# Is 1 determine the interval of convergence show the

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is 1. Determine the interval of convergence. Show 1 . g (d) The Maclaurin series for g , evaluated at 1, x = is a convergent alternating series with individual terms that decrease in absolute value to 0. Show that your approximation in part (c) must differ from ( ) 1 g by less than 1 . 5 (a) ( ) 6 9 12 3 1 3 1 2 3 4 n n x x x x x n + + + + + " " 2 : { 1 : first four terms 1 : general term (b) The interval of convergence is centered at 0. x = At 1, x = − the series is 1 1 1 1 1 , 2 3 4 n " " which diverges because the harmonic series diverges. At 1, x = the series is ( ) 1 1 1 1 1 1 1 , 2 3 4 n n + + + + + " " the alternating harmonic series, which converges. Therefore the interval of convergence is 1 1. x < 2 : answer with analysis (c) The Maclaurin series for ( ) , f x ( ) 2 , f t and ( ) g x are ( ) ( ) 1 3 1 2 5 8 11 1 : 1 3 3 3 3 3 n n n f x x x x x x + = = + + " ( ) ( ) 1 2 6 2 4 10 16 22 1 : 1 3 3 3 3 3 n n n f t t t t t t + = = + + " ( ) ( ) 6 1 5 11 17 23 1 1 3 3 3 3 3 : 1 6 1 5 11 17 23 n n n x x x x x g x n + = = + + " Thus ( ) 3 3 18 1 . 5 11 55 g = 4 : ( ) ( ) ( ) 2 2 1 : two terms for 1 : other terms for 1 : first two terms for 1 : approximation f t f t g x (d) The Maclaurin series for g evaluated at 1 x = is alternating, and the terms decrease in absolute value to 0. Thus ( ) 17 18 3 1 3 1 1 . 55 17 17 5 g < = < 1 : analysis
AP ® CALCULUS BC 2012 SCORING GUIDELINES Question 4 © 2012 The College Board. Visit the College Board on the Web: . x 1 1.1 1.2 1.3 1.4 ( ) f x ¢ 8 10 12 13 14.5 The function f is twice differentiable for 0 x > with ( ) 1 15 f = and ( ) 1 20. f ′′ = Values of , f the derivative of f , are given for selected values of x in the table above. (a) Write an equation for the line tangent to the graph of f at 1. x = Use this line to approximate ( ) 1.4 . f (b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate ( ) 1.4 1 . f x dx Use the approximation for ( ) 1.4 1 f x dx to estimate the value of ( ) 1.4 . f Show the computations that lead to your answer. (c) Use Euler’s method, starting at 1 x = with two steps of equal size, to approximate ( ) 1.4 . f Show the computations that lead to your answer. (d) Write the second-degree Taylor polynomial for f about 1. x = Use the Taylor polynomial to approximate ( ) 1.4 . f (a) ( ) 1 15, f = ( ) 1 8 f = An equation for the tangent line is ( ) 15 8 1 . y x = + ( ) ( ) 15 8 1.4 1.4 1 18.2 f + = 2 : { 1 : tangent line 1 : approximation (b) ( ) ( )( ) ( )( ) 1.4 1 0.2 10 0.2 13 4.6 dx f x + = ( ) ( ) ( ) 1.4 1 1.4 1 f f f dx x = + ( ) 15 4.6 9 1 1 6 .4 . f + = 3 : 1 : midpoint Riemann sum 1 : Fundamental Theorem of Calculus 1 : answer (c) ( ) ( ) ( )( ) 1 0.2 1.2 8 16.6 f f + = ( ) ( )( ) 1.4 16.6 0.2 12 19.0 f + = 2 : { 1 : Euler’s method with two steps 1 : answer (d) ( ) ( ) ( ) ( ) ( ) 2 2 2 20 15 8 1 1 2! 0 1 1 1 8 5 1 T x x x x x = + + + = + ( ) ( ) ( ) 2 15 8 1.4 1 10 1.4 1 19 . . 1 4 8 f + + = 2 : { 1 : Taylor polynomial 1 : approximation
AP ® CALCULUS BC 2012 SCORING GUIDELINES Question 6 © 2012 The College Board.
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