EXAMPLES I 17 



() 



+ x* +1 x* + 3-940J - 5-62 



a; 8 - 5-72* + 4-77 (x +8)(x z -3-5x +4-94) 



(24) Using the Binomial Theorem, find the first five terms in 

 the expansions for (a) (1 + 2x) 7 , (b) (I - x)~ & , (c) (1 + 2x)^, 



(d) (1 + *)"* (e) j-^, (/) ^ l _ a *' 



(25) Give to five places the first and second approximations 



of (a) A/138, (b) \/627, (c) \V734, (d) ^500, (e) \V4000, (/) \V630. 



(26) Using the Binomial Theorem, find the first five terms in 

 the expansion for (1 #)-*. Use your result to calculate the 



value of '/?-= correct to four places of decimals. 

 v oy5 



(27) If I is the length of connecting rod, r the length of 

 crank, and x is the distance of the crosshead from the extreme 



r 2 

 dead point ; then approximately x = r(l cos 0) + 77- (1 cos 20) 



where is the angular position of the crank. Taking r = 1 and 

 1=5, calculate the approximate value of x when = 45. What 

 is the true value, and what is the percentage error ? 



Taking r = 1, / = 3, and = 45, calculate the approximate 

 and true values of x and find the percentage error. 



(28) If a pendulum beats seconds at 15 C. and the rod is 

 brass, find the number of seconds lost per day when the tem- 

 perature increases to 35 C., and find the number of seconds 

 gained per day when the temperature falls to C. Linear 

 coefficient of expansion of brass = 1'9 x 10~ 5 . 



iT 2 X 3 



(29) Using the series log e (l + x)=x + . . . , calculate 



& o 



log e l-2 correct to four places of decimals. 



x z x 3 



(30) Using the series t? = 1 + x + -T-JT- + -rr- + . . . , find the 



If. LSL 



values of Ve, e , and 7= correct to five significant figures. 

 Ve 



(31) Using the series 

 log.(n + l)- 



. . 

 calculate correct to five places of decimals Iog e 2, Iog e 3, Iog e 4, and 



