DHS Maclaurin Series (9758) Topical Revision

6. Maclaurin Series 1 6. Maclaurin Series 1 SAJC/2015Promo/4 Given that the first three non -zero terms of the series expansion of 1 3(1 ) x− are equal to the first three non -zero terms of the series expansion of 3 3 ax bx + + , where a and b are constants, find the values of a and b. [4] 2 RI/2009Prelim/I/2 Given that 2 e 3 5,xyy+ + = 0y , (i) show that ( ) 2 2 2 dd2 2 3 e 0d d xyy yx x  + + + = , [2] (ii) find the first three terms of the Maclaurin’s series for y. [3] 3 RI/2014Promo/7 It is given that ln(1 2 ) 1 xy x += + . (i) Show that 2 22 d d 4(1 ) 2 0d d (1 2 ) yyx x x x+ + + = + . [3] (ii) Find the Maclaurin’s series for y, up to and including the term in 3.x [3] (iii) Verify that the same result is obtained if the standard series expansions for ln(1 2 )x+ and 1(1 ) x −+ are used. [3] 4 YJC/2015Promo/6 It is given that ( ) 2 1f 9 x x = − . (i) Find the expansion of f ( x) in ascending powers of x, up to and including the term in x4. Find the range of values of x for which the expansion is valid. [4] (ii) By letting 1 2x= , find an approximation for 35 , leaving your answer as a fraction in its lowest terms. [2]

6. Maclaurin Series 2 5 DHS/2015Promo/11 (i) Express ( )( ) 2 5f ( ) 1 2 1 xx xx = ++ as 2 .1 2 1 A Bx C xx ++++ [2] (ii) Hence obtain the series expansion of ( )f x up to and including the term in 3.x [3] (iii) Using your result in part (ii), find an approximate value for ( )( ) 1 2 0 d . 1 2 1 x x xx++   Give your answer in its lowest fraction. [2] (iv) Use your calculator to evaluate ( )( ) 1 2 0 d . 1 2 1 x x xx++   Justify whether your answer in part (iii) is an appropriate approximation. [2] 6 RI/2015Promo/8 (a) (i) Expand ( ) ( ) 2 21f 2 1 x x x =− − + as a series in ascending powers of x up to and including the term in 2x . [3] (ii) State the equation of the tangent to the curve ( )fyx= at the origin. [1] (b) Using the standard series given in the List of Formulae (MF 26) or otherwise, show that the first three non -zero terms in the Maclaurin series for 1e x+ can be expressed as ( ) 3e 1 ...px qx+ + + where p and q are constants to be determined. [6] 7 TPJC/2009Prelim/I/7 It is given that ( ) 1tan ln 1yx− =+ . (i) Prove that ( ) 2d11 d yxy x+ = + . [1] (ii) By successively differentiating this result three times, find the Maclaurin’s series for ( )( )tan ln 1 x+ , up to and including the term in 4x . [6] (iii) Show on a sketch the shape of the graph of ( )( )tan ln 1yx=+ for small x, indicating clearly the relationship of the graph to that of yx= . [2] 8 NYJC/2015Promo/9 (Part) (a) Expand ( ) 2 8 2 x− in ascending powers of x up to and including the term in x2, stating the range of value