Assignment 6

Problem 1.  Read sections 10.3 and 10.4

Problem 2.  Find the equation of the plane tangent to the graph of

\displaystyle z = \ln \frac{\cos x}{\cos y}

at the point (\pi/4, \pi/4, 0).

Problem 3.  For this problem we will use the function z = f(x,y) from Problem 2.

(a) Compute the derivative matrix Df.

(b) Compute the linear approximation L(x,y) for f(x,y) at the point (\pi/4, \pi/4, 0).

Problem 4.  Given the function F(x,y,z) = (x^2y - z, \cos(xyz)\,), compute the derivative matrix DF.

Problem 5.  (a) Given a function f:\mathbb{R}^2 \to \mathbb{R}, explain why its derivative matrix Df is a 1 \times 2 “row matrix.”

(b) If instead of arranging the partials of f(x,y) in a matrix we arrange them in a vector, then we do not call the resulting vector expression a “matrix.”  Instead we call it the gradient of f(x,y) and we notate it as

\displaystyle \text{the gradient of } f = \nabla f.

Compute the derivative matrix and compute the gradient, \nabla f, of the function f(x,y) = x^2+y^2.

(c ) Find al points where \nabla f = \vec{0}.

(d) What do the points in part (c) have to do with the graph of z = f(x,y)?


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