1. Let T: C(R) → R be a linear map with domain all continuous functions on R and codomain R. If T(sin(z)) = 8 and T(sin(7x)) = -2, find T(sin(3z) cos(4x)).

Q: The linearization a at a = 0 to √8+7x is A + Br. Compute A and B. A B=

A: To Find: Linearization for fx=8+7x at a=0

Q: compute ∫C F · dr for the oriented curve specified. F(x, y) = (ey^-x, e2x), piecewise linear path…

A: compute ∫C F · dr for the oriented curve specified. F(x, y) = (ey^-x, e2x), piecewise linear path…

Q: 2) Use Green's Theorem to compute f F df where F= (3x +4y)i + 2xyj and C is the triangular path from…

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Q: Find the linearization L(x) at x= a. f(x) = x+ a = - 1

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Q: Ôz and Show 4. Use an appropriate form of the chain rule to find dv (a) z3x²-y tanx, x =u/v, y = u'v

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Q: The mapping $ (y) = fy' : H→H* defined by is not a linear m

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Q: 18.7. Let f(z) = (e + e-2)/2. Evaluate f(2) -dz, where y is any %3D simple closed curve enclosing 0.

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Q: Suppose that P1(t), p2(t), and p3(t) are continuous on R, and that y1, Y2, and y3 are solutions to…

A: Wronskian of the function

Q: Find the linearization at x = a. f(x) = 6x®, a = 1

A: Linearization L(x) of function f(x) at a is given as L(x) = f(a) + f'(a)(x-a)

Q: 7. Find the Jacobian of the transformation T(x, y) = (w sin(v²),w cos(v²)). X=w Sin (v?) w = sin…

A: Disclaimer: Since you have asked multiple questions, we will solve the first question (Q7) for you.…

Q: Find the SPI form of the Boolean expression whose Karnaugh map is below. yz _yz yZ ỹz 1 1 | 1 1 wx…

A: From the given table, the input value of Karnaugh map is…

Q: Find the linearization L (x, y) of the function f (x, y) = y/-x at (-9,4). 2 L(x, y) -x+ 3y – 6 3…

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Q: *) Use the Wronskian to determine whether the functions y₁ = sin(3x) and y₂ = cos(4x) cos(4x) are…

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Q: 12.2.7. Suppose that Q := {(x, y) e R : x > 0 and y > 0} and that f is a continuous function on R?…

A: Sol

Q: Let, T:R' → R³;T(x,y,z)=(2x+ y, y- z,2y+4z) 2. Find the rank and nullity of T.

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Q: Evaluate ∫ ▽f*dr Where f(x,y,z)=cos(πx)+sin(πy)-xyz and C is any path that starts at (1,1/2,2) and…

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Q: 2) Use the Wronskian to show the three functions are linearly independent. f(x) = x, g(x) = cosx,…

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Q: Let f(x, y, z) = ln(x² + z) + y²e®z – cos(2yz). Find fzyn. Zyx ·

A: Given function is  f(x, y, z)=ln(x2+z)+y2exz-cos(2yz)

Q: 3. Let C[0, 1] be equipped with the inner product (f. g) = f(x)g(x)dx, where f = f(x) and g = g(x).…

A:

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: 6. Let f(x,y) = sin(x + y) + arcsin (xy) a. Find the linearization L(x,y) at the point (0, ")

A: We have to find linearization of given function f(x,y): Concept: The linearization of a function…

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: 2) Use Green's Theorem to compute f F. df where F=(3x +4y) i + 2xyj and C is the triangular path…

A: Use green's theorem

Q: 2) Use Green's Theorem to compute f F df where F=(3x + 4y)i + 2xyj and C is the triangular path from…

A:

Q: Determine if the functions Y1=e^{4x}senx and Y2=e^{4x}cosx are linearly independent or linearly…

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Q: Let D, be the linear transformation from C'(a, b] into C[a, b]. Find the preimage of the function.…

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Q: Show that the function f(x, y, 2) = (x² + y² + 2e**)4 is strictly convex on R³.

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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: Compute the discriminant D(x, y) of ƒ(x, y) = x2y2.

A: fx, y=x2y2 Partially differentiating with respect to x, we get ∂f∂x=2xy2 Partially differentiating…

Q: Suppose w=x/y+y/z, where x=e3t, y=2+sin(t)x=e3t, and z=2+cos(6t). A ) Use the chain rule to find…

A: w=xy+yz, where x=e3t, y=2+sint & z=2+cos6t Partially differentiating w=xy+yz with respect to x,…

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>0,y>0 substitute t=1-s therefore, dt=-ds when…

Q: Let D, be the linear transformation from C'(a, b] into C[a, b]. Find the preimage of the function.…

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Q: 2. Let F(x,y) = (cos(x – 2y), –2 cos(x – 2y)). Find curves C and C2 so | that, F. dr = 0 F. dr = 1

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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: If the Wronskian of f and g is t cos t - sin t, and if u = f+ 2g, v = f - g, find the Wronskian of u…

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Q: If z = f(x,y) = tan x =t cose and y =t sin@. Use the Chain Rule method to find y as functions of t…

A: Given that z=f(x,y)=tan-1xy Where x=tcosθ,y=tsinθ Substituting, the values,…

Q: 2t – 2 t2- 1, Let r(t) = - î - tan (t) ĵ+ In(2t) k. %3D (a) Determine the domain of r. (b) Compute…

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Q: Problem 22. Prove that there does not exist a linear map T : R – R' such that range T = null T.

A: Rank nullity theorem: Define a linear transformation Given linear transformation is:

Q: Let f(x,y, z) = In(x²2+ z) + y²e*z – cos(2yz). Find fzyx·

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Q: find the linearization at x = a. y = e√x , a = 1

A: Given function: Using the formula of linearization:

Q: Let w f(t,0) where t and 0 are functions ofr and s, %3D dw is equal to Or then according to the…

A:

Q: 41. Let f(x, y, z) =x?y+ y²z. Use the Chain Rule to calculate af/as and af/at (in terms of s and f),…

A: Since f is function of x, y and z.  For partial derivative f with respect to s and t we using chain…

Q: 1) Given the mapping F(x,y,z)=(3e" ,xzsin 2y ,xy), v=(2,2,-1), using the definition, then the…

A: Let p be a point in ℝn and F:ℝn→ℝm be a map, then the tangent map of F at p is a map between the…

Q: Given that f is an invertible function, if the point (x, y) lies on the graph of y = f(x), then the…

A: We have to find at which point is guaranteed to lies on the graph of  y=f-1(x).

Q: Let z = f(x, y) where g = e° cos t and y = e° sin t. Use the Chain Rule to show: z = e

A: Let z = f(x, y) where x = es cos t and y = es  sin t. We will be using the chain rule to show,…

Q: Given F(x, y, z) = (tan y + cosh x Joi 2xy)i + (x sec² y - x² + 1)ĵ - 2zk, evaluate F. dR where C'…

A: Given that,  F→x,y,z=tany+coshx-2xyi^+xsec2y-x2+1j^-2zk^, Therefore, set fx=tany+coshx-2xy…

Q: Use Green's theorem to evaluate f (x²y) dx + y²dy here C is close path formed y = x and y = x³ from…

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Q: Let c(t) Use the chain rule to compute (foc)'(t). (cos(t), sin(t), t²) and f(x, y, z) = x² + y² +…

A: The objective is to find the derivative.

Q: 2. Let f(t) be a continuous function on the interval [0, 27]. Suppose the 1st order Fourier…

A:

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Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.