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The mass of the part of a metal rod that lies between its left end a point $ x $ meters to the right is $ 3x^2 kg. $ Find the linear density (see Example 2) when $ x $ is (a) 1 m, (b) 2 m, and (c) 3 m, Where is the density the highest? The lowest?

(a) $\rho(1)=6 \mathrm{kg} / \mathrm{m}$

(b) $\rho(2)=12 \mathrm{kg} / \mathrm{m}$

(c) $\rho(3)=18 \mathrm{kg} / \mathrm{m}$

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Campbell University

Oregon State University

Harvey Mudd College

Baylor University

here we have a problem about linear density, and we have an example in the book that describes this. And we learned that linear density is the rate of change of mass as a function of length. And we're told in this problem that the mass is three x squared. And so what we want to do is find its derivative d MD X, and that would be six X, and that would be the linear density. So for part A, we're finding the linear density for X equals one. And for part B for X equals two and for part C for X equal three all right, so D m d X for X equals three. Our X equals one would be six times one, so that would be six. And the units are kilograms per meter and D m d x for X equals two would be six times too. So that would be 12 kilograms per meter. And for X equals three, we have six times three, so we would have 18 kilograms per meter. So what we noticed here is that the highest values are farthest to the right, so the X value tells us how far we're going along the rod from left to right. So the highest values air farthest to the right and logically then lowest values would be the closest to the left.