70 DIFFERENTIAL AND INTEGRAL CALCULUS. 



3. If f(x] and all its differential coefficients be finite 

 and continuous, then will 



f(x+h) =/(*) + hf (x) +...+ -/"" 1 (*) + 



where is a positive proper fraction. 



If /(a?) be a; 4 and w = 3, = J; and if /(a?) be x 5 and 



W = 3, > J and < , . Prove that the limiting value 



of 6 when 7* is indefinitely diminished is - . 



n+ 1 



4. Deduce the series for the expansion of f(x] in a 

 series of ascending powers of x. Prove that 



5. Give a geometrical proof of the equation 



n 



' 



and if f(x] be a + bx + cx\ prove that = | . 



6-j-i j uu ~\~ w I JC- "T" X J f "| i *C ~T" \f I w *T" * 7 ( 



. Expand =-*- -ma series 



of ascending powers of x. 

 The answer is 



x* x* x 6 



is* L* L^ 



7. Expand sin^ sin' 1 a?), and cos(wi sin' 1 ^), in a series 

 of ascending powers of x. 



8. Expand cot" 1 (x + h) in a series of ascending powers 

 of h. 



The expansion is 



0-/Uin0.sin0 + sin' 2 0sin20- sin 3 sin304... , 



12 LI 



where = cot" 1 (j5), (of course being between \IT and |TT). 



