# epef - MAC 2312 Final Exam Prof S Hudson Name Show all your...

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MAC 2312 April 24, 2003 Final Exam Prof. S. Hudson Name Show all your work and explain your reasoning. Don’t use your own paper, a calculator or book. You may ask for paper or for clariﬁcations. 1) [5pts] Find the average value of f ( x ) = x 4 on the interval [0 , 2]. 2) [10pts] Approximate ln 2 two ways (you may leave your answers as fractions): A) Give a Taylor series for ln(1 + x ) and set x = 1 (this should produce a familiar series). Then ﬁnd the 3rd partial sum. Give an error estimate for this answer. B) Give the integral deﬁnition of ln( x ) and set x = 2. Estimate the integral using Simpson’s rule with n = 2. Don’t do an error estimate. 3) [5 pts each] Compute (or explain why no answer exists): R tan 2 x sec 2 x dx R x 2 e x dx R 4 0 | x - 2 | - 1 / 2 dx = R + e dx x ln x 4) [5pts] Use a famous power series to ﬁnd the McLaurin series for f ( x ) = x sin( x 2 ). 5) [10pts] Find the interval of convergence of the series X n =1 3 n x n n 3 6) [10pts] Find the area of the the region inside

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epef - MAC 2312 Final Exam Prof S Hudson Name Show all your...

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