Single Variable Calculus, Chapter 3, 3.7, Section 3.7, Problem 17
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Suppose that the mass of the part of a metal rod that lies between its left end and a point $x$ meters to the right is $3x^2$ kg. Find the linear density when $x$ is (a) 1m, (b) 2m, and (c) 3m. Where is the density highest? Slowest?
$ \begin{equation} \begin{aligned} \text{Linear Density } = \frac{dm}{dx} &= 3 \frac{d}{dx}(x^2)\\ \\ \frac{dm}{dx} &= 3(2x)\\ \\ \frac{dm}{dx} &= 6x \end{aligned} \end{equation} $
a.) when $x = 1m$,
$\displaystyle \frac{dm}{dx} = 6(1) = 6 \frac{\text{kg}}{\text{m}}$
b.) when $x = 2m$,
$\displaystyle \frac{dm}{dx} = 6(2) = 12 \frac{\text{kg}}{\text{m}}$
c.) when $x = 3m$,
$\displaystyle \frac{dm}{dx} = 6(3) = 18 \frac{\text{kg}}{\text{m}}$
Based from the values we obtain, the density is highest when $x = 3m$. On the other hand, the density is lowest at $x =1m$. It means that the density is highest at the right end of the rod while the density is lowest at the left end of the rod.
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