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practicefinala

# practicefinala - comes to rest Problem 4(10 points...

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Final Examination Version A Math 2J Instructions: You have two hours to complete this test. All relevent work must be shown to receive full credit for a problem. No credit will be given for illegible solutions Name: Problem 1 (10 points): (a): Find the second degree Taylor polynomial p 2 ( x ) with center c = 0 for the function cos ( x 2 ) (NOT to be confused with cos 2 ( x )), using only the deﬁnition of the Taylor polynomial. (b): Use the polynomial from part (a) to ﬁnd an approximate value for the deﬁnite integral Z 1 0 cos ( x 2 ) dx

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Problem 2 (10 points): Find the limits of the following sequences. Justify your answers. (a): a n = sin 2 ( n ) n 4 . (b): b n = ne - 2 n
Problem 3 (10 points): The bob of a pendulum swings through an arc of 24 cm on its ﬁrst swing. Each successive swing is 5/6 of the length of the preceding swing. Find the total distance traveled by the bob before the pendulum

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Unformatted text preview: comes to rest. Problem 4 (10 points): Determine which of the following series converges. For any that con-verge, ﬁnd the limit. (a): ∞ X n =1 ln ± 2 n 7 n-5 ² (b): ∞ X n =1 (-1) n 5 2 n (2 n )! (c): ∞ X n =1 ( 2-n-2-3 n ) Problem 5 (10 points): Find the ﬁrst four nonzero coeﬃcients of the MacLaurin series ex-pansion of f ( x ) = ln(1 + x ) 1-x . Problem 6 (10 points): (a): Find the interval of convergence for the power series ∞ X n =1 1 n 5 n ( x-4) n (b): Give an example of two power series ∑ a n ( x-c ) n and ∑ b n ( x-c ) n with identical radius of convergence R such that the radius of conver-gence R of the series X ( a n + b n ) ( x-c ) n satisﬁes R > R ....
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practicefinala - comes to rest Problem 4(10 points...

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