(a) Given a surface function, z(x, y) == + xy with x cos0 and y= sin 0, 2 %3D az and əz (i) find ax az (ii) find by using chain rule.

Q: Find an equation for the tangent plane to the surface z+1 = ry* cos(z) at the point (1, 1, 0)

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Q: 5. Find an equation plane to the surface (u, v) = (3 cos(u) sin(v), 2 sin(u) sin(v), cos(u)) at the…

A: First, we find the point on the surface where the tangent plane would be drawn. Next, we calculate…

Q: Calculate dw/du and dwldv using the following functions: = f(x, y, z) = 3x² – 2ry + 4z2 = x(u, v) =…

A: Apply chain rule

Q: Please do 4

A: 4. (a) To Evaluate: ∫C​x ds, where C is the arc of the parabola y=x2 from (0,0) to (1,1). Let x=t,…

Q: Use hyperbolic functions to parametrize the intersection of the surfaces x² - y2 = 25 and z = 5xy.…

A: Surfaces : ,

Q: Evaluate (2x-y)dx+(x+3y)dy along the path C: curve on y = 1-x² from (0,1) to (1,0).

A: Given the line integralI=∫C(2x−y)dx+(x+3y)dyCurve y=1−x2

Q: Find f, and f, and evaluate each at the given point. (x, y) = arctan), (2, -2) %3D ,(x, y) = ,(x, y)…

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Q: B) r(t) = (4 cos t, 4 sin t, 16)

A: We consider x=rcos(theta) and  y=rsin(theta)  Therefore r^2=x^2+y^2

Q: Use hyperbolic functions to parametrize the intersection of the surfaces x' - y 49 and z= (Use…

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Q: 6. Given f(x, y) = 2x³y=x²y³, a) find the directional derivative of f(x, y) at (1, 1) in the…

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Q: Consider the function (a) Find z = 3e- cos x + 2e* siny. 8²% Әхду 8²z дудх -(0,0) -(0,0) and (b)…

A: Function: z=3e-ycos x+2exsin y To find: a)  ∂2z∂x∂y0,0 and ∂2z∂y∂x0,0 b) The slope of the surface…

Q: f(x, y) = xsin²y + y² cosx, Find fxy a. sin² y - y² cosx b. 2xsinycosy-2ysinxcosy c. 2sinycosy -…

A: We need to find fxy.

Q: Find equations for the tangent plane and the normal line at point Po (Xo Yo-Zo) (5,1,0) on the…

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Q: Let f(x, y) = sin(x y − x − 3y+3). (i) Determine the domain and range of f. (ii) Factor the argument…

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: Find Joy²dx + (xy-x²) dy along C: y = 7x from (0,0) to (1, 7).

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Q: 4. Consider the parametric curve x = cost, y = sin cost, te [-T, π]. (a) Show that (0,0) is a point…

A: Consider the parametric curve given by  and , where  belongs to .a) To show that  is a point of…

Q: 1. The equation r²y³ +2ln(yz) + ye" - 1 determines a surface. Note that you cannot easily write z as…

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Q: a) Find the extreme values of f(x, y) = x' + y' - 3xy +1 by using the Second Derivatives Test. b)…

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Q: For F(x, y,z)=(6xsin y+10xz' )i+(3x cos y-4ye"-2y ze" ) j+(-2y'e* +15x²z² )k , find | a function f…

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