 (3 i j - k ) ( iii ) 2 3 14 ( iv ) - 2 3 i - 2 3 j + 1 3 k Problem 2. Given the planes P 1 :2 ( x - 1) - ( y - 2( z - 2) = 0 ,P 2 :4 x - 2 y +5 z = 3, and the point Q :( - 2 , 7 , 4). (a) Determine whether P 1 and P 2 are parallel, coincident, perpendicular, or none of the preceding. Answer: N 1 · N 2 = 0 implies P 1 P 2 (b) Find an equation for the plane through Q which is parallel to P 1 . Answer: 2( x +2) - ( y - 7) - 2( z - 4) = 0 (c) Determine scalar parametric equations for the line through Q which is parallel to the line of intersection of P 1 and P 2 . Answer: x = - 2 - 9 t, y =7 - 18 t, z =4 Problem 3. The position of an object at time t is given by: r ( t )= e - t i + e t j - t 2 k , 0 t< . (a) Determine the velocity v and the speed of the object at time t . Answer: velocity: v = - e - t i + e t j - 2 k ; speed: ± v ± = e t + e - t (b) Determine the acceleration of the object at time t . Answer: acceleration: a = e - t i + e t j 1
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(c) Find the distance that the object travels during the time interval 0 t< ln3. Answer: 8 3 Problem 4. (a) A curve C in the plane is defned by the parametric equations: x = t 2 +1 ,y = 4 3 t 3 - 1. ( i ) Find the length o± C ±rom t =0to t =2 .
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C3-fin-ex-ans - MATH 2433 Problem 1. FINAL EXAM REVIEW...

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