Q2: If f(x) is Riemann-Stieltjes integrable function with respect to g(x) on [a,b]and if (f(x)dg(x)=0 for every function f(x) which is monotonic on [a,b] . Prove that g(x) must be constant on [a,b].

Q: 33. Let x² - 4y2 if x + 2y f(x, y) = 2y %3D | 8(x), if x = 2y %3D If fis continuous in the whole…

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Q: (3) that кете Use Riemann's Criterion to prove the constant function f. R > R = k given by f(x) 4…

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Q: Define f : Rm \ {0} → R by ƒ(x) = T: Prove that f is uniformly continuous on R" \ B1(0), but not…

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Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

Q: 1. Consider the function: f on [0, 1] by 7 if r1 f(x) |0 if 1 0, 1]. Prove that f is Riemann…

A: To prove that f is not continuous but Riemann integrable

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: If ƒ (t) is an integrable function, then / f (x) dxrepresents a family of functions. O True O False

A: Option B will be the right answer.

Q: Let H(x) integral ,for -2 ≤ x ≤2.Evaluate H'(2).

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Q: Consider the following open Newton-Cotes Rule. | F(x) dx = 4h [2f(x1) – f(x2) + 2f(x3)] 3 x4 – x0…

A: a) The integral to evaluate  ∫0πxcosxdx Here  h=x4-x04   =π-04    =π4 x1=π4,x2=π2,x3=3π4

Q: 4. Let f(x) = x, Vx E [0, b]. Show that f is integrable and that 62 f(x) dx : Answer:

A: Given function is  fx=x, ∀ x∈0,b

Q: 1. Are the following functions Riemann Integrable to their corresponding in- tervals? Explain your…

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Q: (a) Show that if f is Riemann integrable over [0, 1], then | f(x) dæ = lim / f(x) dx c→0

A: According to Bartleby guidelines; for more than one question asked only the first one is to be…

Q: . Apply Jensen's inequality (2) to f(x) = In x, with i 1 = ..= d = 1/n to prove that 1 Vx1x2...X 0,…

A: Given :-  Apply Jensen's inequality (2) to f  x  = ln x  , with λ1 = λ2 = ..........  = λn = 1n to…

Q: 3 Suppose that e(x) and f(x) are differentiable on [0, 1] and that 0 < |e'(x)| < |f'(x)| for all x €…

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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: (m) Suppoe that a >0 and that f is Riemann integrable on [-a, a). If f is even slow that (b) Let /…

A: According to the given information, For part (a) Suppose that a&gt;0 and that f is Riemann integral…

Q: Let H(x) integral ,for -2 ≤ x ≤2.Evaluate H'(1).

A: We have given Hx=∫0x4-t2dt We have to calculate H'(1)

Q: Suppose that |- f(x) is not constant. Il- f(x) is differentiable on 0, o0 II- S f(t)dt = (x +1)f(x)…

A: In the given question we have to choose a function which satisfies the given three condition in the…

Q: Let x > 1, f be continuous and g be a Lipschitz function with constant L satisfying g(1) = 0 defined…

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Q: It is frequently required to solve equations of the form f(x) = 0. When f is a continuous function…

A: To solve an equation fx=0, if f is continuous on an interval a,b such that fa and fb are of opposite…

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

A: Given: ∫05f(x)dx=A∫03f(x)dx=B∫53g(x)dx=C…

Q: Suppose f is Reimann integrable on [a,b]. If f (x)< 0 for all x E [a, b], then S f(x)dx

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Q: Test the function def ined by f (x,y) = - x- 4y for its homogeneity. If it is homogeneous, give its…

A: Find your answer below

Q: Q2: If f(x) is Riemann-Stieltjes integrable function with respect to g(x) on [a,b] and if…

A: It is given that,  f(x) is Riemann-Stieltjes integrable function with respect to g(x) on a, b such…

Q: Suppose f> 0 is integrable on [a, b], continuous at xo € [a, b], and f(xo) > 0. Prove that f>0.

A: Given f≥0 is integral on a,b continuous at x0∈a,b and fx0&gt;0

Q: Suppose f is continuous on the closed interval [a.b]. If Fis an antiderivative of f for all real…

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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: 1. Suppose f(r) is a continuous function on r E [a, b]. Prove f(x)dx < Slf(x)| dx.

A: According to our guidelines we can solve only first question and rest can be reposted.Thank you

Q: Let H(x) integral ,for -2 ≤ x ≤2.Evaluate H(0).

A: Consider the given integral, Hx=∫0x4-t2dt use the substitution method to solve the above integral as…

Q: 1. Determine whether k(x) is continuous at r = 3, where %3D x + 3, x 2.

A: Given  k(x)=x+3,  x&lt;0x2+1, 0≤x≤22x,   x&gt;2 To check the continuity we can use the graph as…

Q: 4. Determine whether or not the following functions are uniformly continuous. (a) f(x) = Vx on (0,…

A: Given the function (a) fx=x on 0,∞ By the mean value theorem Let f=xα and x&lt;y  Since,…

Q: Determine the value of m such that the function 5 + 5 y fi^ - ^ - f(z, y) = is continuous at (0,0).

A: Given that function s continuous at (0,0). We need to find the value of m.

Q: (5) Prove that if f is continuously differentiable on [a, b], then |f(a) – f(b)| a – b max f'(x)|…

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Q: 9. Consider the function f(x, y) = p |xy|. Show that Taylor’s expansion about the point (x, x) is…

A: The Taylor series expansion of a function f(x ,y) about point (a ,b) is given by,…

Q: (b) Let f: [0, 2] R be the constant function f(x) = -2 for all r E [0, 2] and let g : [0, 2] → R be…

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Q: 9. Find the bounds of integration a, b, c, d such that T (a, u) dydar. f(x, y) dædy =

A: x≤y≤4x=y →y=xy=40≤x≤4x=0 to x=4

Q: (3x² -1, x 1, Determine c and d so that f(x) is continuous everywhere.

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Q: 1) If f(x)=x2, find f(x+2)-f(2) .what is bijective h mapping? 2) Does the integral J, dx exist?

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Q: 3. (a) Use the Fundamental Theorem of Calculus to determine a continuous function f: 2. x) -R that…

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Q: Suppose that f and h are integrable and that: L f(x) = -1, f(x) dx = 5, h (x) dx = 4 , Find: %3D %3D…

A: Answer and explanation is given below..

Q: Prove this Let E be an open subset of R" , and let f : E → R™ be a twice continuously differentiable…

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Q: If f.g [a,b] → R are integrable and there exists CE[a,b] so that f(x) = g(x) for all X C, then it…

A: True

Q: Assume that J is continuous on [0, 3 and differentiable on (0, 3) with f (z) > 6 for each z in (0,…

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Q: Let f be the function Jo if x +1 and f(x) = 42 if x = 1. Prove that f is integrable on [0, 2] and…

A: Given :f is  a function such that               fx=0      if x≠1 and42    if x=1Now, on the interval…

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

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Q: If f(x) and g(x) are integrable on the closed interval [a, b], and k is a constant, which of the…

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Q: Assume thatf is differentiable in (-oo, x) and af'(r)N 0 for all a E (-00, 00). Prove that f has the…

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Q: 3. Let f(r) = r sin() if z #0, and f(0) = 0. Show that f is continuous at r= 0 by using e- d…

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Q: Consider the following theorem. b - a If f is integrable on [a, b], then f(x) dx = lim n- 00…

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