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FinalExam-sample - FINAL EXAM Math 251 Name PROBLEM 2 The...

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FINAL EXAM Math 251 Name: PROBLEM 1 (5 pts): Find the equation of the plane containing the points (1 , 0 , 1) , (1 , 1 , 2) and ( 1 , 2 , 9). PROBLEM 2: The two space curves vector r 1 ( t ) = ( t + 1 , t 2 2 t + 2 , 2 t ) and vector r 2 ( s ) = ( 2 s, s 3 2 s + 1 , 5 s 4 ) intersect in exactly one point. a (3 pts): Find the point of intersection and the values t 0 and s 0 that give this point for vector r 1 and vector r 2 respectively. b (3 pts): Find the angle between the vectors vector r 1 ( t 0 ) and vector r 2 ( s 0 ). PROBLEM 3 (12 pts): Find the points of maximum and minimum cur- vature for the curve vector r ( t ) = ( e t ı + ( t on the range 2 t 2. PROBLEM 4 (12 pts): At the point ( 2 , 7) a certain function f ( x,y ) has gradient f ( 2 , 7) = (− 4 , 12 ) . Find two unit vectors (directions) for which the directional derivative of f at P = ( 2 , 7) is equal to 4. PROBLEM 5 (12 pts): Find and classify all critical points of the function f ( x,y ) = xy 2 6 x 2 3 y 2 + 10 .
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