Equations Involving Derivatives

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AP Calculus BC › Equations Involving Derivatives

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The temperature profile of a system is given by

$T(x)=\frac{x^2}{12}+3x$

The heat flux is given by Fourier's law, $q=-k\frac{\mathrm{d} T}{\mathrm{d} x}$ , where k is a constant.

Find the heat flux for the given profile.

$-k(\frac{x}{6}+3)$

$k(\frac{x}{6}+3)$

$-k(\frac{x}{6}+3x)$

$-k(\frac{x}{12}+3)$

To find the heat flux, we must find the change in temperature with respect to x:

$\frac{\mathrm{d} T}{\mathrm{d} x}= \frac{x}{6}+3$

The derivative was found using the following rule:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Plugging this into Fourier's law, we get

$q=-k(\frac{x}{6}+3)$