f(xo+h, yo+h) = f(xo, yo) + [fx (xo, yo)h + fy(xo, yo)k] 1 + xo, Yo)h² +2fxy(xo, Yo)hk + fyy(xo, Yo)k²] + R. 2 [fxx (xo, 3 1. Find the second order approximation of the following functions near the given points using Taylor's Theorem. Your answer should be in terms of x and y. (a) f(x, y) = ln(x + cos y) near (0,0). (b) f(x, y) = x sin(x − y) near (π, 0). (c) f(x, y) = y√1 + x near (0, 1).

Q: Show that the equation Fu 1 u 0x² ² 3t² 0 is satisfied by u=f(x+ct) + F(x-ct) where fand F are…

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Q: The functions f₁(x) = x² and f(x)=x³ are orthogonal on the interval [-3,3] ? True False

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Q: (x, y, z) = 2x²y — xz² + 3y²z. Xyx

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Q: 2 (d) T (x) = Væ + 9Væ7

A: T(x)=x+9x73-2x25Formulaamn=amn1at=a-tddxx=n xn-1

Q: 9: Check Euler's Theorem for the Function

A: We checked Euler's theorem for function.

Q: √x² + y² x-1 (a) Find and sketch the domain of f (x, y). 2. Consider the function f (x, y) = 4 (b)…

A: The given function is  f(x,y)= x2 +y2-4x-1 We know that the domain of a function is the set of…

Q: Suppose that f(z, y) = 10z²y² + 8z? + 2y? then find the discriminant of f.

A: We have to find the discriminant of the function of two variables fx, y. It is given that the…

Q: +タ x2 – 1

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Q: Q (1): Find fxyz for the function: = xe² + ye* + zey a) f(x, y, z) %3D

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Q: Find the derivative of the function. q=V19r - r7 19 - 7,6 B)- -776 1 A)· V19r - r7 2V191 - 17 C)…

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Q: Find and sketch the domain of z = f(x, y) = In(9 - x - y²).

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Q: F(x, y) := x² + xy + y² – 1 y – x2

A: Solve the given problem.

Q: Determine if F = x²yî + (xyz +29)ĵ – x²y²k is conservative or not %3D

A: For vector field F⇀=ui^+vj^+wk^ Curl(F)=i^j^k^∂∂x∂∂y∂∂zuvw   For vector field F⇀ to be conservative…

Q: Consider the difference equation = Xn+1 Xn+bsin(2xn). Determine any fixed points of this difference…

A: Given difference equation isxn+1​=xn​+bsin(2πxn​) -----------------(I)Step 1: First, we find the…

Q: a z = 4x³-6x²y²-y com x= 2sen(t) y=-3cos(t) e

A: #As per the guidelines we are entitled to solve one question at a time, please resubmit the other…

Q: Show that the function u(x,t) = t/2e-*/4 (t > 0) satisfies u, = uxx.

A:

Q: = Evaluate S₁2₁,22,23] f (2) dz where 21 = −i, z2 = 2 + 5i, zg 5i and f(x + y) = x² + iy

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Q: 2. Sketch the domain of the following functions: a) f(x, y) = b) f(x, y, z)= 2-x-y - eª %2 In(4-x² -…

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Q: e +1 Differentiate the function f(x)= In 5x +1 Se+1 3ex+1 Sesz e3x +1 ,5x +1 e 3e3z Sesz e3x +1 5x…

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Q: Find the function r that satifies the following condition. r'(t)=(3t2,4t3,2t); r(1)=(-2,-5,-9)…

A: Given,          r't= 3t2 , 4t3 , 2t  , r1=-2,-5,-9

Q: What does it mean for the differentiability of a function if only one of the Cauchy-Reimann…

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Q: Zez analytical function or not?

A: We check whether w=zez is analytic function or not.

Q: What does it mean for the differentiability of a function if the Cauchy-Reimann equations (Ux = Vy…

A:

Q: if r'y-- 2y + 1.

A:

Q: 9) Find a linear function of the form f(x) = ax +b that satisfy the conditions A-4) = 8 and (3) - 9.

A: f(x) = ax+bf(-4) = 8→ -4a+b = 8.........................1f(3) = 9→3a +b =…

Q: 2. Find whether the function F-(* - y' )i + (v² - 3x)j + (:² - xy is irrotational.

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Q: Determine which of the following pairs of functions are linearly independent. NO_ANSWER 1. f(0) =…

A: The linear dependency of functions can be evaluated using the value of the Wronskian of those…

Q: (iii) For the linearised system in part (i): find all straight line orbits, determine the behaviour…

A: note :As per the instructions we will only solve parts (b.iii) and (b.iv) of the question.

Q: (b) (c) Explain the differentiability of f on R by using definition. 1 Let f' be the first…

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Q: Find those constants a, b, c, d such that f(z) = ax² + bxy + y² + i(x² + cxy + dy?)

A: The given function is : fz=ax2+bxy+y2+i(x2+cxy+dy2)

Q: 8:(- 2, 2]– R, g(x) = ["(1 – 3)(r² + 1))dt

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Q: a) Find the domain and range of the function f(r, y) = Vy – 4+ SUTM 1 UTM x2.

A: