Question 3.9.6: Density of a Thin Rod Suppose that the mass of the first x m...

From (9.1), we have

\rho\left(x_1\right)=\lim _{x \rightarrow x_1} \frac{f(x)-f\left(x_1\right)}{x-x_1}=f^{\prime}\left(x_1\right),

\rho(x)=f^{\prime}(x)=\frac{1}{2 \sqrt{2 x}}(2)=\frac{1}{\sqrt{2 x}} .

Thus, \rho(2)=1 / \sqrt{4}=1 / 2 \text { and } \rho(8)=1 / \sqrt{16}=1 / 4. Notice that this says that the rod is nonhomogeneous (i.e., the mass density in the rod is not constant). Specifically, we have that the rod is less dense at x = 8 than at x = 2.