Real Analysis Answers

Find an approximation to f''(x) using f(x), f(x+h), and f(x+3h).

Show in detail that the integral from 0 to 1 of x^2dx= 1/3.

Prove: If f and g are differentiable and a is in the real numbers, then (fg)'(a) = f(a)g'(a) + f'(a)g(a) and (f/g)'(a) = (g(a)f'(a)-f(a)g'(a))/g(a)^2.

Find the maclaurin series for sinx and cosx and show that they converge for all x to their respective functions.

if f(x)=x^n, where n is an element of N, show that f'(a)=na^n-1 for any a.

if f(x)=x, show that f'(a)=1 for all a.

suppose f and g are differentiable at a. Then, prove f/g is differentiable at a, and
(f/g)'(a)= f'(a)g(a)-f(a)g'(a)/ (g(a))^2.

x belongs to r which satisfy the inequality |x|+|x+1|<2 find all x

Interpret and prove:
(i) o+o=o
(ii) o+O=O
(iii) O+O=O
(iv) o(O)=o[(iv) and (v) represent compositions of functions]
(v) o x O=o

Show that f=O(1) [this is the letter O not the number 0] as x->a if and only if f(x) is bounded on some neighborhood of a.