Applied Analysis Functions Limits Derivatives and Integers Project

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Mathematics

MAUY 4614

MAUY

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Applied Analysis (MA-UY 4614) – Fall 2020 Homework 7 — Due Date: 12/04/2020  x − 1 x ∈ Q ∩ [0, 1] n o , and Pn = 0, n1 , n2 , . . . , nn . (1) Let f : [0, 1] → R be defined by f (x) = 1 − x x ∈ / Q ∩ [0, 1] Compute U(Pn , f ) and L(Pn , f ). Z b f (x) dx = 0, show (2) Suppose f is continuous on [a, b] with f (x) ≥ 0 for all x ∈ [a, b]. If a that f (x) = 0 for all x ∈ [a, b]. (3) Let f : [0, 1] → R be defined by f (x) =  0 x∈Q Z . Compute x x ∈ /Q 1 Z 0 1 f (x) dx. f (x) dx and 0 Is f integrable on [0, 1]? (4) Suppose f (x) is increasing on [a, b]. Let Pn = {x0 , x1 , . . . , xn }, where xk = a + nk (b − a), k = 0, 1, . . . , n. Prove that Z b b−a f (x) dx ≤ (f (b) − f (a)) 0 ≤ U(Pn , f ) − n a Z b f (x) dx = lim U(Pn , f ). and a n→∞ n X k . n→∞ + k2 k=1 (6) Functions f and G are defined by (5) Find the limit lim n2 sin x f (x) = , 1 + x2 Find the derivative G0 (0). Z G(x) = x+π f (x + t) dt, 0 x ∈ R.
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(1) Let f : [0, 1] → R be defined by
(

x − 1, x ∈ Q
,
1 − x, x ∈
6 Q

f (x) =


, 1 . Compute U (Pn , f ) and L (Pn , f ).
and Pn = 0, n1 , · · · , n−1
n
. We have

n
n 
X
1X
1
k−1
n−1
1
1
U (Pn , f ) =
sup f (x) =
1−
=1−
= +
n k−1 ≤x≤ k
n k=1
n
2n
2 2n
k=1
n

n

and
n
X
1
L (Pn , f ) =
n
k=1

n

1X
inf f (x) =
k−1
k
n k=1
≤x≤ n
n






k−1
1
1
− 1 = − U (Pn , f ) = −
+
/
n
2 2n

Rb
(2) Suppose f is continuous on [a, b] with f (x) ≥ 0 for all x ∈ [a, b]. If a f (x) dx = 0,
show that f (x) = 0 for all x ∈ [a, b].
. Assu...


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