## 17Calculus Directional Derivative Application - Temperature

##### 17Calculus

This page covers one application of gradients and directional derivatives, the temperature gradient.

Temperature

When the function $$f$$ is a temperature, the gradient can be interpreted as the change in temperature.

Practice

If we have a material where the heat flow (from hot to cold) is given by $$T = x^3 - 3xy^2$$, determine the direction of maximum decrease of temperature at the point $$(1,2)$$.

Problem Statement

If we have a material where the heat flow (from hot to cold) is given by $$T = x^3 - 3xy^2$$, determine the direction of maximum decrease of temperature at the point $$(1,2)$$.

Solution

### Dr Chris Tisdell - 802 video solution

video by Dr Chris Tisdell

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The temperature, in degrees Celsius, on the surface of a metal is given by $$T(x,y) = 20-4x^2-y^2$$, where $$x$$ and $$y$$ are measured in centimeters.
(a) In what direction, from $$(2,-3)$$, does the temperature increase most rapidly? What is this rate of increase?
(b) A heat-seeking particle is located at $$(2,-3)$$ on the plate. Find the path of the particle as it continuously moves in the direction of maximum temperature increase.

Problem Statement

The temperature, in degrees Celsius, on the surface of a metal is given by $$T(x,y) = 20-4x^2-y^2$$, where $$x$$ and $$y$$ are measured in centimeters.
(a) In what direction, from $$(2,-3)$$, does the temperature increase most rapidly? What is this rate of increase?
(b) A heat-seeking particle is located at $$(2,-3)$$ on the plate. Find the path of the particle as it continuously moves in the direction of maximum temperature increase.

Solution

### Thomas Wernau - 4334 video solution

video by Thomas Wernau

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