f. Is the domain of f(x.y)= In (6x + 5y“) bounded or unbounded? O Unbounded O Bounded

Q: 2. Prove that the function f : R? → IR defined by f(x, y) = 1- a2 – 4y? is neither injective nor…

A: Given the function f:ℝ2→ℝ defined by fx,y=1-x2-4y2

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Q: Find the interval that the fullowing function is continous at In (9 -x) %3D A. [I,3] BL) (3,3) (1,3)…

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Q: Suppose that the function f:(a, b) → R is uniformly continuous. Prove that f: (a, b) → Ris bounded.

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Q: Let BCR be the open ball of radius one centered at the origin, i.e. B = {(x₁,...,xn) € R" : x² + +…

A: As per the question we are given a n dimentional open ball B of radius 1 centered at the origin. And…

Q: 3. Compute the value of F.dr if F = -yi + xj and C is a semicircle from (-1,1) to (1,-1).

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Q: Prove that f: R R where f (x) = |x| is neither injective surjective. nor

A: To prove this theorem, we use the defination of one one and onto.

Q: Show that if u =x - y and v =r+ y, then for any continuous f, e-2s* f(x - y,x + y)dydr = || (a+n)e-…

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Q: e. Choose the correct description of the domain of f(x,y) = In (3x2 + 3y2 ) A. Neither open nor…

A: We can do it by finding domain.

Q: 24. Show that f(x) = mx + b, where m,b e R, is uniformly continuous on R. %3D

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Q: ded on the y-axi

A: In this question, the concept of maxima and minima is applied. Maxima and Minima The extrema of a…

Q: f(r, y) = 3y + 5a² + In (x² + y² – 9) + In (2² + y² – 9)

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Q: Is the following bounded or unbounded and why? f(x,y) = ln(x2 + y2)

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Q: use the formal definition of limit to prove that thefunction is continuous at c.

A: Given: hx=2x-3 To Find: To prove that the given function is continuous at c=2

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Q: (b) Show that the u(x, t) can be written as u(x, t) = 0(x – at) + 4(x + at), where o and are…

A: The multivariable function u(x,t) is a rule that associates or fixes a number to a particular point…

Q: If x and y are elements of an ordered integral domain D,prove the following inequalities:   x2 + y2…

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Q: Determine the are bounded by: y2-8y+8x=0, and x=0

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Q: 3)Evaluate /y² + 1dA , Ris bounded by x = 4 - y² ,x = 0 R

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Q: Show that the functionf (x ,y)=xy³onD: x|<2, ]y|<oodoes not satisfies Lipschitz condition.

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Q: Show that x is continuous everywhere. r(x)-x-0 XX<0

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Q: Show that /(T) is injective (one-to-one) and find (/Y(0). (Hint: To prove /(x) is an injection, ahow…

A: f (x) = ∫0x t2 + a2t4 + a4 atUsing Newton Lebnitt TheoremWe find f'(x) f(x) = ∫m (n)n (n)  g(t) dt∴…

Q: Let f(x, y) = xy - 4x on

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Q: Find the local linear approximation of f(x) = In(5 + x) at xo = -4. %3D In(5 + x)

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Q: Evaluate (x²+y +y²)dA, where F is bounded by the 2. x²+ y t. = 2x

A: To evaluate: The integral ∬F x2+y2dA , where F is bounded by x2+y2 = 2x

Q: Find the Wronskian for the set of functions.{x, ex, sin x, cos x}

A:

Q: Show that x| is continuous everywhere. r(x)- x- 0 xX <0

A: f(x) is continuous, if LHL=RHL=f(x&gt;0), where LHL is left hand limit while RHL is right hand…

Q: iii If the discontinuity of ƒ is removable, redefine ƒ so that it's continuous at 0

A: This question related to the topic continuity of real analysis.

Q: Which of the following interval is f(x) = O (2, +∞0) O [2, +∞0) 0 (-∞0, +∞0) ○ (0, +∞) 1 Vx…

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Q: 7. Show that the function f : R → R* defined by f(x) is not onto. x2 +1

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Q: Show that the function F: C[0, 1] → R defined by F(f) = max f(x) zE[0,1] is continuous.

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Q: Let f(x) = vV72 – x and g(x) = x² – x. Give the domain of f o gusing interval notation.

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Q: Compute Lpf) and Up(f) for the function f(x) = x2 defined on 1= [0,1] where p is the partition…

A: Given that  f(x)=x^2 [0,1] n=5 subintervals

Q: Find aland U so that the function S 7z³ – 3x² + 7, z -2 is both continuous and differentiable. al =

A: To Determine: fins a and b so that the function is continuous and differentiable given: we have a…

Q: 5 Show that the function h(r, y) defined by if r +y? < 1 h(x, y) 0; if r +y 2 1 is discontinuous.

A: For continuity of 2 variable function L.H.L=R.H.L=Function value at (x,y). limx→a+,y→b+…

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Q: show me the follewing funct continuousornot at X-o tion f(x): Ix1 130

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Q: Find all volves of o,b ond c such that the func Ef×メ goo)= てメー1) 1チxg? 21 if x= 1 is continuous at…

A: Given, gx=ax3+bx2+cx+1x-12,  if x≠12,                               if x=1

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A: Given f is differentiable on (a,b). and c lies in (a,b) such that f'(c)=0.

Q: Let f = f(x, y), x = s +t and y = s - t. Show that (9- ) - se se as at ax ду

A: It is given that, f=f(x,y), x=s+t and y=s-t. Show that, ∂f∂x2−∂f∂y2=∂f∂s·∂f∂t.…

Q: Determine the are bounded by: y2-8y+8x=0; and x=0

A: The polynomial y2-8y+8x=0 can be rewritten as x=8y-y28. The function is graphed to understand the…

Q: t. Ost<1 f(t)={-1. 1st<2 -t, 2sts10

A: Topic = Functions

Q: O Show that the function f: N2 N defined by f(x, y) = 2X3Y is injective but not surjective.

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Q: A Let f be a piecewise-desined Xiven by the souowing Determine Valves off m and b that makef…

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Q: e. Choose the correct description of the domain of f(x,y) = In (6x + 5y). O A. Open Region O B.…

A: Solution

Q: Suppose that f is analytic on a domain D andVz € D,f'(z) = af(z) for some constant a. Show that f(2)…

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Q: f(z) = Im z is continuous ylgn O ihi O

A: Use the property of continyity in C which is if real and imaginary part of functiom is continuous in…

Q: Determine local extremum f(x,g) = xy(4-x-). %3D

A:

Q: Let x = s + t and y = s - t. Show that for any differentiable functionf (x, y),

A: Given: x=s+t and y=s-t and a differentiable function fx,y To show: ∂f∂x2-∂f∂y2=∂f∂s∂f∂t,