Show that the function f defined on [0, 1] by the conditions 1 f(x)=2rx when 7=1, 2, 3,...... És integrable orer [0, 1] and that

Q: Show that limit t  appraoaches infinity  integral -t  to t  x dx =0

A: To prove:

Q: Which of the following is an NOT Riemann integrable over the interval [-1,1]? example of a function…

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Q: f is not integrable on [0, 1].

A: It is given that f:[0, 1]→ℝ by f(x)=5x  if x∈ℚ0    if x∉ℚ We know, if lower integral and upper…

Q: Let L2[0, 1] be all bounded Riemann integrable functions [0, 1] → R with inner product (f, g) Split…

A: Consider the provided question, Given that, Hn,kx=2n,    k2n≤x<k+122n-2n,   k+122n≤x<k+12n0,…

Q: If fi and fa are two bounded and integrable functions on [a, b] and there exists a number >0 such…

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Q: f(x) is increasing and bounded on [0, 1], then f(x) is Riemann-integrable on [0, 1].

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Q: sho w that fQ) = x xE To,1] is Riemann integrable amm

A: The given function is: fx=x3, x∈0,1

Q: 3. If f : [a,b] → R is a bounded function such that f(x) = 0 except for x in {c1, C2, .., Cn} in [a,…

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Q: line integral is independent of path by finding a function f such that ∇f = F

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Q: Suppose that f is integrable on [a,b] and define the functions f*(x) = | F(x), f(x)20 [0, and…

A: we have f is integrable on a,b and define the functions f+x=fxfx≥00fx≤0 and    f-x=0fx≥0fxfx≤0 to…

Q: f(x)=0 for x EQ and f(x)=x otherwise. We can conclude f is not integrable on [-1,1] sim

A: In this question we will use basic concepts of calculus i.e concepts of integration and continuity.…

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: Show that I¿(f) = 1, and conclude that f is integrable on

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Q: Show that the function f defined on [0, 1] by the conditions f=2e when<a<를, -1,2, 3, p=1, 2, 3,...…

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Q: (0 if xE Q (1 if x E I If f (x) = show that f is not Reimann Integrable

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Q: 1. Consider the function f defined on [0, 0), f (x) = {x* sin- x + 0 x = 0 where r > 0. Determine…

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Q: Find the local linear approximation of f(x) = In(5 + x) at xo = -4. %3D In(5 + x)

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Q: Let f and g be continuous on the closed interval [0, 5]. Suppose that f(x)dæ = A, | f(x)dx = B ,&nd…

A: We have to find value of integral by using integral properties

Q: Let f be Riemann integrable on J and let f(x) > 0 for x € J. Show that

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Q: Compute the line integral of the scalar function f(x,y) = sqrt(1 + 9xy) over the curve y = x^3 for…

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Q: 19. Let f be the function defined by [1+x, when 0<x<}, <x<I 0, when x=. f(x)= || Show that fis…

A: We know that  A bounded function f which has only a finite number of points of discontinuity in…

Q: Suppose that f and g are integrable and that -1 f(x) dx = 5 h(x) dx = 4 7, dx = %3D Use the rules to…

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Q: Evaluate function given assuming ƒ is integrable on [a, b].

A: We have to calculate,∫abf(x)dx+∫baf(x)dx = ?            ...................(1)As function f(x) is…

Q: Prove Fermat’s Theorem for the case in which f has a localminimum at c .

A: It is given that f has a local minimum value at c. Thus, by the concept of local minimum, fc≤fx when…

Q: The Fourier integrals of two continuous functions f (x) and g(x) defined on R are given by 00 f(x) =…

A: The Fourier integrals of two continuous functions fx and gx defined on R, where…

Q: Are the following functions Riemann Integrable to their corresponding in- tervals? Explain your…

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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: Let f : [0, 1] → R defined by if æ = 0, f (x) - L otherwise is Riemann integrable on [0, 1].

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Q: Let f : [a, b] R be a function defined by 0if rEQ . if r¢Q f(x) = %3D Show that f is not Riemann…

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Q: Set up UP the integrel the Pegioni Th bounded 2x2 and y =l+ x F(x,y) dA in by: y= 2x? 2. 4.

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Q: Suppose that f and g are integrable and that f (x) dx : :-1 | f(x). h(x) dx = 4 Odx = 5 Use the…

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Q: Let f be an integrable function on [-1,3], with the property that for each E [-1,3], we have | F(t)…

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Q: Since f(x) =1/x is continuous on (0, 1), f(x) is integrable on (0,1).

A: This is a false statement.

Q: Show that f(x) = x² is integrable on [a,b].

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Q: Let ƒ be a Riemann integrable function on [0, 1]. n (i). Let g be in (0, 1). Find lim La n→∞…

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Q: Show that if f is Riemann integrable on [a,b] and f(x) ≥ 0 for all x ∈ [a,b], then

A: A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is…

Q: Let f(r. y) = * if (1, y) # (0,0) if (1, y) = (0,0) Show that af/ðr and af /ðy exists at (0,0) but f…

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Q: (1) Give an example of a function f such that | f |is Riemann integrable but f is not.

A: Consider the function f on the interval 0,1 defined by:  fx=1, if x is rational-1, if x is…

Q: Show that if f is Riemann integrable [a, b] and f(r) 0 for all r € [a, b], on f(x) dr 2 0.

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Q: Prove the following result: A function f that is decreasing on [a,b] is integrable on [a,b].

A: Prove the result that,        "A function f that is decreasing on[a,b] is integrable on[a,b]''

Q: Let f(x) = (cosh x)² be defined on (0, 0), find %3D d 1/2) . dy

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Q: Assume that: f:[0,4/3] → R with f(x) = -0. 5 x² 1- Compute U(P,f), L(P, f) 2- compute 1/3 1/3 ! |…

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Q: To prove this function to be Reiman Integrable 0/4 {: 0 if0<x < 1 4 if 1<x < 4 Let f : [0,4] → R be…

A: Given f:0,4→R defined by fx=0    if 0≤x≤14    if 1&lt;x≤4 We use here a theorem which states that…

Q: Suppose that f and h are integrable and that: i fx) = -1, f(x) dx = 5, h (x) dx = 4 ,Find: a. S–2…

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Q: Consider the following theorem. Theorem If f is integrable on [a, b], then [ºf(x) dx = dx = lim…

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Q: Use theorem 5.2 to prove directly that the function f(x) = x* is integrable [0, 1]. on

A: Using the given Theorem 5.2 we have to prove directly that the function fx=x3 is integrable on [0,…

Q: Suppose that f and h are integrable and that: frx) = -1 , , f(x) dx = 5 , ,h (x) dx = 4 ,Find: a. -2…

A: We have to find the given integrals.

Q: Let 2 C = sin dx. As f(x) = sin(1/x) is continuous except at 0 and bounded, f is integrable on…

A: C=∫-11sin1x2dx=-∫-11sin2uu2du is correct because the value of the integral ∫-11sin2uu2du is…