106 PRACTICAL MATHEMATICS 



dA. 1 2 d 2 A. 



- 4 + 23/i - 7h 2 = 

 W - 3-286h + 0-571 - 



h - 1-643 = 1-459 

 h = 0-184 



The root lies between 1-18 and 1-19. 



To get a better approximation for the root 



A = x 3 - I0x z + 40# - 35 = - 0-08097 when x = 1-18 



= 3x 2 - 20# + 40 = 20-5772 when x = 1-18 

 dx 



I = Qx - 20 = - 12-92 when x = 1-18 

 oa; 2 



, dA 1 , . d 2 A 

 But A+ ^ + 2 /i2 ^ = 



- 0-08097 + 20-5772/i - 6'46/i 2 = 

 h* - 3-18536/i + 0-01253 = 



h - 1-59268 - 1-58874 



h = 0-00394 

 The root will be 1-1839. 



Example 2. Solve the equation - = -. 

 The root is evidently between 1 and 2. 



o 



The equation is sin x - x = 



and A = sin x - 0'75 x = 0-0915 when x = 1 

 ,j\ 



= cos x - 0-75 = - 0-2097 when x = 1 

 dx 



J2 A 



- = - sin x = - 0-8415 when x = 1 

 dx 2 



and 0-0915 - 0-2097A - O-4207/i 2 = 



h? + 0-4985A - 0-2175 = 



h + 0-2493 = 0-5287 

 h = 0-2794 



