10. Let f(0) = 0 and f(x) = 1/x for all x € (0, 1]. Show that f is not integrable on [0, 1] by showing that the first term in the Riemann sum, f(x) Ax, can be made arbitrarily large.

Q: Prove that if ƒ and g are Riemann integrable on [a, b] then fg is also Riemann integrable on [a, b].…

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Q: Let f'be continuous on [a, b], a>0, and differentiable on (a, b). Prove that there exists c = (a, b)…

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Q: 3. Let f a nonnegative function defined on R. Suppose that for all n 21 you have to (r)dr <1 Prove…

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Q: Given that the function f is integrable, which of the following is equivalent to: x" f'(x) dr 12 a)…

A: we have to integrate by parts

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Q: sume that f(t) is continuous on [a, b| and that u = g(t) is cont ferentiable on [a, b], then f(u)…

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

Q: Define if x 0 Show that f(x) is continuous on I = [-1, 2] (which means that Extreme Value Theorem…

A: Given, f(x)=2x if x≤01-2x if x&gt;0. First, we show that f(x) is continuous on [-1,2].…

Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

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Q: 14k 14 as a definite integral. 8 Express the limit lim Σ n→00k =1 9 + + 121

A: Known fact: Let f be a continuous function on a,b. Then ∫abfx dx=limn→∞ ∑i=1n fxi ∆x where ∆x=b-an…

Q: Suppose that f and g are integrable. Show that f * g is also integrable.

A: 1 Given that f and g are integrable. We have to show that fg is also integrable. As we know…

Q: a) Show that the function defined as follows 1 <x<. when 2 1 f(x)= 2" where n = 0,1, 2,3,.. on 2"…

A: Given , fx=12n  , when 12n+1&lt;x&lt;12n   where n=0,1,2,3...  onf0=0  is interable in 0,1 has…

Q: Question 3 If f is integrable on the interval [3, 9], then Lf (x)dx = [,'f (x)dx a) True b) False

A: Given query is to find the statement is true or false.

Q: 9 Prove that if m(x) is differentiable on (-∞, 0) and its derivative m'(x) is bounded then m is…

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Q: . The average value of f(x)=√(4-x^2) on [-2, 2] is 1/2 True False The function f(x) * f(x) = x² + 1…

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Q: Every function which is Riemann 0/3 integrable on [a,b] is continuous. * True O False

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Q: -2 x< -1 Q1)a) For what values of a and b is g(x) = ах — b -1<x<1, x21 Continuous at x= -1

A: The suitable formula for continious       LEFT HAND LIMIT = RIGHT HAND LIMIT

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: 7. Show that if f is differentiable on (a, b), and f'(c) > 0, where ce (a, b) and if f' is…

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Q: 6. Let f(x) = x2/3. (a) Then fis uniformly continuous on [-1,1]. Why? (b) Show that there does not…

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Q: 5. Determine the interval on which f(x) = x+2 is integrable. %3D (-00, 2) [-2, co0) (-00,-2) U (-2,…

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Q: Suppose f(z) is defined on all of C, is differentiable at z = a, and satisfies f(a) = a. Show that…

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Q: 6. Let f, g, h : [a, b] → R satisfy f(x) < g(x) < h(x) for all x E [a, b]. Suppose that f and h are…

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Q: Isg continuous on (0,1]? * if x 4.

A: Solution:Given  g(x)=xif x≤03if 0&lt;x≤13-x2if 1&lt;x≤4x-3if x&gt;4To find:g is continuous at (0,1]

Q: Is the function f(x) = ( 1, x not equal 1)             ( 0, x equal 1) integrable over the interval…

A: Here, we will show that the given function is integrable. Let us take a point c ∈ (0,1).

Q: ¹).. Suppose that fn are integrable and ffn du <∞. Show that, if fn 1 f, then f is integrable and f…

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Q: ppose that f is a continuous function and the followi ƒ(2) = 2, ƒ(4) = −1, ƒ(6) = 6, ƒ(8) = = culate…

A: We have to choose the correct option from the given options according to the given information

Q: Find the maximum modulus of the function f(z) = -iz + i, |z| < 5.

A: Find the maximum modulus of the functionfz=-iz+iz≤5

Q: 1. Which of the following functions are in the Hilbert space with indicated interval? (a) f(z) = e*,…

A: Hilbert spaces allow generalizing the methods of linear algebra and calculus from Euclidean vector…

Q: Only one option can be true

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Q: 10. If f(x) is continuous on [a,b], then there exists a c such that: f'(c) = ( f(b) - f(a) ) / (b -…

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Q: Suppose that f (0) = 2 and f' (x) ≤ 3 for x &gt; 0. Apply the MVT to the interval [0, 4] to prove…

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Q: Suppose f(x) is continuous on [0, 1], with f(0) = 2 and f(1) = 0. Then there must be a value x in…

A: Here given that f(x) is continues.

Q: f(x + a) dx = f(1) dæ = | f(x) d.r. -T+a

A: Proving of integral of periodic function