Exam 13: Introduction to Functions of Several Variables

The surface of a mountain is modeled by the equation $h ( x , y ) = 4000 - 0.005 x ^ { 2 } - 0.007 y ^ { 2 }$ . A mountain climber is at the point $( 400,600,4390 )$ . In what direction should the climber move in order to ascend at the greatest rate? Round all numerical values in your answer to one decimal place.

$\text { Find } \frac { \partial w } { \partial t } \text { using the appropriate Chain Rule for } w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \text { where }$ $x = 8 t \sin s , y = 8 t \cos s$ , and $z = 5 s t ^ { 2 }$ .

Assume a rule that is one of several methods used to determine the lumber yield of a log (in board-feet) in terms of its diameter d (in inches) and its length L (in feet). The number of Board-feet is $N ( d , L ) = \left( \frac { d - 2 } { 2 } \right) ^ { 2 }$ . Find the number of board-feet of lumber in a log 28 inches in Diameter and $10 \text { feet in length. }$

Find the total differential of the function $z = - \frac { 1 } { x ^ { 5 } + y ^ { 10 } }$