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MATH291 (DB425) Advanced Engineering Mathematics

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Solve the problem.

Find the derivative of the function f(x, y) = (x) with superscript (2) + xy + (y) with superscript (2) at the point (6, 7) in the direction in which the function decreases most rapidly.
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Compute the gradient of the function at the given point.

f(x, y) = (tan) with superscript (-1) (-7x/y), (3, -7)
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Use implicit differentiation to find the specified derivative at the given point.

Find (∂z/∂y) at the point (6, 1, -1) for ln ((yz/x)) - exy+z2 = 0.
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Use implicit differentiation to find the specified derivative at the given point.

Find (∂y/∂x) at the point (1, 4, e6) for ln(xz)y + 2y3 = 0.
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Use the chain rule to find the given partial derivative.

Evaluate (∂w/∂u) at (u, v) = (1, 4) for the function w = xz + yz - z2; x = uv, y = uv, z = u.
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Use the chain rule to find the given partial derivative.

Evaluate (∂z/∂v) at (u, v) = (5, 3) for the function z = xy2 - ln x; x = eu+v, y = uv.
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Find the domain and range for the function f(x,y).

f(x, y) = square root of (16 - (x) with superscript (2) - (y) with superscript (2))
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Solve the problem.

Find the equation for the tangent plane to the surface (x) with superscript (2) - 7xyz + (y) with superscript (2) = 9(z) with superscript (2) at the point (-1, -1, -1).
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Solve the problem.

Find parametric equations for the normal line to the surface z = -10(x) with superscript (2) + 7(y) with superscript (2) at the point (2, 1, -33).
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Solve the problem.

Find the derivative of the function f(x, y) = (x) with superscript (2) + xy + (y) with superscript (2) at the point (6, 7) in the direction in which the function decreases most rapidly.
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