MTH409 - Riemann Integration and Series of Functions Assignment

Assignment Task

1. Suppose ƒ is a bounded function on [a, b] with |f|≤ B for some B > 0. Let P and Q be two partitions of [a, b] with Q having at most m more points other the points of P then show by induction 0 ≤ U(f, P) – U(f, Q) ≤ 2mB||P||, where ||P|| denotes the norm of the partition P.

2. Show that if f is integrable on [a, b] then ƒ is integrable on every interval [c, d] C [a, b].

3. Let f be a continuous function on R and define rx+1 F(x) = [ f (t)dt for R. Show that F is differentiable on R and compute F'.

4. Suppose that for all xa, 0 ≤ f(x) ≤ g(x). Show that if then so does 9(t)dt converges, [f(t)dt and in that case, [f(t)dt ≤ [g(t)dt. Also prove that if f(t)dt diverges then so does [g(t)dt.

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