Unformatted text preview: f ( x ) = cf ( x ) for all x ∈ R then f ( x ) = e cx f (0) for all x ∈ R . 5. Using the identities (sin x ) = cos x and (cos x ) =sin x , ﬁnd the derivative of tan x . Use the chain rule and the identity arctan(tan x ) = x to show that the derivative of f ( x ) = arctan x is equal to (1 + x 2 )1 . 6. Use the mean value theorem to obtain the best upper and lower bounds on arctan(1 . 1) that you can. Note that arctan(1) = π/ 4. 7. Use an appropriate theorem to ﬁnd the range of values of x ∈ R for which each of the following power series converges: ( i ) ∞ X n =1 n 91 n 8 + 1 x n . ( ii ) ∞ X n =0 2 n + 3 n 4 n + 5 n x n . ( iii ) ∞ X n =0 (2 n )! x n ( n !) 2 . 1...
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 Spring '09
 Calculus, Derivative, Mathematical analysis, minimum values, mean value

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