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Please do exercise 129:
Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N → N be the unique function that satisfies f(0) = 2 and f(next(n)) =r(f(n)) for all n E N. 102 1. Prove that f(3) = 8. 2. Prove that 2 <f(n) for all n E N. Exercise 129: Define r and f as in Exercise 128. Assume that x + y. Define r' = {(x,y),(y,x)}. Let g:N + {x,y} be the unique function that...
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Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
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Let h : X −→ Y be defined by
h(x) :=
f(x) if x ∈ F
g
−1
(x) if x ∈ X − F
Now we must prove that h is injective and bijective. Starting
with injectivity, let x1, x2 ∈
X such that h(x1) = h(x2). Assume x1 ∈ F and x2 ∈ X −F. Then h(x1)
= f(x1) ∈ f(F)
and h(x2) = g
−1
(x2) ∈ g
−1
(X − F) = Y...
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Please solve the exercise 3.20 .
Thank you for your help !
⠀
Review. Let M be a o-algebra on a set X and u be a measure on M. Furthermore, let PL(X, M) be the set of all nonnegative M-measurable functions. For f E PL(X, M), the lower unsigned Lebesgue integral is defined by f du sup dμ. O<<f geSL+(X,M) Here, SL+(X, M) stands the set of all step functions with nonnegative co- efficients. Especially, if f e Sl+(X,...
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31 (a) If fis integrable, prove that fa is integrable. Hint: Given e>0, let h and k be step functions such that h f k and j (k-h) < ε/M, where M is the maximum value of Ik(x) +h(x)]. Then prove that h and k2 are step functions with h' srsk (we may assume that OShSSk since f is integrable if and only if I is-why?), and that I (k2 - h2) <e. Then apply Theorem 3.3. (b) If fand...
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Given the integral below, do the following. 2 cos(x2) dx Exercise (a) Find the approximations T4 and M4 for the given interval. Step 1 The Midpoint Rule says that b f(x) dx = Mn Ax[f(+1) + f(22) + ... + f(n)] with ax = . b - a + n a 1 We need to estimate 6 2 cos(x2) dx with n = 4 subintervals. For this, 1 - 0 Ax = 4 = 1/4 1/4 Step 2 Let žų...
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Consider the vector field: f (x, y)= «M(x, y), N(x, y)= v promet Let C be any simple, positively oriented, closed curve that encloses the origin. Show that: F. do 21. We will solve this problem by completing the following steps: STEP 1 Let C be a positively oriented circle of radius r with the center at the origin. Letr be so small that the circle Člies within the region enclosed by the curve C(see figure below) Compute the integral...
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prove e
EOLU Exercise 4.1.1. Prove Theorem 4.1.6. (Hints: for (a) and (b), use the root test (Theorem 7.5.1). For (c), use the Weierstrass M-test (Theorem 3.5.7). For (d). use Theorem 3.7.1. For (e), use Corollary 3.6.2. The signale UI tre rauUS UI CUNvergence is the IUIUWII. Theorem 4.1.6. Let - Cn(x-a)" be a formal power series, and let R be its radius of convergence. (e) (Integration of power series) For any closed interval [y, z] con- tained in (a...
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Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
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Exercise 25: Let f: [0,1R be defined by x=0 fx)/n, m/n, with m, n E N and n is the minimal n such that z m/n x- m/n, with m,n E N and n is the minimal n such that x a) Show that L(f, P) = 0 for all partitions P of [0, 1]. b) Let m E N. Show that the cardinality of the set A :-{х є [0, 1] : f(x) > 1/m} is bounded by m(m...
![that h(mn ) h ( m)n, h ( ) and that if m < n then h ( m ) < n ( n ) = . Exercise 2.7.4. [Used in Theorem 2.7.1.] Complete the](http://img.homeworklib.com/images/96b86e71-bf15-4e1b-be36-9a7a4ce54709.png?x-oss-process=image/resize,w_560)
Please do exercise 129:
Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N → N be the unique function that satisfies f(0) = 2 and f(next(n)) =r(f(n)) for all n E N. 102 1. Prove that f(3) = 8. 2. Prove that 2 <f(n) for all n E N. Exercise 129: Define r and f as in Exercise 128. Assume that x + y. Define r' = {(x,y),(y,x)}. Let g:N + {x,y} be the unique function that...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Let h : X −→ Y be defined by
h(x) :=
f(x) if x ∈ F
g
−1
(x) if x ∈ X − F
Now we must prove that h is injective and bijective. Starting
with injectivity, let x1, x2 ∈
X such that h(x1) = h(x2). Assume x1 ∈ F and x2 ∈ X −F. Then h(x1)
= f(x1) ∈ f(F)
and h(x2) = g
−1
(x2) ∈ g
−1
(X − F) = Y...
Please solve the exercise 3.20 .
Thank you for your help !
⠀
Review. Let M be a o-algebra on a set X and u be a measure on M. Furthermore, let PL(X, M) be the set of all nonnegative M-measurable functions. For f E PL(X, M), the lower unsigned Lebesgue integral is defined by f du sup dμ. O<<f geSL+(X,M) Here, SL+(X, M) stands the set of all step functions with nonnegative co- efficients. Especially, if f e Sl+(X,...
31 (a) If fis integrable, prove that fa is integrable. Hint: Given e>0, let h and k be step functions such that h f k and j (k-h) < ε/M, where M is the maximum value of Ik(x) +h(x)]. Then prove that h and k2 are step functions with h' srsk (we may assume that OShSSk since f is integrable if and only if I is-why?), and that I (k2 - h2) <e. Then apply Theorem 3.3. (b) If fand...
Given the integral below, do the following. 2 cos(x2) dx Exercise (a) Find the approximations T4 and M4 for the given interval. Step 1 The Midpoint Rule says that b f(x) dx = Mn Ax[f(+1) + f(22) + ... + f(n)] with ax = . b - a + n a 1 We need to estimate 6 2 cos(x2) dx with n = 4 subintervals. For this, 1 - 0 Ax = 4 = 1/4 1/4 Step 2 Let žų...
Consider the vector field: f (x, y)= «M(x, y), N(x, y)= v promet Let C be any simple, positively oriented, closed curve that encloses the origin. Show that: F. do 21. We will solve this problem by completing the following steps: STEP 1 Let C be a positively oriented circle of radius r with the center at the origin. Letr be so small that the circle Člies within the region enclosed by the curve C(see figure below) Compute the integral...
prove e
EOLU Exercise 4.1.1. Prove Theorem 4.1.6. (Hints: for (a) and (b), use the root test (Theorem 7.5.1). For (c), use the Weierstrass M-test (Theorem 3.5.7). For (d). use Theorem 3.7.1. For (e), use Corollary 3.6.2. The signale UI tre rauUS UI CUNvergence is the IUIUWII. Theorem 4.1.6. Let - Cn(x-a)" be a formal power series, and let R be its radius of convergence. (e) (Integration of power series) For any closed interval [y, z] con- tained in (a...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
Exercise 25: Let f: [0,1R be defined by x=0 fx)/n, m/n, with m, n E N and n is the minimal n such that z m/n x- m/n, with m,n E N and n is the minimal n such that x a) Show that L(f, P) = 0 for all partitions P of [0, 1]. b) Let m E N. Show that the cardinality of the set A :-{х є [0, 1] : f(x) > 1/m} is bounded by m(m...