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# fin06fk - MAC 2311 Final Exam and Key Prof S Hudson 1(20...

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MAC 2311 Dec 13, 2006 Final Exam and Key Prof. S. Hudson 1) (20 pts) Compute and simplify. Include an arbitrary constant, if appropriate; 3 x 2 1+ x 3 dx = 3 1+ x 2 dx = sin 2 t dt = e 3 x +2 dx = 2) (20 pts) Compute; d dx x x d dx (cos 2 x tan 2 x ) d dx log 2 x 3 Find dy/dx , given that xy 2 + x 2 y = y . 3) (10 pts) Solve this initial-value problem; find a formula for y ( t ) so that y ( t ) = 6( t + 1) 2 and y (0) = 7. 4) (10 pts) Compute; lim x 0 + (1 + 3 x ) 1 /x lim x 1 ln( x ) sin( πx ) 5) (10 pts) Answer TRUE or FALSE: The slope of the graph of tan - 1 ( x ) is positive for all x . The domain of sin - 1 ( x ) is [ - 1 , 1]. Every polynomial has exactly one antiderivative whose graph contains the point (3,4). The function f ( x ) = sec(2 x ) has an inverse on [0 , π/ 6]. If F is an antiderivative of an antiderivative of f , then F ( x ) = f ( x ). 6) (10 pts) Given parametric equations, x ( t ) = t 2 + 1 and y ( t ) = t 3 + 1, find the equation of the tangent line at the point where t = 2. 7a) (5pts) State the Intermediate Value Theorem. 1

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7b) (5pts) State the Extreme Value Theorem. 8) (10 pts) CHOOSE ONE; A) Prove the Product Rule. B) Prove (as in Ch2) that lim x 0 sin x x = 1 C) Prove (using ) that lim x 0 4 x + 7 = 7. Bonus (5pts): Use Newton’s method to approximate 3 (a positive root of x 2 - 3 = 0), starting from
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fin06fk - MAC 2311 Final Exam and Key Prof S Hudson 1(20...

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