Partial Derivatives

If f(x,-y) = x^3 + cosy, determine fxx and fxy.

Chain Rule

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unit vector

The temperature of a plate at the point (x,y) is given by T(x,y) = 300+ 3x^2 -2y^2. A heat hating ant is located at the point (3,2). In which direction will the ant begin to walk? (Give a unit vector in that direction.)

directional derivative

) Find the directional derivative of f(x,y) = 2x^3-y^2+xy at the point (1,2) in the direction of the vector (1,3). Be careful: That direction vector isn't a unit vector!

local max and min

Find the local maximum and minimum values and the saddle points of f(x,y) = 2xy + x-y.

Use Language Multipliers

Use Lagrange multipliers to determine the smallest value of the function f(x,y) = 2x^2-x+3y^2 for points (x,y) on the circle x^2 + y^2 = 1.

Find an equation for the plane

Determine an equation for the plane tangent to the ellipsoid x^2 +2y^2 +3z^2 = 20 at the point (3,2,1) on the surface.

iterated integral

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Compute the volume of a solid

A solid has a rectangular base in the x,y-plane with sides parallel to the coordinate axes and with opposite comers at points (0,0) and (4,6). The solid's sides are vertical, and the top is determined by the plane z = 3x + y. Compute the volume of the solid.

integration to evaluate

please see attached file