{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

C3-fin-ex-ans

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

This preview shows pages 1–3. Sign up to view the full content.

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 + 2 j - k , c = i + 2 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 + 2 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 + 1) - 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(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 defined by the parametric equations: x = t 2 + 1 , y = 4 3 t 3 - 1.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}