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mid2samp1soln

# mid2samp1soln - Math 1A Spring 2008 Wilkening Sample...

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Math 1A, Spring 2008, Wilkening Sample Midterm 2 You are allowed one 8 . 5 × 11 sheet of notes with writing on both sides. This sheet must be turned in with your exam. Calculators are not allowed. 1. (1 point) write your name, section number, and GSI’s name on your exam and write your name on your sheet of notes. 2. (4 points) Find the equation of the tangent line to the curve y 2 = x 3 + 3 x 2 at the point (1 , - 2). Answer: d dx ( y 2 ) = d dx ( x 3 + 3 x 2 ) 2 y dy dx = 3 x 2 + 6 x 2( - 2) dy dx = 3(1) 2 + 6(1) = 9 dy dx = - 9 4 so the equation of the tangent line is y + 2 = - 9 4 ( x - 1). 3. (5 points) Find the relative maxima, minima and inflection points of the function f ( x ) = xe - x 2 / 2 Answer: f ( x ) = x ( - x ) e - x 2 / 2 + e - x 2 / 2 = (1 - x 2 ) e - x 2 / 2 f ( x ) = 0 if x = ± 1. (because (1 - x 2 ) = 0 if x = ± 1) f ( x ) > 0 if - 1 < x < 1. (because (1 - x 2 ) > 0 if - 1 < x < 1.) f ( x ) < 0 if x < - 1 or x > 1. (because (1 - x 2 ) < 0 if x < - 1 or x > 1.) (note: e - x 2 / 2 is always > 0) so f has a local minimum at x = - 1 and a local maximum at x = 1 f ( x ) = (1 - x 2 )( - x ) e - x 2 / 2 + ( - 2 x ) e - x 2 / 2 = ( x 3 - 3 x ) e - x 2 / 2

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f ( x ) = 0 if x = 0 or x = ± 3 f ( x ) > 0 if - 3 < x < 0 or x > 3 f ( x ) > 0 if x < -
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mid2samp1soln - Math 1A Spring 2008 Wilkening Sample...

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