Let f : [0,1] → IR be a continuous function. Prove that και 15') - - lim ko k Σ'() lim ko k j=0 Σ'() = f ra) dr. f(x) j=1

Q: (d) (e) Determine the range f(A) where A = (a, 0] for some fixed a < 0,a # Determine the supremum…

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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: 2. Let f(x) be a continuous function on R. The following is known about f(x): • The domain of f(x)…

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Q: ,10 If f is a continuous function and F'(x)=f(x) for all x € R, then A 2[F(5) – F(1)] B 2[F(10) –…

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Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = r? + y (x, y) # (1, –1) x + y is not…

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Q: → R be a function defined by 3x² + x+5 f(x) = not continuous at x = = 0.

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Q: Suppose f is defined on [a, b] and g is defined on [b, c] with f(b) g(b). Then define { f(x) if a <…

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Q: fn. (1,00) Evaluate lim

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Q: If f (s Continuous then on -fex) True or felse?

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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 /: X →Y be a function, A, BC X and C, DCY. Show that: 1. f(AUB) = f(A) uf(B) 2. If AC B, then…

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Q: How would you remove the dis . z46 f(r) =

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Q: Let f(x, y) = sin (x - y) [x] + [y] ' 0, |x | + (x, y)

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Q: Suppose that X P(X < 3Y). unif(-2, 2) and Y ~ unif(-1,2) are independent continuous RVs. Find P (|X…

A: Given: X ~ U (-2, 2) Y ~ U (-1, 2) X and Y are independent.

Q: b) If the function f(x), g(x) and h(x) are continuous on [a, b] and derivable on (a, b), show that…

A: The given functions f,g and h are continous on [a,b] and differntiable on (a,b). So intermediate…

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: 7. Show that if f is differentiable on (a, b), and f'(c) > 0, where ce (a, b) and if f' is…

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Q: Give a pair of functions f: R →R and g : R →R such that g is not tangent to f at 0 and yet lim,0…

A: To give a pair of function f : ℝ→ℝ and g : ℝ→ℝ such that g is not tangent to f at 0 and yet limx→0…

Q: Use the simplex method to solve the linear programming problem. Maximize subject to: O B. There is…

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Q: If f is differentiable at xo, prove that f (xo + h) – f (xo – h) – f' (x0) - lim h-0 2h

A: Given that: f is differentiable at xo   To prove: limh→0f(x0+h)-f(x0-h)2h=f'(x0)

Q: Calculate of of and for the function f(x, y) defined as : ax' dy sin (xy) 1 + y f(x, y) =

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Q: d) Let f be a continuous function. Then 1 [ S f(x,y) dady = f*f* f(x,y) dyda. 0

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Q: 10 If f is a continuous function and F(x)=f(x) for all x € R, then x)dx=? A © FS) - F)] B ©2[F(5) –…

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Q: Let f be a differentiable function such that f(h) f(x + y) = f(x) + f(y) + 2xy and lim = 15. Find…

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Q: Let f(x) be a differentiable function with the properties that f(1) = 8 and lim f(x) = 5. [ªfa f'(x)…

A: f(x) be a differentiable function with the propertiesf(1)=8 and limx→∞f(x)=5.

Q: 5. Assume f: [0, 1] → [0, 1] is continuous with f(0) = f(1). Prove there are x, y = [0, 1] with |x -…

A: We have to assume f: [0, 1] → [0, 1] is continuous with f(0) = f(1). and prove that there are x, y ∈…

Q: If f', and y are continuous on [a, b] and derivable on Ja, b[, then show that there is a value c…

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Q: If fCxl is continuous on [2,7],then Si f(u) dvz f (7)-(2) True e or false?

A: we have given that f(x) is continuous on[2,7].

Q: 3. Define f(x) so that S(2) = {ap(). f(x) = 0, |피 1 (a) Explain why the function f(x) must be…

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