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Spring 2004 - Nordgren's Class - Exam 1

# Spring 2004 - Nordgren's Class - Exam 1 - Solutions to...

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Solutions to midterm 1 By H˚ akan Nordgren Question 1: Find the most general anti-derivative of 1. f ( t ) = 3 cos t + t 2 and 2. g ( t ) = 2 t + e t . Answer: 1. We need to find a function with we can differentiate to obtain 3 cos t and one which differentiates to t 2 . The functions 3 sin t and 1 3 t 3 work. Thus the most general anti-derivative of f is F ( t ) = 3 sin t + 1 3 t 3 + constant. Make sure that you always check your answers. 2. This time, we need to find a function with we can differentiate to obtain 2 t and one which differentiates to e t . The functions 2 2 3 t 3 2 and e t work. Thus the most general anti-derivative of g is G ( t ) = 4 3 t 3 2 + e t + constant. Question 2: A particle is moving along a straight line with velocity v ( t ) = sin t - cos t meters per second, and has initial displacement s (0) = 0 meters. Find its position at time t . Answer: So we know that ds ( t ) dt = v ( t ) = sin t - cos t . Thus we can integrate with respect to t to obtain s ( t ) = - cos t - sin t + C , where C is a constant. Since we know that 0 = s (0) = - 1+ C we must have C = 1. Thus the position of the particle at time t is s ( t ) = - cos t - sin t + 1. Question 3: With the aid of a sketch, evaluate the following integral by interpreting

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