Let f be continuous on S(0,1) = {z : |2| = 1}. 1. Show that f(2) (-)dz = - -dz. S(0,1) s(0,1) Use the identity g(t)dt = S" g(t)dt which holds for any continuous function g : [a, b] → C.

Q: Is differentiable on (a,b) and F(x) = | f(t) dt , a <x< b, F is continuous on [a.b]. The function d…

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Q: For the function f(t) graphed at the right, let g(x)=| f(t)dt. Find :a) g(-4); b) g(3); c)g'(3) f(t)

A: Consider the graph of function f(t)

Q: Q.Let g: [1,4] → R be continuous function. Define G(x) = g(t)dt. Then dx dG a. G(x) is…

A: Here given g:1, 4→ℝ be a continuous function. Also, define Gx=∫ax2g(t)dt

Q: 2. Let f,g: R → R³ be differentiable functions. Show that d dt (f(t) × g(t)) = d d f(t) × g(t) +…

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Q: Let f(x) be a continuous function on R-{0} such that (f(x) dx exists for all a > 0.If g(x)= |- dt, t…

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Q: Let f(t) = 7t – 7t + 7e'. Then f'(t)

A: The function is given by ft = 7t4-7t+7et To evaluate: The derivative of the function.

Q: Let f(r) = re. Find f'(r).

A: f(x)=xex2

Q: Let u(1) = (x(1), y(y), z(1)) be a curve in 3-space, i.e. a function u: R→ R³, and consider its…

A: Let ut=xt,yt,zt be a curve in 3d space and its derivative is dudt=dxdt,dydt,dzdt. Given that…

Q: Let f EBV (Co,1]). Define F(x) = St Show that F E BV([0,1). [*f(t) dt, x € (0,1], x = 0.

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Q: Let y= f(u) =u and u= compute the derivative h'(1.494) at the point z 1.494. g(z) -1-. Define the…

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Q: Let f(t) be a continuous function on (-∞,). Show that the particular solution to the equation U…

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Q: A. T(f(t)) = |B. T(ƒ(t)) = f(−t) from Pg to Pg = tª ƒ' (t) from P₂ to P5 = f(t)ƒ' (t) from P7 to P13…

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Q: 6. Show that (an)neN C R is bounded if and only if lim supn→∞ an < +∞o.

A: To Prove:  ann is bounded sequence if and only if limsupan&lt;+∞.

Q: Prove that if f is continuous, then f(t)(x-t) dt = f(u) du ) dt

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Q: Let f(t) be a continuous function with f(0) 5 and f(4) = 3. Compute F'(2), where cx² F(x) = [" .…

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Q: Exercise 3: Let f be a continuous function. d (a) Compute | tf(t) dt. dx (b) Compute | xf(t) dt.…

A: a)To solve this differential use Leibnitz formula as.

Q: a) Let F(x) = | x· f(t)dt. Compute F'(x). 0, b) Assume that f is continuous. Prove that f(t)dt ) du.

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Q: A continuous function f satisfies the equation 1 f (x) = x – t+ 2 f(t) dt 0, For all x e [-1, 3].…

A: We have to use leibiniz rule of  differentiating definite integral  d∫f(x)g(x)h(t)…

Q: R. Suppose f € C" with f(1,1)=1; 1, f(1, 2)). Compute (1) and (1). fi(1,1)= a; f₂(1, 1) = b.

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Q: Define f:c[-1, 1] → C as ƒ(x) = L₁x(1)dt – fx(1) di Find f.

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Q: Let f(x)= I Determine f'(x). sin² (³+1) dt, x= [1,1].

A: Given: f(x)=∫-xx2sin2(t3+1)dt

Q: r(t) = ti +j+ tk is continuous at t = 0. اختر احدى الدجابات True False

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Q: Q4(a)- If fE R(a)on [a, b, a < x < b and F(x) = S* f(t) dt] then F(x)is continuous on [a, b].

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Q: Let f(x) = S V4 – t? dt. - (a) Compute f'(x) (b) What is the domain of f(x)?

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Q: 1. Let f (-1,1) R, f(x) = (1+x³3)-3. Compute g': (0,1) → R where : (a) g(x) = få ƒ. 0 (b) g(x) = [ x…

A: To compute g′(x) using the formula for Differentiation Under the Integral Sign, we'll need to apply…

Q: [2] (A) f'(a) = √ 1 – f ± (a) for all a ∈ (0, (B) f'(a) = = true (0) 2 -- for all a E (C) f(a) f'(a)…

A: It satisfies option A

Q: Let f EBV (Co,1]). Define F(x) = { ["f(t) dt, x € (0,1], St Lo, x = 0. Show that F E BV([0.1).

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Q: Let A(z) – S. f(t) dt, with f(x)

A: