Prove that there is a unique continuous function y : [0, 1] → R such that y(r²) x E (0,1]. y(x) = e" + 2

Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = tan x, a = π

A: The linearization of a function f(x) at x = a is, Now differentiating  given function with respect…

Q: Find an upper bound for ||L|| where L: R² → R³ with L(x, y) = (2x + y,x + 3y, 3x – 2y) -

A: Here we use the definition of boundedness of a linear operator.

Q: Let fn(x) = E s;(x) Y; be a linear smoother. Find a condition on the weights s;(x) so that fn(x) is…

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Q: Consider the Cauchy Problem y 0 = a(x) arctan y, y(0) = 1, where a(x) is a continuous function…

A: We have the given Cauchy problem is y'=a(x)tan-1y,  y(0)=1   …..(1) where a(x) is a continuous…

Q: Find the linearization L(X,Y) of the function  f(x,y)=y-1/x+1 at(0,0).

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Q: Define h(0,0) in such a way that h is continuous at origin       h(u,v)=ln(nu2-u2v2+nv2/u2+v2

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Q: If f: (a,b]-R has a local minimum at pe (a,b) and f is differentiable at p, then prove that f'(p) =…

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Q: The function f (x, y) = x3y + x2 + y2 − 2x − y + 4 has a stationary point (1, 0). Compute the…

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Q: In Exercises 23-28 explain why fis continuous. 26. f(x, y, z) = sin (xyz – 1) 27. f(x, y, z) = ln(e"…

A: The function is continuous if all partial derivatives exist.

Q: A necessary condition for the functional J defined by X2 J(Y) = | dx f(x, Y(x), Y'(x)) to have a…

A: The function J(y) is defined and a necessary condition for the functional J is defined…

Q: Define f : X → K by f(x) = x'(1), x € X. Again, the linear functional f is not continuous. By 6.4,…

A: Given: f:X→K by fx=x'1, x∈X To prove f is not continuous using the given corollary.

Q: please answer thank you!

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Q: x+y if (x, y) # (0,0) (1 if (x,y) (0,0) Q.3 ( 3+3+4) (a) Prove that the function f(x, y) is not…

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Q: Find the linearization L(x,y) of the function f(x,y) = e (X cos (5y) at the points (0,0) and 0,- 2

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Q: f(z) =|z| is continuous in C 0 صواب ihi O

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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: Let z be defined implicitly as a function of x and y by the e e-zy əz find O UTM + cos (xy) = (4 –…

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Q: Let z = = g(x, y) = e* cos y +y In(x – 1).

A: Let we havez=g(x,y)=excosy+yln(x-1)If x=s-t and y=s2-t2

Q: Let g(x) V1+ 8t dt. Use the Fundamental Theorem of Calculus to find g'(x). g'(z) =

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Q: let f(x) = { ax2+x+1 if x is less than or equal to 1                 { bx if x >1   find a and b…

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Q: 1. Consider the function z = x In(1+ y²), if x 2-t and y= Vf, use an appropriate form %3D dz at t…

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Q: Let g(x) = kf(x) for all x E/= {x: asxsb}. Show that Jglx)dx = kJ f(x)dx if k<0.

A: To prove the above result we use the definition of lower and upper Riemann integrals. For the proof…

Q: Let f(x, y, z) = ln(x2 + z) + y2exz − cos(2yz). Find fzyx.

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Q: 2. Show that the function g defined by VVT y² + √² 2, is not continuous at (0,0). (z,y) = (0,0)…

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Q: find the linearization L(x, y) of the function ateach point. ƒ(x, y) = ex cos y at a. (0, 0), b. (0,…

A: Consider the following function: fx,y=excosy Differentiate fx,y with respect to x taking y as…

Q: 11. Show that the Beta function, defined by B(x,y) = (1-t"-1 dt , for x> 0, y > 0, satisfies the…

A: the given function is βx,y=∫01tx-11-ty-1dt for x&gt;0,y&gt;0 substitute t=1-s therefore, dt=-ds when…

Q: Prove that the function f(z2) defined by x*(1+ i) – y°(1 – 1) , (z ± 0) x² + y? f(z) = [0, (z = 0)…

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Q: Determine whether the function f (z)= Įz² – e3z I0

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Q: - Find the total differential df of f (x, y) = (x² + y²) 3/2.

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Q: Let f(x, y) = x² + y² – 2a°y- if (x, y) # (0,0), - (14 + y²)² ' and f(0, 0) = 0. Show that f is…

A: Given: fx,y=x2+y2-2x2y-4x6y2x4+y22 if x,y≠0,0 and f0,0=0.   To show: (a.) f is continuous at the…

Q: 6. let h be the function defines by h(x) = 2e***dt, Find h'(2) %3D

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: Find the linearization of the function f (x) = x° at z = 1.

A: Given, f(x)=x6

Q: A necessary condition for the functional defined by X2 = dx f(x, Y(x),Y'(x)) to have a local minimum…

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Q: Verify Young's Theorem S. f(x, y) = = e(2x+y) In(xy)

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Q: Consider a regularizing term defined by where k!2* and the linear differential operator D is defined…

A: We are given the following regularizing term.                    ∫ℝN0DF(x)2 dx=∑k=0∞ak∫ℝN0DkF(x)2 dx…

Q: Suppose that z is a differentiable function of x and y with dz (1, 1) = 162 and(1, 1) = 6. dz ду dx…

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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: 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 f(x, y)= g(x)h(y), where g and h are continuously differentiable functions of one variable…

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Q: For a function h and an interval I, the oscillation of h on I is defined by w(h, I) = sup |h(æ) –…

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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…

Q: z is a function of x and y implicitly defined by Find and i at (x, y, z) = (0, 2, -1) əz Əx Əy y³…

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Q: Let T: C* (R) → C (R), T(f) = f' – 2f be a (linear) differential operator on the space of infinitely…

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Q: 2 find holomorphic function f(z) = u(x, y) + jv(x, y) if  v(x, y) = 2*x*y + 3*x f(0) = 0

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Q: 2-3. If f is integrable on [a, b], then ("f(x) dx = lim E f(x) Ax i=1 • In the theorem above, define…

A: We know that integration is equivalent to summation as mentioned above. Here, △x is the small step…

Q: If f:[a, b] →R has a local minimum at p E (a, b) and f is differentiable at p, then prove that f'(p)…

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Q: Show that there exist constants 0 < A < B <∞ such that, for every y, z ER with A В 1 dx < ly – z|3…

A: We need to prove that there exist positive and finite constants A≤B such that, Ay-z13≤∫ℝ1x-y23…

Q: Show (in terms of e - 6) that a function f : R → R defined by f(r, y, 2) = (2x + 3y + 42) is…

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Q: Study the continuity of f(x,y) = xe-Vx at p(1,0) ey.

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