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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?

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(a) $\rho(1)=6 \mathrm{kg} / \mathrm{m}$(b) $\rho(2)=12 \mathrm{kg} / \mathrm{m}$(c) $\rho(3)=18 \mathrm{kg} / \mathrm{m}$

00:32

Amrita Bhasin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 7

Rates of Change in the Natural and Social Sciences

Derivatives

Differentiation

Campbell University

Baylor University

University of Michigan - Ann Arbor

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

02:48

The mass of the part of a …

01:41

01:25

A rod of length 2 meters a…

01:56

A rod of linear density $\…

03:31

The mass of the first $x$ …

02:15

The linear density of a 3 …

02:30

01:31

01:18

A rod of length 3 meters w…

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.

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