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17Calculus Directional Derivative Application - Temperature

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Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

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.

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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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