Download Math 3210-1 HW 16 Properties of the Riemann Integral

Math 3210-1
HW 16
Due Wednesday, July 25
Properties of the Riemann Integral
1. A function f on [a, b] is called a step function if there exists a partition P = {a = u0 < u1 < · · · <
um = b} of [a, b] such that f is constant on each interval (uj−1 , uj ), say f (x) = cj for x ∈ (uj−1 , uj ).
Rb
(a) Show that a step function f is integrable and evaluate a f .
R4
15
if 0 ≤ x < 1
(b) Given P (x) =
, evaluate 0 P (x) dx. Note: P (x) is called the
15 + 13n if n ≤ x < n + 1
postage-stamp function. Do you see why?
2. Find an example of a function f : [0, 1] → R such that f is not integrable on [0, 1] by |f | is integrable
on [0, 1].
Rb
Rb
3. Suppose that f and g are continuous function on [a, b] such that a f = a g. Prove that there exists
x ∈ [a, b] such that f (x) = g(x).
The Fundamental Theorem of Calculus
4. Calculate
Z x
2
1
(a) lim
et dt
x→0 x
0
Z 3+h
2
1
et dt
(b) lim
h→0 h
3
5. Let f be defined as follows: f (t) = 0 for t < 0; f (t) = t for 0 ≤ t ≤ 1; f (t) = 4 for t > 1.
Rx
(a) Determine the function F (x) = 0 f (t) dt.
(b) Sketch F . Where is F continuous?
(c) Where is F differentiable? Calculate F ′ at the points of differentiability.
6. Let f : [0, 1] → R be a continuous function with continuous second derivative f ′′ , and f (0) = f ′ (1) = 0.
Z 1
f (x)f ′′ (x) dx = 0, then f ≡ 0.
Prove that if
0