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Graphs of the velocity functions of two particles are shown, where $t$ is measured in seconds. When is each particle speeding up? When is it slowing down? Explain.

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00:58

Felicia Sanders

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 7

Rates of Change in the Natural and Social Sciences

Derivatives

Differentiation

Harvey Mudd College

Baylor University

Idaho State University

Boston College

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.

01:44

Graphs of the position fun…

03:23

Graphs of the $ position $…

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15:27

02:59

Graphs of the velocity fun…

0:00

For the particle described…

01:34

The graphs in Figure 11 re…

01:35

The graphs of three positi…

02:00

$s(t)$ is a position funct…

here we have the graph of velocity function, and we're going to figure out when the particle is speeding up and when it's slowing down. So what we need to know is that a particle is speeding up when the velocity and the acceleration have the same sign. So that means they could be both positive or both negative. And the particle is slowing down when the velocity and the acceleration have different signs. So one positive one negative. So let's take a look at the graph and let's break it down into the changing, um, intervals. So it looks like it's doing something on the interval from 0 to 1. And then it's doing something different on the interval from 1 to 2. And then it's doing something different on the interval from 2 to 3. So let's analyze the velocity and the acceleration on each of those intervals. Okay, so when the graph is above the X axis, the velocity is positive. So we see that for the first interval, the velocities positive and for the second interval of last year's positive, When the graph is below the X axis, the velocity is negative. So from 2 to 3 of the velocity is negative. When the velocity graph is increasing, the acceleration is positive. So we can see the velocity graph increasing from 0 to 1. And when the velocity graph is decreasing, the acceleration is negative. So we can see that happening from 1 to 2 and from 2 to 3. Okay, Notice that we have the same sign from time zero to time one and the same sign from time 32 time from time to time. Three. So that means that the particle is speeding up for those times speeding up 01 and 2 to 3 and then notice that we have different signs on the interval from 1 to 2. So the particle is slowing down from the on the interval from 1 to 2. Okay, we can do part B the same way. And we're interested in times from 0 to 4. So let's break this down. Something's happening from time 01 Something else is happening from 1 to 2. Something else is happening from 2 to 3, and something else is happening from 3 to 4. Now we want to take a look at what's happening to the velocity and what's happening to the acceleration for each interval. So the velocity is positive from time zero to time. Three because the graph is above the X axis. So we have positive, positive, positive, negative, and the acceleration is positive whenever the velocity is increasing, so that would be from 1 to 2 and the velocity. The acceleration is negative whenever the velocity is decreasing. So that would be from 0 to 1 from 2 to 3 and from 3 to 4. All right, so let's look for where we have the same sign. We have the same sign from 1 to 2 and 3 to 4, so that means it's speeding up on those intervals and we have opposite signs from 0 to 1 and from 2 to 3, so it's slowing down on those intervals.

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