152pretestIIsols

# 152pretestIIsols - [ 2 ] (F) Determine the radius of...

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MATH 152 FALL 2009 CALCULUS II FOR THE MATHEMATICAL AND PHYSICAL SCIENCES PRETEST II ON DECEMBER 9, 2009 Name: Sample Solutions , Total score: 20 [ 20 ] (A) Write down the 7th Maclaurin polynomial T 7 of sin ( x ) . T 7 ( x ) = x - x 3 3! + x 5 5! - x 7 7! [ 4 ] (B) Using the 7th Maclaurin polynomial above, write down an approximation to sin ( 1 ) . sin ( 1 ) 1 - 1/3! + 1/5! - 1/7! [ 2 ] (C) Estimate the error between your approximation and the exact value of sin ( 1 ) . K = max u [ 0,1 ] | f ( 8 ) ( u ) | ≤ 1 | T 7 ( 1 ) - sin ( 1 ) | ≤ 1 · | 1 - 0 | 8 8! = 1/8! [4] (D) Compute the ﬁrst four terms of the Maclaurin series S ( x ) of e x 2 . S ( x ) = 1 + x 2 + x 4 2! + x 6 3! + . . . [ 4 ] (E) Write down a closed formula for S ( x ) . S ( x ) = n = 0 x 2 n n !
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Unformatted text preview: [ 2 ] (F) Determine the radius of convergence of S ( x ) , and write down the interval of con-vergence of the series. Brieﬂy explain your reasoning. R = ∞ , interval of convergence = ( ∞ , ∞ ) [4] We use the ratio test and consider ρ = lim n → ∞ | a n + 1 a n | with a n = x 2 n / n !. We ﬁnd that ρ = lim n → ∞ | x 2 n + 1 | = 0 regardless of the value of x . Since ρ < 1 for all values of x , the ratio test tells us that the series converges everywhere....
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## This note was uploaded on 02/29/2012 for the course MATH 152 taught by Professor Sc during the Fall '08 term at Rutgers.

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