practiceFinal2solutions - Math 1A, Spring 2008, Wilkening...

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Math 1A, Spring 2008, Wilkening Sample Final Exam 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. 0. (1 point) write your name, section number, and GSI’s name on your exam. 1. (3 points) give precise definitions of the following statements or expressions: (a) f ( x ) is neither even nor odd (b) R f ( x ) dx (c) R b a f ( x ) dx Solution: (a) There exist numbers x 1 and x 2 such that f ( - x 1 ) 6 = f ( x 1 ) and f ( - x 2 ) 6 = - f ( x 2 ). (b) R f ( x ) dx is any antiderivative of f ( x ), i.e. a function F ( x ) such that F 0 ( x ) = f ( x ). (c) the definite integral is defined as Z b a f ( x ) dx = lim max Δ x i 0 n X i =1 f ( x * i x i , where the limit is over all partitions a = x 0 < x 1 < ··· < x n - 1 < x n = b of the interval [ a, b ] into subintervals of length Δ i = x i - x i - 1 , and x * i is a sample point in the i th interval [ x i - 1 , x i ]. 2. (4 points) Show that the tangent lines to the curves y = x 3 and x 2 + 3 y 2 = 1 are perpendicular where the curves intersect. Solution: The slope of the tangent line of the first curve is m 1 = y 0 = 3 x 2 . For the second, differentiate implicitly: 2 x + 6 yy 0 = 0 y 0 = - x 3 y When the curves intersect, we have y = x 3 , so m 2 = y 0 = - 1 / (3 x 2 ). Since m 2 = - 1 /m 1 , these tangent lines are perpendicular.
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3. (3 points) Evaluate Z 1 0 tan - 1 x 1 + x 2 dx . Solution: Let u = tan - 1 x . Then du = dx 1 + x 2 and the limits of integration become x = 0 u = 0 , x = 1 u = π 4 .
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practiceFinal2solutions - Math 1A, Spring 2008, Wilkening...

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