2. (Linearity of the integral) Let I= [a, b] x ... x [an, bn] be a generalized rectangle in R". Suppose that the function f: IR and g: I→ R are integrable, and a, 3 are real numbers. Prove that the function af + Bg: IR is integrable and [(as. (af ++ Bg) = a[ƒ + B [₁g. 9. Show your argument step by step.

Q: Let f : R → R be the function defined as if x 1. Use Theorem 32.5 to prove that f is Darboux…

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Q: Find the field of quotients of the integral domain J [i] = {a + ib | a, b € l}.

A: We have to find the field of quotients of the integral domain

Q: Exercise 9. Let f be any function such that the integral of f over C is 4. Then the integral of – f…

A: Given f be any function such that the integral of f over C is 4.

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A: (a) Define f:ℝ×ℝ↦ℝ×ℝ by fx,y=2x-y, x-2y. Prove that f is one-one. Find f-1 and f-16,3. Show that…

Q: Consider the function f : [0,3] –→ R defined by f (x) = 1 for x E [0, 1), ƒ(x) = 2 for x E [1,2),…

A: We are given a function f:[0,3]→ℝ defined as,   f(x)=1 if 0≤x<12 if 1≤x<23 if 2≤x≤3 We need to…

Q: 9. Let D be an integral domain such that (x + 1)² = x²+ 1 in D[x]. Char D = а) 0 b) 1 c) 2 d) 3 O a)…

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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: Prove that the function f : R? → R, f(x, y) = sin(/x² +|sin y|) is uniformly continuous on R?.

A: The time interval between two waves is known as a Period whereas a function that repeats its values…

Q: F[x₁,...,n] is an integral domain.

A: Solution:-

Q: ) Let (x, y) €Rx R. Consider the following functions from R2 to R? r(x, y)) = (-y,x) and s(x, y) =…

A: Since you have asked for multiple parts, solving a) and b) , kindly post the remaining as a separate…

Q: Let (f (u, v,w) and (u = u(x, y, z), v = v(x, y, z), w = w(x, y, z) be differentiable functions)…

A: Given function f(u, v, w) and u = u(x, y ,z),  v = v(x, y, z) , w = w(x,y,z) are differentiable…

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A: We have to check

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

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Q: If f is bounded and integrable on [a, b] sch that 1f(x) | 0).

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Q: Ja - 2, x < 0, a+ 2, x20 9. Let f: R R be defined by f (x) = then f-(]1, 3() (the inverse image of…

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Q: Find the relative extrema of f|S in Exercises 8 to 11. 11. f: R3 → R. (x, y, z) → x² + y² + 2, s =…

A: Given that: f:ℝ3→ℝ, x,y,z↦x2+y2+z2,  S=x,y,z|z≥2+x2+y2. We have to find the relative extrema of f|S.

Q: Theorem 10 Let (a, b] c R and f: [a, b] → R be a function of bounded variation on [a, b]. Then f is…

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Q: If f is bounded and integrable on [a, b] such that | f(x) | 0).

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Q: B). For what values of r, reR is the function [(x+y+z) ƒ(x,y,z) = { x² + y² +z² 0, continuous on R.…

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Q: 1. Suppose f: [a, b].→ R is C¹ and that f(a) = f(b) = 0. Show that 2 (S f(x)dx) a 6³ - a³ 3 rb […

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Q: Let a and b two real numbers such that a < b. A region R is bounded by a curve C and two different…

A: Given: A region R is bounded by a curve C and two parallel lines to x-axis and C does not cut y-axis…

Q: Let a and b two real numbers such that a < b. A region R is bounded by a curve C and two different…

A: Please see below picture for detailed solution.

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Q: 54. Use the function f in the figure below and the function g defined by g(x) = | f(t) dt. 0 |1|2|…

A: Here we can note that the function g(x) denote the net area under f(x) from 0 to x as its formula is…

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Q: Let f(x) be any function in C[a, b] with the values given in the following table. Evaluate the…

A: Note - the answer is not in the option.

Q: Exercise 2. Consider the function f: [0, 1] → R defined by O ifx e Q f(x) =- Vx ifx eR\Q Show in…

A: The given function is f:0,1→ℝ defined by: fx=0if x∈ℚxif x∈ℝ\ℚ (i). We know that a function f:0,1→ℝ…

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