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Unformatted text preview: Exam 2a 1)(15 points) Find the radius of convergence of 3n (x − 1)n √ n (−1)k+1 2k . Does it: 2)(15 points) Look at k! a) Converge absolutely? b) Converge Conditionally? 3)(50 points) Let f (x) = x 3 ; a = 1. i) Find T3 (x) ii) Find an algebraic form for the error term. 1 iii) Use i) to compute .98 3 iv) Use ii) to ﬁnd the error in iii) v) How many decimal places of accuracy does ii) give? 4)(20 points) Find a power series for cos(x2 ) dx 1 Exam IIa 1)(40 points) Find a power series in x and the radius of convergence of the following. Write your answers longhand and using summation notation: 1 a) x sin(x) dx b) (1 − x2 )2 2)(40 points) Let f (x) = x− 3 , a = 1. a) Find T2 , R2 . 2 b) Use T2 to estimate (1.03)− 3 . c) Use R2 to estimate the error in b) d) How many decimal places of accuracy is this? Exam IIb 3)(20 points) Use three terms of your answer in 2a to compute 1)(40 points) Find a power series in x and the radius of convergence .01 of the following. Write your answers x) dx x sin( longhand and using summation notation: 0 2 b) tan−1 x a) x cos x What is the error?
1 2)(40 points) Let f (x) = 1−x , a = 0. a) Find T2 , R2 . b) Use T2 to estimate 1.1 . 03 c) Use R2 to estimate the error in b) d) How many decimal places of accuracy is this?
2 3)(20 points) Use three terms of your answer in 2b to compute tan−1 (.02) Use alternating series to ﬁnd the error. Exam 2b Calculators Required 1)(30 points) a) Find a power series in x for the integral below, writing your answer longhand (three terms) b) Write your answer using summation notation. c) Find the radius of convergence of the series in b) 1 Exam II dx 1 + x4 1)(10 points) Find the radius of convergence of: 1 2)(50 points) Let f (x) = 1 , a = 0. (1−x) 3n ∞ 3 (x + 2)n a) Find T3 . √ 1 n b) Use T3 to estimate 1. n=1
(1.03) 3 c) How big can the error be in b) d) How many decimal places of accuracy is this? 2)(30 points) Find a power series in x and the radius of convergence of: the following. Write your answers longhand and using 3)(20 points) Use two terms of your answer in #1 to compute summation notation: .01 1 1 2 ) a) sin(x ) dx+ x4 bdx 1 0 (1 − x2 )2 3)(40 points) Let f (x) = x− 3 , a = 1. a) Find T2 , R2 . 1 b) Use T2 to estimate (.98)− 3 . c) Use R2 to estimate the error in b) d) How many decimal places of accuracy is this? 4)(20 points) Use three terms of your answer in 2a) to compute What is the error?
.01
1 What is the error? sin(x2 ) dx 0 Quiz 6b 25min 1) Let x = sin2 t, y = − cos t. Convert to a Cartesian equation and use that to sketch the curve. Do NOT get your sketch by plotting points!6a Quiz 25min 1) Let x = 1 − cos2 t, y = sin t. Convert to a Cartesian ¯ equation and 2, that to sketch 0, curve. (− NOT get ¯ ¯ 2) Let a = (1,use −1), b = (−2, the−1), c =Do 4, 1, −2) your sketch Which are parallel, be vectors. by plotting points! perpendicular, or neither? Why? Show work, don’t just give yes or no.
Find the area under the parametric curve x = sin2 t, y = cos t for 2)≤Do the points P (1, 2), Q(2, 3), R(1, 1) lie on the same 0 t ≤ π. Sketch the parametric curve x = et , y = e−t . Find the area under the parametric curve x = et , y = e−t for 0 ≤ t ≤ 1. Find the area under the parametric curve x = 2 cos t, y = sin t for 0 ≤ t ≤ π. 2 Find the length of the parametric curve x = 1 + 2t2 , y = 1 − t2 for 0 ≤ t ≤ 1. Find the length of the parametric curve x = 1 + 2t2 , y = 1 − t2 for 0 ≤ t ≤ 1. Find the length of the parametric curve x = cos3 t, y = sin3 t for 0 ≤ t ≤ π. line? Use vector techniques to ﬁnd an answer. ...
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This note was uploaded on 02/24/2010 for the course M 408d taught by Professor Sadler during the Fall '07 term at University of Texas at Austin.
 Fall '07
 Sadler
 Algebra

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