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Chapter 12: Q. 27 (page 976)
In Exercises 27–30, use the result from Example 4 to find the distance from the point P to the given plane.
,
The distance is.
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In Exercises, by considering the function subject to the constraint you will explore a situation in which the method of Lagrange multipliers does not provide an extremum of a function.
Optimize subject to the constraint for nonzero constants a and b. Are there any nonzero values of a and b for which the method of Lagrange multipliers succeeds?
Partial derivatives: Find all first- and second-order partial derivatives for the following functions:
Construct examples of the thing(s) described in
the following.
Try to find examples that are different than
any in the reading.
(a) A function z = f(x, y) for which ∇f(0, 0) = 0 but f is
not differentiable at (0, 0).
(b) A function z = f(x, y) for which ∇f(0, 0) = 0 for every
point in R2.
(c) A function z = f(x, y) and a unit vector u such that
Du f(0, 0) = ∇f(0, 0) · u.
In Exercises , use the partial derivatives of and the point specified to
find the equation of the line tangent to the surface defined by the function in the direction,
find the equation of the line tangent to the surface defined by the function in the direction, and
find the equation of the plane containing the lines you found in parts and .
Use Theorem 12.33 to find the indicated derivatives in Exercises 27–30. Express your answers as functions of two variables.
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