Find that F = 12x²yi + (4x + 8yz²)j + 8y² zk is conservative, and find a function f such that F = Vf, and use the result to evaluate F • dr, where C is any curve from A = (0, 0,0) to B = (-3,2, 3).

Q: Find and classify all stationary points of the function f(x,y) = x³ + xy + y² − x.

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Q: Use the gradient to find the directional derivative of the function at P in the direction of Q. f(x,…

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Q: Consider the function f(x, y) = 3x²+y²-5 x²-2y A. Find and sketch the domain of f. B. Find and…

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Q: Find (url and dive rgence the of F(xy2)=Xy2 i+ 2x Y j t z(os xyk.

A: Given: F(x, y, z) = xyz i + 2xy j + 2cosxy k To find: Curl and divergence of the given function.

Q: -2+z) ds over t e point (0, 0, 0)

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Q: 9. Determine an expression for the gradient of f =Y x+y

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Q: F=(x-y², x²•y +3) at A-(1, 1), B=(2, –1) and C=(0,-2). %3D

A: The divergence of a vector field F = <P,Q> is defined as the partial derivative of P with…

Q: Let f (x, y) and g(x, y) be functions of two variables withthe property that ∂ f/∂x= ∂g/∂x and ∂…

A: Given: f(x, y)  and g(x, y) be functions of two variables with the property that ∂g∂x = ∂g∂x and…

Q: The x and y coordinates of the inflection point(s) of the function f(x) = x + 21x* is (are): A. No…

A: Solution :-   Given fx = x3+21x2We have to find inflection point for this, we put f''x = 0

Q: Evaluate F.dr where F(x, v)=xyi (4, 0).

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Q: Find the directional derivative of f(x, y, z)= = in the direction from P to Q(2, 4, -1). NOTE: Enter…

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Q: Consider the function f(x) = In (x - 4) a.) Use the First Derivative Test or the Second Derivative…

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Q: Verify that ƒxy = ƒyx, for ƒ(x, y) = xey.

A: Given function is fx,y=xey We have to verify fxy=fyx Solution: fxy means that first we find partial…

Q: Let f(x, y) = 3x² + 4y? – axy. (a) Find the values of a such that the graph of f increases when…

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Q: Determine the magnitude and direction of the largest change 2 2 (x² + y²)z at the point M (1, -2,1).…

A: Introduction How sharply a curve rises determines its magnitude. It is simplest just examine the…

Q: Calculate fz(3, 6, 9), where f(x, y, z) = xyz. fz(3, 6, 9) =

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Q: Give two distinct linear functions ƒ and g that satisfy ƒ'(x) = g'(x); that is, the lines have equal…

A: It is given that fx and gx two distinct function.

Q: Use Lagrange quadratic polynomial to estimate the value of y at x = 1.5 from the data points (0,…

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Q: 7] Sketch a graph of the function below. Include calculations of all key features. 1 f(x) : ーx-3 %3D…

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Q: Find the gradient of f(m, n) = m³ +n³. grad f = i +

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Q: Find Vf(a), given the following data. Calculate 91, 92 and g3 such that Vƒ(a): 91 92 93 = || || = 91…

A: The given problem is to find the grad f for the given data of directional derivative of function f…

Q: Find the gradient of the function y = x³ + x² – 2x at points where - a. It crosses the x -axis b. It…

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Q: Consider an F-curve with d f = (2, 14). Determine F0.05.

A: The F-curve 0.05 area with degrees of freedom 2 and 14.

Q: Sketch the domain of f(x, y) = √4x² - y² x - y² on the ry-plane.

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Q: Compute ƒxxx and ƒxxy for ƒ(x, y) = x3y.

A: Given function - f(x, y) = x3yConcept - While partially differentiating with respect to x, other…

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: