13 - 5.) Find the general solution of the following...

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CALCULUS II EXAM 1 Show all your work to recive full credit. Calculators are not allowed. Closed book, closed notes. .. GOOD LUCK ! NAME : 1.) Calculate the following limits ( a ) lim x →∞ ( x 3 + 1) 1 ln( x ) , ( b ) lim x 0 ( 1 sin( x ) - 1 x ) .
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2.) a.) Determine whether or not the following improper integrals converge or diverge. To receive full credit indicate what tests you are using and show all your steps. ( a 1 ) Z e 1 x (ln( x )) 2 dx, ( a 2 ) Z 1 0 e x x dx, 2
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b.)Determine whether or not the following infinite series converge or diverge. To receive full credit indicate what tests you are using and show all your steps ( b 1 ) X k =1 2 k + k k 3 + k , ( b 2 ) X k =1 2 k k ! k k , ( b 3 ) X k =1 ( - 1) k k + 2 k 2 + k . 3
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3.) Expand g ( x ) = x ln( x ) in powers of x - 2 . 4
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4.) a.) Find the interval of convergence for the power series X k =1 ( - 1) k k 2 ( k + 1)! ( x + 3) k . b.) Sum the given series X k =0 1 k ! x 3 k +1 . 5
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Unformatted text preview: 5.) Find the general solution of the following dierential equation xy-y = 2 x ln( x ) . 6 6.) Consider the function(transformation) from R 2 to R 2 dened by f ( x y ) = x 2-y 2 x-2 . a.) Find all solutions of f ( x y ) = . b.) Is f invertible? If so, nd a formula for the inverse. If not, explain why not. 7 7.) Consider the following transformations from R 2 to R 2 . Which ones are linear? Explain your answers and for those that are linear, write down the corresponding matrix. a.) f ( x y ) = y + x y-x . b.) g ( x y ) = | x | y-x . c.) h ( x y ) = x 2-y 2 x 2 + y 2 . 8...
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This note was uploaded on 04/27/2008 for the course MATH 1502 taught by Professor Mcclain during the Fall '07 term at Georgia Institute of Technology.

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13 - 5.) Find the general solution of the following...

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