EXAMPLES VII 118 



(9) If y - e* tan x, prove that ^ - (1 + 2 tan <r) -J+2/ (tan x 

 - cot #) - 0. 



(10) liy = xd*, find - ^- T ^ and hence find -j^- 



ctr etc 2 cte 3 d# n 



(11) Using Maclaurin's Expansion, find the first five terms in 

 the expansions for sin nx and cos nx. 



(12) If y = tan (x + y), show that f| - - i^ and ^ 



2 i + 



y* 



(13) Find the first five terms in the expansion for tan -1 x. 



(14) If y = e* tan n x, find -j-, and use the result to find the first 



CLX 



five differential coefficients of & tan x. Hence find the first five 

 terms in the expansion for e* tan x. 



(15) If _KI. prove that y - (1 -*)|, S* - (1 - x) g, 



3 ft , (i _ , ft and hence show that *S = i* 

 6tr 2 7 (ic 3 dx n (i _ a;) n 



(16) If _ prove that $ - 1^. ft - - -p^- S. 



+ 1 oi 1 + or d^ l + ^da; 



&y 3 d 2 ?/ . d"y lx +i|w(l-7/) 



-=4 = -- -=-2 and hence show that -j-^- = ( - 1) l =l - . 

 dx 3 1 + x dx 2 dx n (1 + X ) n 



v + 1 du v 1 d 2 ?/ 2 dw ^V 



,17, If , __, prove that J = f-^, J = T _ J, J 



. _!_ ft and hence show that *S . Ijte" 1 ) 

 1 a; ctr* or" (i _ a;) n 



(18) If x = a(0 - sin 6) and y = a (1 - cos 0), find -j- and -^|, 



giving each result in terms of 6. 



(19) Using Taylor's Expansion, find the approximate root of 

 the equation tan x = x, knowing that the root lies between 4 

 and 4-5 radians. 



(20) Using Taylor's Expansion, find the approximate root of 



x 2 



the equation 3 sin - + - = 3-7, knowing that the loot lies between 

 2 x 



0-5 and 1-0 radians. 



(21) Using Taylor's Expansion, find the approximate root of 

 the equation x* + 5x* - 7x z + lOx - 12 = 0, knowing that the root 

 lies between 1 and 2. 



H 



