Optional - Math 210 Kouba Optional Power Series Problems 1...

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Unformatted text preview: Math 210 Kouba Optional Power Series Problems 1 1 — as standard shortcuts (substitution, polynomial multiplication, polynomial division, differen- tiation, and integration) to find the first three nonzero terms of the Maclaurin series for each of the following functions. 1.) Use the known Maclaurin series for em, Sings, cosm, , and ln(1 + 93) together with (I: 2 2 a.) exln(1+m) b.)i:—(1+_—”;) c.)(sin(a:/3))2 d.) m 1 e.) Elf—W (HINT: Start with 1 _ a: and use differentiation.) f.) m3 arctan(a:2) (HINT: Start with 1 + 2 and use integration.) as 2.) Use the absolute error for an alternating series, IRnI, to estimate the exact value of each integral, which cannot be solved using the Fundamental Theorem of Calculus, with error at most 0.0001. 1 1/2 2 a.) / cos(w4) dm b.) / e‘w da: 0 0 3.) Use the absolute Lagrange error (remainder) for a Taylor series, |R(:I:; a)! = (n + 1). where an is between a: and a, to estimate the exact value of each number with error at most 0.0001. a.) 60'7 b.) sin(1.1) c.) ln(0.8) 4.) Write each Maclaurin series as a simple function. 3 5 7 9 {13 IL’ {E T a9§‘§+fi‘§+” 531133 55:35 57307 c.)2+3.2-m+4.3.m2+5-4-x3+6-5-x4+~- a: x2 (1:3 m4 d”“§+n"a+§‘ e)m3_i+m_9_fi+fi_... ' 2 3 4 5 .fln+n(qa<w—-arfia 3 ...
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