final exam review 2419 111806

# final exam review 2419 111806 - MATH 2419 FINAL EXAM REVIEW...

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Unformatted text preview: MATH 2419 FINAL EXAM REVIEW 18 November 2006 Complex Numbers DeMoivre's Theorem: If ) sin (cos i r z + = and n is a positive integer then n n i r z )] sin (cos [ + = ) sin (cos n i n r n + = The n roots of ( 29 sin cos i r z + = are given by + + + = n k i n k r u n k 2 sin 2 cos for 1 ,..., 1 ,- = n k 1. Compute the indicated power by using DeMoivre's Theorem (a) 3 1 + i i (b) ( 29 4 3 i- 2. Find the cube roots of i 27 3. Find the fourth roots of i 3 3 + Chapter 8, Section 8 1. Evaluate the improper integral or determine that it diverges. (a) 3 3 4 2 ) 9 (- x x dx (b) 6 4 2 2- x x dx (c) 9 2 2 + x dx (d) x tan dx (e) 2 1 2 1 2 2 ) 4 ( 1- x x dx Chapter 9, Sequences and Series 1. Determine whether the series is absolutely convergent, conditionally convergent, or divergent. (a) = 1 k k 3 tan (b) = 3 k ( 29 2 ln 5 k k (c) = 1 n n n n 2 ) 1 (- (d) = 2 n 1 1 ) 1 ( 2 2 +-- n n n (e) = 1 n 2 n e n (f) = 2 n n n ln ) 1 (- (g) = 2 n n n n n 2 ln ) 1 ( 2- (h) = 1 n 2 ) ! ( 3 )! 2 ( n n n (i) = 1 k ) 1 4 ( 11 7 3 ) 2 3 ( 11 8 5- + k k 2. Find the 5th degree Maclaurin polynomial of x x x f cos ) ( = . 3 Find the interval of convergence of the power series. Page 1 of 9 (a) = k ) 4 ( 2 ) 5 ( ) 1 ( +-- k x k k k (b) = 1 n n n n n x 3 ) 2 ( ) 1 ( 2-- 4. Let f be the function = ) ( x f = 1 n n x n n n 4 ) ( ) 1 (- . Find the interval of convergence for dx x f ) ( . 5. Find a power series centered at 2- = c for the function x x f- = 3 3 ) ( and identify the interval of convergence. 6. Find a power series centered at 1 = c for the function 2 4 3 ) (- = x x f and identify the radius of convergence. 7. Let 2 5 ) ( 2--- = x x x x f (a) Find the partial fraction decomposition of f . (b) Find the power series centered at zero for each term of the decomposition. (c) Write the power series for f and identify the interval of convergence. 8. Given the Maclaurin series = x sin = n )! 1 2 ( ) ( ) 1 ( 1 2--- n x n n .... ! 7 ! 5 ! 3 7 5 3 +- +- = x x x x for x in ) , ( - (a) Find the Maclaurin series for 2 sin x (b) Use the first 4 terms of the series found in part (a) to approximate 3 . 2 sin x dx (c) Find the Maclaurin series for x cos (d) Given the identity 2 2 cos 1 cos 2 x x + = and the series in part (c), determine the validity of the following statement; + = 1 cos 2 x = 1 n )! 2 ( 2 ) 1 ( 2 1 2 n x n n n-- . Chapter 10 1. Let C be the curve represented by the parametric equations 1 + = t x and t y- = 3 (a) Find the corresponding rectangular equation for C by eliminating the parameter....
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final exam review 2419 111806 - MATH 2419 FINAL EXAM REVIEW...

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