) (15 points) Find the linearization (tangent plane approximation) of the function f (x, y) = y?x +x² at the point (1, 1). Use the linearization to compute an approx- imation for f(0.9, 1.1).

Q: (a) Find the directional derivative of f (x,y) = In(xy) at (-) in the direction of (1,1).

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 2. Determine the directional derivative for u = log,y x2lnz if it is defined by the direction at dx…

A: The given function is, u=logyx2lnz Given that, dxdz=1,dydx=2. From the above, x=z+c,y=2x+d Then, the…

Q: F(x,y) = (1+ 2xy?)î+ (2x°y +2y)}

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Q: An image is projected onto a wall, and the irradiance of the light on the wall is (3) given by f(x,…

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Q: Find the derivative of the function at Po in the direction of A. h(x, y, z) = cos xy + e + In zx,…

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Q: 1. Find one normal vector of the tangent plane to f(x, y) = cos(x² + y²) at point P(1, 1, cos 2).

A: Please check step 2 for the solution.

Q: Let f(x, y) = 3x2 – xy+ 2y2. Then an equation for the tangent plane to the graph of f at the point…

A: Given f(x,y)=3x2-xy+2y2

Q: Compute f(X.y) o xI the directional derivative of ait the point (1,1) in a direction ormal to the…

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Q: 0. for the funchion f (x,4) = 3-x?- u?, find a unit tangent vector to the level curve at the point…

A: Given: fx,y=3-x2-y2-3 at P3,5​​​​​​​

Q: Q5-let F(x,g.2) =x -x 22 @ Fined VE B write the equution of tangent Plane t- the graph of at Point…

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Q: Q4:- Find the derivative of the function at Po in the direction of the vector AC. f(x, y, z) = x² +…

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Q: A differentiable function f(x,y) has the property that f(2,4)=2 and fx(2,4)=5 and fy(2,4)=−6. Find…

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Q: 5. Find the linear approximation (i.e. the tangent plane) of f(x, y) = Vx/y at the point (9, 4) and…

A: The Local Linear approximation of f at (x,y) using the partial derivative f at (x0,y0) is given by…

Q: ind the directional derivative of the function f(x, y, z) = Vx² + y² + z² at point (2, -3, 6) in the…

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Q: Find equations for (a) the tangent plane and (b) normal line at the point P on the given surface 1)…

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Q: A line passes through the point Po(xo, Yo) and has gradient m = tan a. Show that a general point of…

A: Gradient, or slope of a line characterizes the direction of the line and is found out by dividing…

Q: . Use the second-derivative test for local extrema (: extrema for the function. Identify whether the…

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Q: 5. Consider the saddle surface r = y- 2, find the point where the tangent plane is parallel to x+y+z…

A: If two vectors a1, b1, c1 and a2, b2, c2 are parallel, then it can be written that a1, b1, c1=ka2,…

Q: 8. (a) Show that F(x, y) = (ye + sin y)i+ (e" + x cos y + 2y)j is conservative and find a function f…

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Q: Find the local linear approximations L to the specified function f (x, y)= xsiny at the point P…

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Q: Let f(x, y, z) = x In y - exz.Calculate the directional derivative of fat point (-5, 1, -2) and…

A: In this question, we find the directional derivative of the function f(x,y,z)=xln(y)-exz3The line…

Q: ) Find the linearization L(x, y) of the function f(x, y) = ez+y at (0,0). Use it to estimate f(0.01,…

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Q: Verify Stoke's Theorem for F =( x²)i +(xz)j + (xyz)k and C is the curve of intersection between z=1…

A: Given: F =( x²)i +(xz)j + (xyz)k and C is the curve of intersection between z=1 and z² = x² + y².

Q: Using an appropriate linearization, the local linear approximation of f(x) n+ cos a at is %3D

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Q: Let f(x,y)= e, Compute the cartesian equation of the tangent plane r,y) at the point (2.2).

A: The equation of tangent plane

Q: 6. Find an equation of the tangent plane and a set of symmetric equations of the normal line to the…

A: Let's find.

Q: F(x, y) = 2xyi + (y² – x²)j (x2 + y2)2 and C is any positively oriented simple closed curve that…

A: Given  F(x,y) = 2xyx2+y22i+y2-x2x2+y22j To Evaluate  ∫CF(x,y)dr

Q: (2x-y²)² (x,y) + (0,0) Let f(x,y) = x²+y4 2, (x, y) = (0,0) Find the slope of the tangent line to…

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Q: 7. Given (x, y)= x' + y² a) Find the derivative at P(1,1) in the direction of " = u,i +u ;j b) In…

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Q: 6. Let f(x, y, z) = e¬ª cosh y + z and P: (-1,0, 0). Find the derivative of f at the point P in the…

A: Given: f(x,y,z)= e-xcoshy+z.

Q: P. let f-uti and fcz)=z= Prove that U and v Stadify lasplare equation

A: A function f(x,y) satisfies Laplace equation if fxx+fyy=0 Given function is f(z)=z3 Substitute…

Q: Let f(x, y, z) = xyz Find the direction in which the function increase most rapidly at the point…

A: Given: fx,y,z=xy2z3 The gradient of a function if given by ∇f(x,y,z) =∂f∂x,∂f∂y,∂f∂z.

Q: 3. Find the derivative of the function f(r, y, ) = xy + yz+ the direction A = 3i + 6j – 2k. zr at…

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Q: Q1: Find the derivative of f(x,y,z)=x^3-xy^2-2z at point P_o (1,1,0) in the direction of vector…

A: Solution:-

Q: Let f (x,y)=x2+5xy−4y2. Then a standard equation for the tangent plane to the graph of f at the…

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Q: 4. Compute the directional derivative of f(r, y) = at the point (1, 1) in a direction normal to the…

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Q: 4. Let g(x, y) = e"/v, Find an equation for the tangent plane at (x, y) = (2, 1).

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Q: Use a linear approximation to estimate the change in f(x,y, z) = e2" cos(5yz) when moving from the…

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Q: Integrate f(x,y) =x+y over the curve C: x² + y = 49 in the first quadrant from (7,0) to (0,7).

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Q: 3. Let f(x, y) = xy² + x°y². Find an equation for the tangent plane at (x, y) (-1,2). %3D

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Q: Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x,…

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Q: 1. For the function f(x, y) = In root (2 + xy), find the equation of the tangent plane at the point…

A: Note: As per the guidelines we are supposed to answer only one question because this is not a…

Q: Using an appropriate linearization, the local linear approximation of is f(x) = T+ cos r at 2.

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Q: 2. Use suitable linearization to find the approximate value for the function at point P. f(x,y) (x-…

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Q: (b) Find the tangent plane and normal line to f(x, y, z) = x²y+y?z+ z2x at the point (1, – 1, 1).

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Q: : @ use suitable linearization to find the approximate value for the Function at point p. f (xy) =…

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Q: Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) =…

A: Given that, ƒxx(0, 0) = -6  ƒyy(0, 0) = -3 ƒxy(0, 0) = 4

Q: z=f(x, y) = 2x +3 y at the point (2, 2, 20) (ii) Use a linear approximation to estima

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Q: (2) Find an equation for the tangent plane to the graph of z = where z = f(x, y) is implicitly…

A: Given: The function is fx,y,z=xy+xz+yz-lnz2+1-4. Calculation: Obtain thethe value of  fxx,y, fyx,y…

Q: (#10) The implicit derivative formula on a surface F (x,y, z) = 0 is: az Fr F, Evaluate dz/ax on the…

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