3. Verify that u(x, y) = x² + 4.x – y² + 2y is harmonic and find %3D - a) the harmonic conjugate function v(x, y) of u(x, y).

Q: 8y'' - 12y' + 5y = 6 cos t

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Q: 4x Find the linearization L(x,y) of the function f(x,y) = e cos (5y) at the points (0,0) and 0,

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Q: Find g'(x) if g(x) = t² cos 6t dt %3D 2

A: We can apply fundamental theorem of calculus here.  The second fundamental theorem of calculus…

Q: Find af af and f(x, y) = etry In(6y) af 4yln(6y)e^(4xy) af 1/ye^(4xy)+4xIn(6y)e^4xy dy f(x, y) = cos…

A: In the first part the answer is correct but you have mistakenly not taken bracket for 1/y . You can…

Q: EXAMPLE 2: dz 1 Given that z = x+y,where x = and y 2r. Find by: dt 1 a. Substituting x= and y = 2r'…

A: From the given problem:   z=x2+y2 ; x=1t2 ; y=2t3 a). Now, substitute x=1t2 ; y=2t3 in z=x2+y2 .…

Q: (b) Given that, z (r, y) = a²y + y cos r+y x = sin t y = In(t"), find an expression for dz using…

A:

Q: Let f(r, y, z) = ln(x² + z) + y²e#² – cos(2yz). Find fzyr- %3|

A: Consider,                                                                 f(x, y, z) = ln(x² + z) +…

Q: 4x Find the linearization L(x,y) of the function f(x,y) = e cos (3y) at the points (0,0) and0,

A: Answer is given below

Q: Find the linearization L(x) of y = e In(x) at a = 1. L(x) = %3D

A: Given a function f(x)=y=e3x ln(x) To find the linearization L(x) at a=1 The linearization L(x) is…

Q: (b) Compute the partial derivative fy at the point (x, y) = (1, 1), or show that it does not exist.…

A: Let f:R2→R  be the function   F(x,y)=x-1.y-113(b) compute the partial derivative fy at the point…

Q: t2, in whicht= x – y. Use chain rule to compute dz az and 1. a) Consider the function z dz and show…

A:

Q: Let dy/dx =f(x, y)= x^2 -y Use Euler’s method ( with step size ℎ = 1) to approximate the value of…

A: Given, dydx=f(x,y)=x2-y y(0)=1 h=1

Q: dw where w = cos 8x sin 3y, x=- dt 5. and y=t". Use the Chain Rule to find 2

A:

Q: Find the partial derivatives of the function f(x, y) = integral from x to y of cos(-6t^2 + 2t -…

A: Given: To Find: The partial derivative of f(x,y).   Fundamental Theorem of Calculus:

Q: 10. Find the linearization, L(x,y,z) of the function f (x, y, z) = e* + cos(y + z) at Po (0,",").

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Q: Suppose that u and v are differentiable at x = 2 and that u(2) = 4, v(2) = -9, u'(2) = 5, and v'(2)…

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Q: Consider the finction f(x, y) = 6y² + 4x² – 12xy- 4y. Then: (1,1) is a critical point and f has a…

A: The given function is f(x,y)=6y2+4x2-12xy-4y

Q: Verify Young's Theorem y? x2 z = In(2x + y) – + e(x+2y) + - y

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Q: 1(a) Given that f (x,y) = 6x²y³ – cosx + 5y – siny + 10x, establish | the fact that, the order in…

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Q: For the relation siny – y² =x²y– x+x , is dx 3x²y+1 А. cos y – 2y –x T – 1 3x² + 2y – cos y cos y –…

A:

Q: For f(x,y,z)=3xy+2y+8z - cos(x²+y²–z), the partial derivative fxy(1,2,5)=

A: Given function is fx,y,z=3xy+2y+8z−cosx2+y2−z. We have to find the partial derivative fxy1,2,5.…

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: dz at t = dt if z = f(x, y) = x² – 3xy+ 2y², x = x(t) = 3 sin 2t, y = y(t) = 4 cos 2t. Use chain…

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Q: Q2// B :- If u+v3e"cos y and u- v=e' sin y find the Jacobain of the function alu,v at dx.yl (x.y)…

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Q: Compute the indicated partial derivative. g(x, y, z) = x³y^z° 9xxyz %3D

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Q: If f(x, y, z) = 2x5 + 3y5- 8x2y?z?, what is the partial derivative of f with %3D respect to z? fz =…

A:

Q: Let f(x, y, z) = ln(x² + z) + y²e#z – cos(2yz). Find fzya.

A:

Q: 1. (a) Let f(r,y)- cos(z + 2y). Compute the diectional derivative Df(r,9), where u- (b) Compute ff,…

A: Solve the following

Q: Given that e2y, 10x2+4. f (x, y) = sin Enter a 10 significant figure approximation to the partial…

A: We will use matlab to find the answer

Q: Given the functions f(x, y) = 4 cos(4x - 6y) and g(x, y) = -6 cos(4x - 6y). af The partial…

A: Here we have to use the concept of partial derivatives

Q: Find the first partial derivatives for f(x,y) = x cos (5xy). of = cos (5xy) - 5xy sin (5xy) %3D of…

A: The solution for the above question is as shown below.

Q: Find the quadratic approximation at (0,0) for f(x,y) = ln(1+x^2 + y^2).

A:

Q: 4. Given that c(t) = t² + 2, y(t) = t³ – 2t and f(x,y) = x² sin (xy). Using the df and evaluate it…

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Q: O from the equation z = In(x² + y²) + Zyy Show that zxx %3D

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Q: Find the partial derivatives of the following when z = 0 and y = 1: (sin(5r²y+ 2xy)). (sin(5a'y +…

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Q: The conjugate harmonic function v(x, y) of u(x, y) = In (x2 + y2) is *() + 1. 2tan -1 2tan-1 + C ()…

A:

Q: Let z = x*y + 3xy* where x = sin 2t and y dz find = cos t. Use the chain rule to %3D when t = 0. dt

A: z=x2y+3xy4where x = sin 2t and y = cos t We need to find dzdt when t =0 Let's use the chain rule :…

Q: 2(a) Given that f(x, y) = 6x²y³ – cosx + 5y – siny + 10x, establish the fact that, the order in…

A: Given that, fx,y=6x2y3-cosx+5y-siny+10x

Q: Assume x and y are functions of t. Evaluate dy for 4xy - 7x+4y3 = - 48, with the conditions dt dx =…

A: Explained below

Q: 2 6.xy + 2z how that G(r, y, 2) = (3y? + ( 3y2 + 2x , x, y, z > 0 is conservative %3D 22 a) without…

A: Scalar potential function

Q: Find the first partial derivatives of the function. z = (7x + 8y)7 dz %3D dx %3D dy

A:

Q: Let z(x, y) be the function implicitly defined as the solution to 2x + y + z + sin(xyz) = 4 + that…

A: ddxsinnx =cosnxddxnx =ncosnx

Q: Let z = f(x,y) = 6x - 9xy + 4y. Find the following using the formal definition of the partial…

A: Authoring guidelines: As per the authoring guidelines, we can solve only the first three parts of…

Q: If f (x, y) = 4e4*cos (y - 1) , then the linear approximation at (0, 1) is %3D (A) 4+ 16x (B) 4+4x…

A:

Q: Q2: Find the Partial Derivatives of the functions with respect to each variable x and y and z: 1.…

A: partial derivative

Q: Find the partial derivative of z with respect to x дz (2x−y) дх Н дz 2 О 0 = 3 - 1 " дх Т 2y 2y дz…

A: Just use the rule of partial derivative and find the required solution.Final solution is in the box.

Q: Let ƒ(x, y) = 3x2 + y3.a. Compute ƒx and ƒy.b. Evaluate each derivative at (1, 3).c. Find the four…

A: Given:                      f(x, y)=3x2+y3   To determine:               a) The partial derivatives…

Q: Verify that y = x[cos (Inx)] is a solution of x²y" – xy' + 2y = 0

A:

Q: 1. if z = a² sin[e!n 2y] + 3xy cos[e!n2x] – 4a?y², find the partial derivative of z with respect to…

A: