Assume g, f, f' and g' are integrable on [a, b]. Prove fo f(x)g'(x) dx = ƒ(b)g(b) — ƒ(a)g(a) – Så f'(x)g(x) dx. A - -

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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: If f(x) is continuous at x = c then f'(c) exists. True False

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Q: Consider the function f(x)=1/x+4. a) As x→−∞, f(x)→ b)As x→∞, f(x)→

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Q: Assume that f"(x) is continuous on [0, 1] an ƒ(0)=1,ƒ(1) = 3, ƒ'(1) = 7. Find [xf"(x) dx.

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Q: Suppose f, g:C → C are differentiable. Prove that fg is differentiable and that (fg)' = f'g + fg'.

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Q: ·b f(b) − ƒ(a) = f'(a)(b − a) – ſå ƒ"(x)(x − b)dx - a

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Q: Let f(x) = sin? (x). Define F(x) | f(t) dt. What is F" -) ?

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Q: Suppose that f is differentiable on [a, b]. Prove that if f'(a)f'(b)< 0 then there exists a point c…

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Q: Let f(x) = x?sin (-) for x + 0 and f(0) = 0. c) Show that f'is not continuous at x = 0.

A: Given function  f(x)=x2 sin1x      for x≠00                  for x=0 we have to show that f' is not…

Q: Suppose f(z) is defined on all of C, is differentiable at z = a, and satisfies f(a) = a. Show that…

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Q: 6. Let f be a continuous function on [0, 2] whose second derivative exists on (0, 2). Assume f(0) =…

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Q: Let I be and interval and let f : I →R and g : I → R be functions differentiable at c. If h : I → R…

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Q: If f' is continuous on [a, b], show that 2 [* f(x)f'(x) dx = [f(b)J* – [f(@}J°

A: Proof:            Given: f' is a continuous…