DHS Maclaurin Series (9758) (Revision Solutions)

Y5 Topical Revision Package 1 6. Maclaurin Series (solutions) 1 6. Maclaurin Series (solutions) 1 3 1 ) 1 (x− + −−+ − + = 23 2 3 1 3 1 ) (! 2 ) () ( 1 xx 211 391 xx= −− +  ------(I) 1) 3 )( 3 (−+ +bx ax 1 3 11 ) 1 ( ) 3 )( 3 (−− ++ = bxax       +− −+ − + + = 2 3 1 3 1 3 1 ) (! 2 ) 2 )( 1 () )( 1 ( 1 ) 1 ( bxbxax )1 ( ) 1 ( 2 2 9 1 3 1 3 1 + + − + =x b bx ax + − + − + =2 9 12 9 1 3 1 3 1 ) ( ) ( 1x ab b x b a + − + − + =22 9 1 3 1 ) ( ) ( 1x ab b x b a ------(II) Comparing (I) & (II), 1− = −b a ----(1) 12 − = −ab b -----(2) From (1), 1ab= − Subst a into (2), 1 ) 1 (2 − = − −b b b Thus b = –1, a = –2 2(i) (ii) 2 e 3 5 (1)xyy+ + = −−−−−−−−− Differentiating (1) with respect to x, dd2 e3 0dd d(2 3) e 0 (2)d x x yyy xx yy x ++ = + + = −−−−−−− Differentiating (2) with respect to x, ( ) 2 2 2 2 2 dd d2 (2 3) e 0dd d dd2 2 3 e 0 (Shown) (3)d d x x yy y yxx x yy yx x    + + +=      + + + = −−−−−  When x = 0, 2 3 40 ( 1)( 4) 0 yy yy + −= − += (from (1)) ⇒ y = 1 or − 4 (NA since y > 0)

Y5 Topical Revision Package 1 6. Maclaurin Series (solutions) 2 d1When 0, d5 yx x= =− (from (2)) 2 2 d 27 d 125 y x =− (from (3)) 21 271 ...5 250y xx= −− + 3(i) (1 ) ln(1 2 )xy x+=+ … (*) Differentiate (*) w.r.t. x : d2(1 ) d 12 yxy xx+ += + … (1) Differentiate (1) w.r.t. x : 2 22 d dd 2(1 ) 2d d d (1 2 ) yyyx x xx x  −+ ++=  + 2 22 dd 4(1 ) 2 0d d (1 2 ) yyx xx x+ ++ = + (shown)… (2) (ii) Differentiate (2) w.r.t. x : 32 32 3 d d 16(1 ) 3 0d d (1 2 ) yyx xx x+ +− = + … (3) When x = 0 From (*) : ln1 0y= = From (1) : d 2d y x = From (2) : 2 2 d 8d y x =− From (3) : 3 3 d 40d y x = Hence, y = 238 400 2 ...2! 3!xx x−++ + + = 23 202 4 ... 3xx x−+ + (iii) 1(1 ) ln(1 2 )yx x −= ++ = ( 21 ...xx−+ + ) 23(2 ) (2 )2 ...23 xxx −++ = 2x + ( 2 2)−− 2x + 8 223  ++ 3x + …

Y5 Topical Revision Package 1 6. Maclaurin Series (solutions) 3 = 23 202 4 ... 3xx x−+ + (verified to be same as part(ii)) 4(i) ( ) ( ) 2 1 2 2 1 2 2 222 24 1f 9 9 1 139 11 111 221 ...3 2 9 2! 9 1 1 ...3 18 216 x x x x xx xx − − = − = − = −    − −−       = +− − + − +     = ++ + Range of validity: 2 2199 33 x x x − <⇒ < ∴− < < (ii) When 1 2x= , 24 2 11 11 221 ...3 18 21619 2      = +++ −  ( ) 1 1 3505 3 345635 4 3 3456 2073635 23505 3505 ≈  = ×= 5(i) ( ) ( ) 22 5f( ) 12 112 1 x A Bx Cx xxxx += = +++++ Solving 2, 1, 2A BC= −== , ie, 2 22f( ) 12 1 xx xx += −+++ (ii) ( ) ( ) 1 23 23 2 21 212 2 1 2 4 8 ... 2 4 8 16 ... xx xx x xx x − − = −++ = −−+ − + = −+ − + +

Y5 Topical Revision Package 1 6. Maclaurin Series (solutions) 4 ( ) ( ) 12 2 2