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Calculus II Notes Week 14

by: Zachary Hill

Calculus II Notes Week 14 MATH 1220

Marketplace > Tulane University > Mathematics (M) > MATH 1220 > Calculus II Notes Week 14
Zachary Hill
GPA 3.88

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About this Document

These notes cover some review for Taylor series and the ratio test.
Calculus II
Benjamin Klaff
Class Notes
Math, MATH1220, Calculus, calculusii, Klaff
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This 4 page Class Notes was uploaded by Zachary Hill on Sunday April 24, 2016. The Class Notes belongs to MATH 1220 at Tulane University taught by Benjamin Klaff in Spring 2016. Since its upload, it has received 14 views. For similar materials see Calculus II in Mathematics (M) at Tulane University.

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Date Created: 04/24/16
MATH 1220 Notes for Week #14  18 April 2016  ● Find lim sin(x.  x→0 tan(x) ○ lim sin(x= lim ssin(x)lim sin(x)cos(= limcos(x) = 1  x→0 tan(x) x→0 cos(x) x→0 sin(x) x→0 ○ Recognize that the limit of the ratios of these functions near x = 0  is 1 because  that exhibit the same characteristics (slope of 1 in this case) near x = 0 .  ● Find the 3rd degree Taylor polynomial to f(x) = tan(x) near x = 0  ○   (0) f (0) = tan(0) = 0  y 0 0  (1) cos (0)+sin (0) 1 f (0) = (tan(0)) ′ = cos (0) = cos (0)=  y 1 x 1!x  2              sec (0) = 1  (2) 2 0 2 f (0) = (sec (0)) ′= 2sec(0)(sec(0)) ′=  y 2 x + x 2!x              2sec(0)((−2in(0) =  sin(0)0)             2sec(0)cos(0)(cos(0) =              2sec(0)tan(0)sec(0) =               2(1)(0)(1) = 0  f (0) = (2sec (0)tan(0)) ′=  y 3 x + x = x + x   1 3 2 3! 3              (2sec (0′tan(0) +               2sec (0)(tan(0′=               2(2sec(0))(sec(0)′tan(0) +               2sec (0)sec (0) =               4sec(0)tan(0)sec(0)tan(0) +  4 2 2              2sec (0) = 4sec (0)tan (0) +                2sec (0) = 4(1)(0) + 2(1) = 2  ● Exercise assigned for outside practice   1 ○ Find the first four nonzero terms of f(x) = 1+x2 near x = 0 by  (n) ■ Computing f (0). . .  ■ Manipulating the geometric series. . .    ○ Find the first four nonzero terms of f(x) = arctan(x)  near x = 0 by  (n) ■ Computing f (0). . .  ■ Manipulating the geometric series. . .          19 April 2016  Explored various number systems that will not be covered on the exam. This included modular  systems and the complex numbers.        20 April 2016  ● Find the first four nonzero terms of the Taylor series for e  centered at (or near) x = 1 .  2 3 f0(1) f1(1) f(2(1) 2 f3(1) 3 ○ T 3x) = c +0c (x1− 1) + c (x 2 1) + c (x − 3) = 0! + 1! (x − 1) + 2! (x − 1) + 3! (x − 1) = e + e(x − 1) + (x − 1) + (x − 1)   3 2! 3! ● Watched  ○ Khan Academy “Taylor Series”  ○ Khan Academy “Central Limit Theorem”  ○ Ben Lambert Central Limit Theorem Proof Part 1 on Youtube        22 April 2016  We practiced recalling the ratio test and the comparison test, and we came up with some series  for which the ratio test shows convergence and some for which it does not.  ● Series that converge according to the ratio test  ∞ xn ○ ∑ n!  n=0 ∞ n ○ ∑ (−1)   n=0 2n ● Series that don’t converge according to the ratio test  ∞ ○ ∑ nn  n=0x ∞ 2n ○ ∑ (−1) n n=0


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