ep1 - 92-131 Calculus 1 Anti-Derivatives Find general...

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Unformatted text preview: 92-131 Calculus 1 Anti-Derivatives Find general antiderivatives for the following functions: 1) f ( x) = e 3 x + x 2 2) f ( x) = e - x / 2 - x 2 3 3) s (t ) = cos t + sin t - sec 2 t ( x + 1) 2 x 4) g ( x) = 1 1 - 2 x x 1 1- x2 + 1 1+ x2 5) g ( x) = 6) f ( x) = 7) f ( x) = sin(3 x) + cos( x) 8) s (t ) = 2t cos(t 2 ) 2 4x3 9) g ( x) = 4 + xe x x +1 10) Suppose a robot is moving on a linear track so that its position (in meters) is given by: t4 s ( t ) = 16 t - , for t [ 0 , 4 ] , 4 where t is time in seconds. a) What is the robot's velocity at t = 2 seconds? b) When does the robot reverse direction? c) What is the robot's acceleration at t = 3 seconds? 11 )Suppose a particle is moving on a linear track so that its acceleration (in m/s2) is given by: a ( t ) = 4 - sin 4t , for t [ 0 , 4 ] , where t is time in seconds. If the particle's initial position is s (0) = 2 m, and initial velocity is v (0) = -2 m/s, what is its position at any given time? 12) Suppose a particle is moving on a linear track so that its acceleration (in m/s2) is given by: a ( t ) = 4 t + sin 2 t + e -3t , for t [ 0 , 10 ] , where t is time in seconds. If the particle's initial position is s (0) = 2 m, and initial velocity is v (0) = -2 m/s, what is its position at any given time? Solutions 1 1 1) F ( x) = e 3 x + x 3 + C 3 3 3) S (t ) = sin t - cos t - tan t + C 2) F ( x) = -2e - x / 2 - 33 5 x +C 5 4) 1 G ( x) = - - 2 x + C x so G ( x) = x2 + 2 x + ln x + C 2 5) g ( x) = ( x + 1) 2 x 2 + 2 x + 1 1 = = x+2+ x x x 7) 6) F ( x) = arcsin x + arctan x + C 1 1 f ( x) = - cos(3 x) + sin( x) + C 3 G ( x) = ln( x 4 + 1) + 1 x2 e +C 2 8) s (t ) = sin(t 2 ) + C 9) 10) a) s ( 2 ) = v ( 2) = 16 - t 3 c) s ( 3 ) = - 3t 11) s(t ) = 2t 2 + ( 2 t =3 ) = -27 m/s t =2 = 8 m/s 2 b) When s ( t ) = 16 - t 3 = 0 , or t = 16 3 1 9 sin(4t ) - t + 2 4 16 2 1 1 -3t 7 17 12) s ( t ) = t 3 - sin 2 t + e - t + 3 4 9 6 9 ...
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This note was uploaded on 02/13/2012 for the course MATH 92.131 taught by Professor Staff during the Fall '09 term at UMass Lowell.

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ep1 - 92-131 Calculus 1 Anti-Derivatives Find general...

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