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 values of x for which the expansion is valid. [4]

6. Maclaurin Series 3 (b) Given that cot 2 4yx =+  , show that ( ) 2d 21d y yx =− + . [2] By further differentiation, find the Maclaurin series for y, up to and including the term in x3. [5] Hence show that 2tan 2 4x a bx cx +  + + where a, b and c are constants to be determined. [4] 9 NJC/2015Promo/3 Given that x and y are related by ( ) 224 3,d d yxx x −−= where 22 x−   , and that 2y= when 0,x= (i) find y in terms of x, [3] (ii) by differentiating further, or otherwise, find the first three non -zero terms of the Maclaurin’s series for y. [3] 10 SAJC/2009Prelim/I/3 Given that 13tane xy − = , show that ( ) 2 d13 d yxy x+= . By repeated differentiation of this result, find the Maclaurin’s expansion for y, up to and including the term in 3x . [5] Hence, deduce the Maclaurin’s expansion for 12 3tane xx −+ , up to and including the term in 2x . [2] 11 RVHS/2015Prelim/I/8 Given that  )1ln(sin 1 += − xy , show that 1 1 d dcos += xx yy and 2 2 2 2 )1( 1 d dsind dcos +−=    − xx yyx yy . [2] (i) By further differentiation, find the Maclaurin series for y, up to and including the term in 3x . [3] (ii) Find the set of values of x for which the value of y is within 0.1 of the value found by its Maclaurin series. [3] (iii) Deduce the series expansion for 2))1(ln(1)1( 1 +−+ xx up to and including the term in 2x . [2]

6. Maclaurin Series 4 12 DHS/2009Prelim/II/3(a) It is given that 1tan 2 de d 1 xy x x − = + , where 1tan x− denotes the principal value. (i) Find an expression for y in terms of x given that y = 1 when x = 0. [2] (ii) Show that ( ) 2 2 2 dd1 (2 1) 0 dd yyxx xx + + − = . [2] (iii) By further differentiation of the result in (ii), find the Maclaurin series for y up to and including the term in x3. [4] (iv) State the series expansion for 1tan 2 e 1 x x − + up to and including the term in x2. [1] 13 RVHS/2015Promo/4 Given that 2e sin 1xyx=+ , show that d e sin 2d xy x y ax = + + and 2 2 dd 2 2e cos 2dd xyy y x bxx− + = + , where a and b are constants to be determined. [3] (i) Hence find the Maclaurin series for y, up to and including the term in 2x . [2] (ii) Write down the first three terms in the series expansion for 1(1 ) x −− . Deduce the Maclaurin series of 22e sin 2 1 1 x x x − + − up to and including the term in 2x . [3] 14 JJC/2015Prelim/I/4 Two ground spotlights P and Q are shining at the top of a tower 'TT of height h m. P is due west and Q is due south of the tower. The angle 'TPT is 6  radians and the angle 'TQT is 4 x + radians, where x is small. Show that ( )( ) 1 1 1 .QT h x x − = − + [3] Hence, by using the standard results in MF26, show that ( ) 2 2 2 14 2.xxPQ h −+ [3]

6. Maclaurin Series 5 15 ACJC/2015Promo/3 The diagram above shows triangle ABC which has fixed points A and C and a variable point B such that AB kBC= , where k is a constant such that 01 k . The angles BAC and BCA, measured in radians, are x and y respectively. (i) Show that sin siny k x= . [1] (ii) By successively differentiating the equation in part (i), show that 22 2 ddcos sin sindd yyy y k xxx − =−  . [1] Hence find the Maclaurin series for y , up to and including the term 3x . [3] 16 IJC/2015Promo/6 In the triangle ,ABC 1AC = , angle 2 3BAC = radians and angle ACB = radians (see diagram). (i) Show that 3 3 cos sin BC  = − . [3] (ii) Given that  is a sufficiently small angle, show that 21 3 BC p   + + , where p is a constant to be determined. [4] 17 IJC/2015Promo/10 (i) Given that 1sin 3e xy − = , show that ( ) 2 2 2 dd1 9 9 9 dd yyx x y xx− − = . [3] (ii) By further differentiation of the result in part (i), find the Maclaurin series for y in ascending powers of x, up to and including the term in 3x . [4] (iii) Hence find an approximate value of 2e − , giving your answer as a fraction in its simplest form. [2] A B C B C

6. Maclaurin Series 6 18 MI/I/2015Promo/10 (a) Given that ( )f sin 2 4xx =+  , find the exact values of ( ) ( ) ( )f 0 , f 0 and f 0  . Hence, write down the first three non -zero terms in the Maclaurin series for