2. Let f be a function such that f' is continuous on [a, b). Show that f(t)f'(t) dt = ; (f(b)² – f(a)*) . a

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A: We have to find the answer using the definition of uniform continuity.

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A: R- integrable function

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A: Please refer the attached image for complete solution.

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A: Given,A(x)=∫0x f(t) dt

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Q: If f(x,y,z) = z In(x²ycosz): then fz is

A: Here, the given function is: f(x,y,z)=z ln(x2ycos z) Using the properties of the natural logarithm:…

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Q: 2. Let F(x) = re* %3D VE dt, find F'(x). t2+sint

A: Solution is given below:

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A: To prove: ∫ab∫xbftdtdx=∫abx-afxdx

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A: Linearization

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Q: 4. Let f € L'(R). Prove that the function F(x) = S f(t)dt is continuous.

A: Let, f∈L1ℝ. Since,  L1=f:ℝ→ℂ∫ℝftdt&lt;∞∫ℝftdt≤∫ℝftdt&lt;∞Therefore,∫ℝftdt&lt;∞ ∫ℝftdt is a finite…

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A: Given : F(x)=∫-xxcos(t2+1)dt Aim: to calculate using the fundamental theorem of calculus ,the…

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A: We have to give an example of a nonzero continuous function f  on [0,1] such that  ∫01f(x)dx = 0

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Q: assume x and y are functions of t.  Evaluate dy/dt for the following: y3=2x4+6; dx/dt=2, x=1, y=2

A: The given function is  y3 = 2x4 + 6    ...... (1) dx/dt = 2, x = 1,   y = 2

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A: The Given function is: The function f(x) is continuous if its limit exists and the limit value and…