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Unformatted text preview: Review Solution for Midterm II, MATH 20C, 2009 1 Chapter 13 1. A parking ramp follows the curve in space traced by the function r ( t ) = < cos t, sin t,t > for 0 < t < 4 π. What is the curvature function κ ( t ) of the ramp? Ans. Note that the curvature formula is κ ( t ) =  r ( t ) × r 00 ( t )   r ( t )  3 . r ( t ) = < sin t, cos t, 1 > , r 00 ( t ) = < cos t, sin t, > , r ( t ) × r 00 ( t ) = < sin t, cos t, 1 > . So,  r ( t ) × r 00 ( t )  = √ 2.  r ( t )  = √ 2. Hence, κ ( t ) = √ 2 ( √ 2) 3 = 1 2 . 2. (13.5) Santa Claus’ sleigh loses power after taking off at 64 ft/s from a 32 foot high roof at an angle 45 ◦ . How many seconds elapse before he hits the ground? (For this problem, assume that Santa exists and that his magic is not powerful enough to alter the earth’s gravitational acceleration of 32 ft/s 2 ). Ans. a ( t ) = < , 32 > , v (0) = 64 < cos45 ◦ , sin45 ◦ > , r (0) = (0 , 32). v ( t ) = Z t a ( u ) du + v (0) = < 32 √ 2 , 32 t + 32 √ 2 >, r ( t ) = Z t v ( u ) du + r (0) = < 32 √ 2 t, 16 t 2 + 32 √ 2 t + 32 > . We want to find t such that 16 t 2 + 32 √ 2 t + 32 = 0, or t 2 2 √ 2 t 2 = 0 and the positive root of this equation is t = 2 + √ 2 . 3. (13.5) At t = 0, an airplane takes off. At that moment, its position vector is < , , > and its velocity vector is < 1 , 2 , > . Find its position vector at time t = 6, if the acceleration of the airplane is a ( t ) = < 1 , ,t > . Ans. r (0) = < , , > , v (0) = < 1 , 2 , > , a ( t ) = < 1 , ,t > . v ( t ) = Z t a ( u ) du + v (0) = t, , 1 2 t 2 + < 1 , 2 , > = t + 1 , 2 , 1 2 t 2 , r ( t ) = Z t v ( u ) du + r (0) = 1 2 t 2 + t, 2 t, 1 6 t 3 . Hence, the position at t = 6 is r (6) = < 24 , 12 , 36 > . 2 Chapter 14 1. (14.2) Compute the following limits. If the limit does not exist, explain why. (a) lim ( x,y ) → (2 , 3) x 2 + xy + 2 y 2 1 x 2 y 2 + 4 (b) lim ( x,y ) → (0 , 0) x 2 + y 2 x 2 + xy + y 2 Ans. (a) For this limit, we can simply plug in the values x = 2 and y = 3 because the rational function x 2 + xy +2 y 2 1 x 2 y 2 +4 is defined and continuous at (2...
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This note was uploaded on 01/26/2010 for the course MATH Math 20C taught by Professor Lunasin during the Fall '08 term at UCSD.
 Fall '08
 Lunasin
 Math

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