2.61 Let fn(x) = n 1. Prove: fn 0 uniformly on R.

Q: Let f: R R be defined as f(x) D 1 + x2 Let /= [- 1, 2), then f(1) = Select one: а. 1] b. (0, d. None…

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Q: Let R={(x,y) ERxR: |x| = |v|). Then R a function on R. оо True False

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Q: Exercise 2. Prove that dim(null T") = dim(null T) + dim W – dim V and dim(range T") = dim(range T),…

A: We prove dim null T*=dim null T + dim W-dim V and dim range T *=dim range T for every T L(V·W).

Q: 27. Let f(x)= [x²/3]. Find /(S) if a) S-2, 1, 0, 1, 2, 3). b) S= (0, 1, 2, 3, 4, 5). c) S = (1,5. 7.…

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Q: #: Let X₁, X₂, ..., X₁ ~ f(x) = {0x0-1 0<x< 1 find Suff. Comp. St. " e.w.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Q. 7. Let f (x) be derivable over [a, b] such that f' (a) = 10 and f'(b) = 20. Prove that the re…

A: Since the function is differentiable. So in the interval (a,b) f'(x) will attain all values from 10…

Q: 5. Let f be defined for all real numbers and suppose f(x)-f(y)| ≤ (x - y)² for all real numbers, x…

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Q: (a) Prove that Σ³ = n²(n+1) ². j=1 (b) Evaluate fr³ de directly from the definition, without using…

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Q: Prop (1.23 ≤ x ≤ 1.72) = Prop (-2.50 ≤ x ≤ 0.5) = Prop (x ≥ 3.00)

A: here use normal distribution table

Q: 6. Show that there does not exist a sequence of polynomials converging uniformly on R to f if (i)…

A: A sequence is said to be convergent sequence if it eventually reaches a certain real value as the…

Q: Find Pr (|w| < 0.2)  where W ~ N (-1,4)

A: From the given information, W follows normal distribution with mean µ=-1 and variance σ2=4.

Q: be a sequence of F-integrable functions on [a, t ow f is F-integrable and

A: Let  be a sequence of integrable functions whose choose  are arbitrary.Since  is uniformly…

Q: 4. Let f: [a, b] → R be integrable. Define F: [a, b] → R by F(x):= == f(t) dt Show that F is of…

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Q: ) Let f(x) = 2x - x². Prove that f is integrable on [0,2] using either the definition or the -…

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Q: 2 +3. Determine c such that f(c) is the average value of f(x) f(x) = Let on [1, 4].

A: Topic: application of integration

Q: 5. Show that the functions x,x? -,x³ -x x are mutually orthogonal in the interval [-1,1].

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Q: A function f : A->R is given to be differentiable. A is an open interval. Let a1 and a2 (a1 < a2)…

A: For part (i) Given that:

Q: 5. Determine whether f is a function from Z to R if a) f (n) = ±n. b) f (n) = v n2 + 1. c) f (n) =…

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Q: Σ 1 is continuous function in R. Prove f(x) = 4 n=1 n' +x*

A: The given function is                                     f(x)=∑n=1∞1n4+x4 To show that the above…

Q: 46

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