FDWKCalcSM_ch9

FDWKCalcSM_ch9 - Section 9.1 377 55. Continued xe x dx = xe...

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Section 9.1 377 55. Continued xe dx xe e dx xe e C xx x −− =− + + Area + →∞ →∞ lim lim b b b bb xe e be e 0 1 1 1 11 + + = →∞ →∞ lim lim b b b b b e e 56. (a) xe dx x 2 0 (b) lim b x b xe dx →∞ 2 0 (c) Note that dx x 2 can be found by parts: xe dx x e e dx x 22 2 () + 24 e C . Area = lim lim b x b b b xe dx xe e 2 0 0 + = lim . b e 0 4 4 57. (a) ππ xd y dy y 2 2 0 0 1 = + (b) lim b b dy y + π 1 2 0 (c) Volume = + + lim lim ( ) b b dy y y 1 1 2 1 0 0 + + = lim b b 1 1 1 58. Note that dx x can be found by parts: xe dx x e e dx xe e C x =− −− = − − + . So xe dx xe dx xe e x k x k k k == lim lim 0 0 0 = −−+ lim k kk k ee 1 1 By L’Hopital’s rule, lim lim . k k k k k = 1 0 Therefore, xe dx k x k = lim 1 1 0 =−+= 0011 The integral converges to 1. Chapter 9 Infinite Series Section 9.1 Power Series (pp. 473– 483) Exploration 1 Finding Power Series for Other Functions 1. 1 23 −+ − + +− + xx x x n ±± . 2. xx x x x nn −+−++ + + 234 1 1 . 3. 12 4 8 2 ++ + ++ + x n . 4. 1 1 −−+− −− ++ − + ( ) . x x This geometric series converges for –1 < x 1 < 1, which is equivalent to 0 < x < 2. The interval of convergence is (0, 2). 5. 1 3 1 3 1 1 3 1 1 3 1 1 3 +− + ()() x ± −+ n n x . 1 ± This geometric series converges for –1 < x 1 < 1, which is equivalent to 0 < x < 2. The interval of convergence is (0, 2). Exploration 2 Finding a Power Series for tan –1 x 1. 246 2 + xxx x . 2. tan = + 1 2 1 1 x t dt x 0 + + + + =−+− (( ) ) 35 0 62 ttt t d t t x 7 2 1 0 7 1 21 ++ + + =− + − + n n x t n x x x n n n 72 1 7 1 + + . 3. The graphs of the first four partial sums appear to be converging on the interval ( 1, 1).
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378 Section 9.1 4. When x = 1, the series becomes 1 1 3 1 5 1 7 1 21 −+−+ + + + ±± () . n n This series does appear to converge. The terms are getting smaller, and because they alternate in sign they cause the partial sums to oscillate above and below a limit. The two calculator statements shown below will cause the succes- sive partial sums to appear on the calculator each time the ENTER button is pushed. The partial sums will appear to be approaching a limit of π /4 (which is tan ( )), 1 1 although very slowly. Exploration 3 A Series with a Curious Property 1. fx x xx x n n ( !! ! . ) =+ + + + + + 1 23 2. f . 0100 1 =+++ = ± 3. Since this function is its own derivative and takes on the value 1 at x = 0, we suspect that it must be e x . 4. If then and when yf x dy dx yy x == = = , . 10 5. The differential equation is separable. dy y dx dy y dx yxC yK e Ke K ye k x = = =+ = =⇒ = ∴= ln .
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This note was uploaded on 09/26/2010 for the course MATH 135 taught by Professor Noone during the Summer '08 term at Rutgers.

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FDWKCalcSM_ch9 - Section 9.1 377 55. Continued xe x dx = xe...

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