4. (i) Let f : [a, b] → R continuous. Use the Fundamental Theorem of Calculus to show that there is CE (a, b) with f(c)(b – a) f(x) dx (ii) Assume that there exists cE (a, b) with 0. f(x) dx > f(x) dx < Show that there exists ro E (a, b) with f(xo): = 0. Hint: Apply the previous result to [a, c] and [c, b] and use the Intermediate Value Theorem.

Q: 12. Consider x²y f(x, y) = g4 + y? for (x, y) (0,0) and zero otherwise. Is f continuous at (x, y) =…

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Q: f(x+y)=f(x)+f(y) and f is continuous at 0. Prove that f is continuous at any a.

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Q: 2. Let f(x, y) = In(x² + y²). Show that fre + fyy = 0. %3D

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Q: Theorem 11. Let f be a continuous function on R such that f(x+y)Df(x)+fy) vx, yER. Show that…

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Q: 2. If f is continuous on an open interval containing x, and fchanges its direction of concavity at…

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Q: 3 Suppose that e(x) and f(x) are differentiable on [0, 1] and that 0 < |e'(x)| < |f'(x)| for all x €…

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Q: 2.) Determine the set of points at which the function is continuous. а. F(x,у) — 1+x²+y² 1-x2-у? ху…

A: Here, to find the set of points at which the function is continous.

Q: If f is continuous on [-1, 1] and f(x)dx = 0, then there exists some ce [-1,1] for which f(c) = 0.

A: Given: f is a continuous function on the interval -1, 1 and ∫-11f(x) dx=0. Prove: There exists some…

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A: Given, ∫06fxdx=9∫36fxdx=5∫30gxdx=-7

Q: * (0,0) is not continuous if (x, y) Q.3 ( 3+3+4) (a) Prove that the function f(x,y) = {2x+y (i if…

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Q: Suppose that y is a differentiable function of x and z = cos (x5y6). dz in terms of x, y, and dY dx…

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Q: It is frequently required to solve equations of the form f(x) = 0. When f is a continuous function…

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Q: Suppose f(x) is continuous on R such that F(x) is an antiderivative of f(x). Furthermore i Find F(1)…

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Q: (3) Determine the values of a for which x4 + y* , if(x, y) # (0, 0), (x² + y?)ª f(x, y) = ,if(x, y)…

A: solution.

Q: 3) Examine the limit of 1 (x² +y²).sin- x*+ (x, y) (0,0) f(x, y) = (0, (x, y)=(0,0)

A: Solution:Givenf(x,y)=x2+y2 sin 1x2+y2To Eamine the limit of a functionFormula:limx→0 sin xx=1

Q: 7. Let f(2) = f(x, y) = + i , = 0. when z +0 and let f(0) %3D 5xº+y² (a) Show that lim f(z) = f(0) =…

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Q: If z is a function of x and y implicitly defined by əz əz find and əx ду at (x, y, z) = (0, 2, -1).…

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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: Let the function f: R R be differentiable at x = 0. Prove that lim %3D = 0.

A: Given the function f:R→R be differentiable function at x=0. We have to prove that limx→0fx2−f0x=0.…

Q: Prove that there is a unique continuous function y : [0, 1] → R such that y(r²) x E (0,1]. y(x) = e"…

A: Differentiate the equation wrt x:  ddxyx=ddxex+yx22y'x=ddxex+ddxyx22y'x=ex+xy'x2

Q: 4. For a function f:D → R and a point xo in D, define A = {x in D x ≥ xo} and B: {x in D x ≤ xo).…

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Q: Determine if a function f : R2 → R defined by f (x; y) = { x sin(1/x2+y2) , if (x, y) ≠ (0,0) and 0…

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Q: 1. Suppose f(r) is a continuous function on r E [a, b]. Prove f(x)dx < Slf(x)| dx.

A: According to our guidelines we can solve only first question and rest can be reposted.Thank you

Q: At what points (x,y,z) in space are the functions continuous? a. h(x,y,z) = In (5z° - x-y² - 3) 1 b.…

A: In this problem, we have to check condition for having the function continuous.

Q: 3. Prove by using the Squeeze Theorem, or otherwise: 1 t2 dt is continuous at 0. T sin t (a) (b) 1+…

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: I. Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y…

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Q: 4. If f is a differentiable function of x and y with w = f(x, y), x = r cos 0 and y = r sin 0, show…

A: Given, w=f(x,y) x=rcosθ and y=rsinθ

Q: Find fy for f(x, y) = (In x) (sin x² y). O fy = (In x)(cos x²y)(2xy) O fy = (cos x2y)(x²) O fy = (In…

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Q: Suppose that f is a continuous function on R and, g and h are differentiable functions on R.…

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Q: Consider the function { that is defined on R as: f(x)=-x² +4x-3. Designate by (C) its representative…

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Q: Let f be a differentiable function with f(0) = 1 and / f'(1) dt = x². Then Of(x) = COS X Of(x) = (x+…

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Q: Find the antiderivative F for f(x) = | sin z|, -T <x <0 satisfying F(0) = 1. (1) F(x) = - cos x – 1…

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Q: (3x² -1, x 1, Determine c and d so that f(x) is continuous everywhere.

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Q: 1. Find if the function is continuous in the given points (x, y) = (0,0) (x, y) = (0,0) x + y = 0 x…

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Q: If z=f(x,y), x= r cos θ and y= sin θ, where f is a differentiable function expressing az/ax and…

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Q: If F(x)=[ cos(t² +1)dt , calculate using the fundamental theorem of calculus derived version.

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Q: 2. i) Give an example of a nonzero continuous function f on [0,1] such that f(x)dx = 0.

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

Q: 3. Show that there does not exist a non-constant continuous function f : R2 → R such that f(x, y) =…

A: Given:- f: R2→R

Q: 3. Evaluate the limit r(t +h) - r(t) lim h→0 h for r(t) = (t-1, sin(t), 4).

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Q: 3. Let f(r) = r sin() if z #0, and f(0) = 0. Show that f is continuous at r= 0 by using e- d…

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Q: Consider the sequence of functions {fn} defined on the interval (1, 0), where fn (x) 1 Vx E (1, 0),…

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Q: if (x, y) # (0,0) (1 if (x,y) (0,0) x+y (a) Prove that the function f(x, y) =2x+y is not continuous…

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Q: 6x2 - x²y +6y2 x2 + y2 Define f(0, 0) in a way that extends f(x, y) = to be continuous at the…

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Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as: