TAYLOR'S EXPANSION 105 



Putting x ~ 1, we get 



64. Application of Taylor's Expansion to the solution of equations. 



d\ i- 



But if y is taken as a small quantity 



dA 1 a d 

 then/(# + y) - A + t/ ^ + - t/ 2 ^5 approximately. 



Thus if we are solving an equation and we find by trial that 

 a root of the equation lies between two definite limits, say xx-^ 

 and x = x z . 



Then calling the equation f(x) = 

 and when x = x v f(x^ = a 



also when x - x z , f(x 2 ) =- b 



Then, if a is nearer to than b, the actual root of the equation 

 can be x^ + h and h will be small. 

 But f(x l + h) = 



,dA 1 . , 



and hence A + h -r- + - h z -r-? = 



dx 2 dx* 



dA d 2 A 

 where A =f(x] and A, -r , and -7-5 have the values when x^x t . 



This gives a quadratic equation for h. 



Next, if 6 is nearer to than a, the actual root of the equation 

 can be x 2 k, and k will be small. 



But f(x t -k) = 



. dA l, 9 



and hence A k -y- + -k z -3-5 = 



dx 2 dx z 



where A =/(#) and A, -^-, and -T-T have the values when x~x z . 

 dx dx z 



This gives a quadratic equation for k. 



Example 1. To find the root of the equation x 3 10# 2 + 40# 

 - 35 = 0, knowing that it lies between 1 and 2. 

 Then A or f(x) = x 3 - Wx 2 + 40* - 35 



when x = !,/(#) = 4, and when x = 2, /(#) = 13. 

 Hence the root is nearer to 1 than it is to 2 



and A = x 3 - I0x 2 + 40# - 35 = - 4 when x =1 



-j- = &r 2 - 20x + 40 = 23 when x = 1 

 da? 



d 2 A 



-j-5 = Go; - 20 = - 14 when x = 1 



ete 2 



