Let $z = f ( x , y )$ $x = r \cos \theta$

question 218

Essay

Let $z = f ( x , y )$ , $x = r \cos \theta$ , and $y = r \sin \theta$ .(a) Show that $| \nabla f | ^ { 2 } = \left( \frac { \delta f } { \delta r } \right) ^ { 2 } + \frac { 1 } { r ^ { 2 } } \left( \frac { \delta f } { \delta \theta } \right) ^ { 2 }$ (b) Let $z = \ln \left( x ^ { 2 } + y ^ { 2 } \right)$ compute $| \nabla f |$ by using the formula in part (a).

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Definitions:

Marginal Revenue

The augmented income from the sale of one more unit of a good or service.

Pure Competition

A market structure characterized by a large number of buyers and sellers, free entry and exit, and a homogeneous product, leading to price taking behavior.

Competitive Firm

A business that competes in a market where it has some influence on the prices of its products or services but faces competition from other firms.