8. (a) Show that F(x, y) = (ye + sin y)i+ (e" + x cos y + 2y)j is conservative and find a function f so that F= Vf. (b) Find the work done by F (F as in part (a)) in moving an object along a curve C from (0, 1) to (2, 7).

Q: b) Show that F - dr is independent of its path, given that y F(r, y, 2) = 1+. + tan-2) i+ tanrj+ k.…

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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: 26) Find the partial derivative of of f= y cos x+ 2x sin z + 5y with respect to y.

A:

Q: (b)Find the derivative of g in the direction of a= 2i+ curl w where at point (0,-1,1) g = In- 4 + 2…

A: Given, w=cosxyzi+xeyz+1j+xyzk Then, Curl…

Q: 3. find the derivative of the function at Po in the direction of u. f (x, y,z) = 3e* cos yz,…

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Q: find the directional derivative of f(x,y,z)=3e^xcos(yz) at P(0,0,0) in the direction of A=2i+j-2k

A: First find out the partial differentiation of f(x,y,z) and then put the values (0,0,0) and then for…

Q: Show that if the function f(x, y) = ex siny + ecosx, satisfies Laplace Equation.

A: We can solve this as follows:

Q: Use linear approximation to find y = L (x) of the function f (x) = = COS X about the point xXo = JT…

A: Use formula of  linear approximation

Q: Find the directional derivative of f(r,y, z) = e² cos(Tyz) at (0, 1, ) in 2-1 -y-1= I-2 the…

A: To Find: Directional derivative

Q: 8. Find the directional derivative off at the p in the direction of v = () of f(x, y, z)=x²exx, p =…

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Q: (2) Let f(x, y) = x² – y². Find the tangent line to the level curve f(x, y) = -3 at (1,2).

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Q: Find the orthogonal trajectories of the family of curves: y = k (csc x + cotx) Ⓒy² = 2 (C- cos x) O…

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Q: 2/6 Consider the upper half of the ellipsoid f(x.y) = 1--n and the point P-- 3 on the level curve…

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Q: (b) Find the derivative of g in the direction of a=2i+curl w at point (0,-1,1) where u u +2 g= In. J…

A:

Q: 1. Give an equation of the tangent plane to the surface z = In(4y – x) at the point (3,1,0). 2. Use…

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

Q: directional derivative

A:

Q: Find the directional derivative of the function f(x,y,z) = tan (xýz) At point (1,2,3) İin the…

A: The solution is given as

Q: What is the linearization L(x, y) of a function f (x, y) at a point (a, b)? How is L(x, y) connected…

A: Linearization L(x,y) of a function f(x,y) is given by the formula

Q: Given that f (x, y) = sin Enter a 10 significant figure approximation to the partial derivative a10…

A: When the partial differentiation is done with respect to a specific variable, then all other…

Q: 8. Find the directional derivative of f (x, y, z) = xyz3 at P(2, 1, 1) in the direction of Q(0,…

A: Ans=1 I have considered image question bcz I think you have done mistake while typing... Now go…

Q: Express the curve by an equation in x and y given x(t) = sin(t) and y(t) = 8 + cos² (t). a) O-x² +y…

A:

Q: f(x,y)=ln(5x+4y). Find the directional derivative of f at (1,0) in the direction of the line…

A:

Q: Find the orthogonal trajectories of the family of curves: y = k (csc x + cotx) Ⓒy²= 2 (C- cos x) O y…

A: This question is related to differential equations, we will solve it using given information.

Q: 1. Consider the function g(x, y) = cos(xy) + sin(x – 2y) (a) Determine g (c) Determine (c) What is…

A: Given function is  g(x,y)=cos(xy)+sin(x-2y) we have to find the second order partial derivatives…

Q: Find an equation for the tangent plane to the graph of z = xe' cos z – 2 - at the point (2,0,0). 1.…

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Q: 2. Find the derivative of f(x, y, z) = x³ - xy2 - z at Po(1,1,0) in the direction of V = 2ỉ – 3j +…

A:

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: A line passes through the point P0(x0, y0) and has gradient m = tan α. Show that a general point of…

A:

Q: The second order partial derivative of f with respect to z (mean fz) where f(x, y, z) = x'0 sin(3y +…

A: Given: fx,y,z=x10 sin3y+10z To determine: fzz

Q: Compute F. dr where F (x, y) : (7xy + x cos x) i+ (-yev + 6x²) j and C is a piecewise closed curve…

A:

Q: Find the derivative of g in the direction of a = 2i+ curl w at point (0 where น u+2 8 = In- J u =…

A: Given: the function g=lnuu+2, u=z(xy+z2)2, w=cosxyzi+xeyz+1j+xyzk To find: the directional…

Q: 3x The directional derivative of f(x,y,z) = ln (주)을 1+yz from P to Q(-1,-1,2) is given by...…

A: We can solve this using given information

Q: Find the derivative of the function at Po in the direction of A. f(x, y, z) = xy + yz + zx, Po(1,…

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Q: 3x The directional derivative of f(x,y,z)=ln at the point P (2,0,-1) in the direction 1+yz z² from P…

A:

Q: 1. Find the value of the directional derivative at the particular point Po for the given function in…

A: Given that gx,y=y2tan2x ; U→=-32 i^+12 j^ ; P0π3,2We need to find the value of the dorectional…

Q: 5. (а) Find the directional derivative of f at P in the direction of ā if f(x,y) = y In(x) ; P(1,4);…

A: To find The directional derivative of f at p in the direction of a.

Q: 2. Let closed curve C consist of the line segment from (0, 0) to (2, 1), followed by the line…

A: We will solve both parts of this question using basic knowledge of line integrals and definite…

Q: A heat-seeking particle is located at the point (2, −2) on a metal plate whose temperature at (x,y)…

A:

Q: 3. Evaluate the gradient of S(1, y, 2) = In(cosh z+ y) - arctan| at (1,y) = (7/2, 1).

A: Given that fx,y,z=lncosh x+y-z2 arctanx2y2+z2 To find the gradient of f(x,y,z) :…

Q: Let C denote the curve of intersection between x = -5z° and y = 6z", starting from A(0,0,0) and…

A:

Q: The maximum directional derivative of f (x, y, z) = x² sin(zy) – yx + 10 | at (-1, 2, 0) is

A:

Q: ) Compute the directional derivative of f(x, y) = x2 y at the point (1, 1) in a direction normal to…

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Q: 8. Find the directional derivative of f (x, y, z) = xy²z³ at P(2,1,1) in the direction of Q(0, –3,5)

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Q: 2 - What is the directional derivative of the function f(x, y, z) = x° - 2xy + z point A, = (1,1,0)…

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Q: 23. Find the directional derivative of f(x, y) = *+y at P(1,0) in the direction of Q(-1,-1).

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Q: 1. Find the directional derivative of f(x,y) = y²e²" at (0, 1) in the direction of v =.

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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: Find the gradient for the curve defined by y + xy = 3 at the point (1,2)

A: Gradient to given curve

Q: 1. Find the directional derivative of the function f(x, y, z) = 3e" sin(ry) in the direction of v =…

A: