MIDTERM1 20C B

# MIDTERM1 20C B - tangential and normal components 5 FORMULA...

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1 MIDTERM I NAME MATH 20C (please print) TA’S NAME (please print) FORM B Problem 1. (14 pts.) Find the equation of: (a) the line through the point P = (1 , - 1 , 0) and parallel to vector v = (0 , 1 , 1) . (b) the plane through the points P 1 = (1 , 0 , 0) , P 2 = (1 , 2 , 0) and P 3 = (2 , 1 , 1) . Problem 2. (14 pts.) Let v 1 = (1 , 1 , - 1) , v 2 = (2 , 1 , 0) , v 3 = (0 , 2 , 1) be three given vectors in R 3 . Find: (a) the area of the parallelogram spaned by v 1 and v 2 , (b) the volume of the parallelepiped spaned by v 1 ,v 2 ,v 3 .

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2 Problem 3. (14 pts.) Let r 1 ( t ) = ( t 2 ,t, 0) , r 2 ( t ) = (0 ,t,e t ) . Find: ( a ) d dt ( r 1 ( t ) ˙ r 2 ( t )) ( b ) d dt ( r 1 ( t ) × r 2 ( t )) Problem 4. (14 pts. ) Compute the following limit. Justify your answer. lim ( x,y ) (0 , 0) x 4 - y 4 x 2 + y 2
3 Problem 5. (16 pts.) An object thrown from the ground level at the angle of 45 lands 144 ft away. Find the initial speed of the object. Problem 6. (14 pts) Find the arc length of the given curve: r ( t ) = ( 5 t, cos 2 t, sin 2 t ) , 0 ± t ± π 2 .

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4 Problem 7. (14 pts) A particle moves according to the following trajectory r ( t ) = (4 t, cos 3 t, sin 3 t ) . Find the decomposition of the acceleration into the
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Unformatted text preview: tangential and normal components. 5 FORMULA SHEET 1. The trajectory of an object thrown in the air is given by the formula: r ( t ) = r + v t-1 2 g t 2 where g = (0 , 32) , r is the initial position and v is the initial velocity. Additionally v = || v || is the initial speed. 2. The arc length of a curve given by the parametrization r ( t ) , a ± t ± b, is equal to: Ú b a || r Í ( t ) || dt. 3. The tangential and normal components of the acceleration for the trajectory given by the parametrization r ( t ) . The acceleration has a decomposition a ( t ) = v Í ( t ) T ( t ) + v Í ( t ) || T Í ( t ) || N ( t ) , where v ( t ) = || r Í ( t ) || , T ( t ) = 1 || r Í ( t ) || r Í ( t ) and N ( t ) = 1 || T Í ( t ) || T Í ( t ) ....
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## This note was uploaded on 08/14/2011 for the course MATH 20 C taught by Professor Ronevans during the Spring '08 term at UCSD.

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MIDTERM1 20C B - tangential and normal components 5 FORMULA...

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