For any irrational number a show that the fractional parts of an, which is denoted by {an), are uniformly distributed in the interval [0, 1].

Q: 12. For each n € N, define A, = [0, 1+] and B₁ = [0,1]. - =1 (a) Prove that U1A,, is an interval,…

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Q: Suppose that f(x) is convex and continuous on an interval I. Show that dif(x1) + ·.+ A„f(xn) + Anf…

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Q: Q2. | For what values of a, b and c does the function 6, x = 0 f(x) = {2x2 + a, bx + c, 0 <x < 2 2…

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Q: Let X~N(10,9).

A: The mean of the distribution is 10. The standard deviation of the distribution is 9.

Q: Using the Mean Value Theorem, prove that √1 + x² 0. Make sure to check all the necessary conditions…

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Q: Let X-Uni(60,90) find P(x <= 75) 1 none O 0.7 0.5

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Q: If X is uniform in the interval [-3,3] compute X(i.e. E{X}) and o² (i.e.E{(X-X)²}).

A: Given that X ~ U[-3,3] to find mean and variance

Q: 2.58 Given an example of f E C (a, b) for which the range of f is a) a finite open interval. b) an…

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Q: Show that the interval [a,b] is complete

A: Given

Q: How many finite functions exist of the form f: {1, 2, 3,...,m} {1, 2, 3,..., n}? How many of them…

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Q: A function f is continuous on [-3, 5] and contains the points (-3, -4), (0, ƒ(0)), and (5, –3). If ƒ…

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Q: Find a one-to-one correspondence between [0, 1] and (0, 1). Find a one-to-one correspondence between…

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Q: 1. Let a be a positive real number and n be a natural number. Define S = {x €R: x >0 and x" a?…

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Q: Show that there is a correspondence between the two intervals [0, 1) and [0,1].

A: Given interval 0≤x&lt;1 is a subset of  0≤x≤1. In this situation ,there is an obvious injective…

Q: Let a1, a2, ...,a, be real numbers that are all contained in the interval [-7/2, 7/2]. Use the…

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Q: Find numbers m and M Such that m<f'(1)< M figure in (1.4,1.3) l1,2 (1.6, 1,4) +

A: Given: The graph of f as  To find: The numbers m and M such that m&lt;f'1&lt;M

Q: Show that the nondegenerate interval [2, 5] is uncountable.

A: Here first we have to show that  is not countable.If possible let,  is countable ,so A can be…

Q: Let fand g be the functions defined by f(x)= x+et-&x + 1, and g(x) ==- 4x + 2. Let Q and R be the…

A: Hi! All the sub-parts are quite big. As nothing is specified, we are solving (a). kindly re-post the…

Q: 1. Let a and b be real numbers such that a < b. Apply the Mean Value Theorem to f(x) = In(cosh x) to…

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Q: if p is a prime, what is 10/p as a function of p?

A: We have to tell  What is 10/p as a function of p   It is given that p is prime

Q: Determine the values a, b, c, and d such that the function f satisfies the hypotheses of the Mean…

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Q: Use the Mean Value Theorem from §4.2 to prove that | sin(b) – sin(a)| < 16 – a| for any interval [a,…

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Q: Let f(x) = x³ - 10, x= [-15√3, 15√3]. Find the value of c guaranteed by the Mean Value Theorem on…

A: The given function fx=x3-10, x∈-153, 153. We have to find the value of c guaranteed by the Mean…

Q: Which of the following intervals will contain all of the function values of f (x) on the interval-4,…

A: We can solve it by cross check from given options  Let we select 0 from interval [-4,1] at x=0 f(0)…

Q: Which of the following sets are countable? ORIN {7,99, 512}. 1 1 1 .}. " 10 100 1000 The set of all…

A: We know R is uncountable and N is coutable.  So, R \ N is uncountable [as R = (R \ N) U N, if R \N…