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(a) A particle starts by moving to the right along a horizontal line; the graph of its position function is shown in the figure. When is the particle moving to the right? Moving to the left? Standing still?
(b) Draw a graph of the velocity function.
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(a) The particle is moving to the right when $s$ is increasing; that is, on the intervals (0,1) and $(4,6) .$ The particle is moving to the left when $s$ is decreasing; that is, on the interval $(2,3) .$ The particle is standing still when $s$ is constant; that is, on the intervals (1,2) and (3,4). (b) Velocity function graph is slope of the position vs time graph. The velocity is 4 m/s on (0,1); 0 m/s on (1,2) and (3,4); -3 m/s on (2,3); 1 m/s on (4,6)
Calculus 1 / AB
Limits and Derivatives
Derivatives and Rates of Change
October 26, 2020
Daniel J., thanks this was super helpful.
This will help alot with my midterm
Harvey Mudd College
University of Michigan - Ann Arbor
University of Nottingham
Idaho State University
In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.
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.
(a) A particle starts by m…
particle starts by moving …
A particle is moving along…
Particle Motion A particle…
A particle $P$ moves on th…
The graph of the position …
The accompanying figure sh…
all right here we have a particle moving um and on a axis and notice that it is um in red we can see the position of the particle with time. So as it moves along the axis. Um Okay so basically we need to study this graph uh and we're gonna look at trying to figure out when the particle is moving right, moving left or standing still and then draw velocity graph. So first of all to move right, that is when velocity is greater than zero. So we need velocity to be greater than zero and velocity is a slope. So basically we need to slow the position graph to be positive. I'll just write slow posit maybe that's easier. Um I'll just say positive slope of S S. Graph. Okay, so if I look at the graph I see a positive slope from 0 to 1. So that's one of our intervals. I'll write it down here. So zero 21 and I think we're doing them as open intervals. Let's do that. So we are moving to the right Between zero and 1 and I'll kind of market you can see that that's this here. We're increasing. Um And we're also increasing from 4 to 6. Right, the slope is positive. So 01 and for six those would be the two intervals where we are moving right, make a little division here so we can see it. Okay good. All right moving left is the opposite case. That's going to be when Our velocity is less than zero which we have a negative slope uh on our espera and we'll make little division there. Okay, so let's look at when we have a negative slope, it's really just this interval here where we're going to decreasing our function is decreasing. So we'll go from 2-3. So just the one Interval to the three and standing still is when you have velocity is zero, Or you have a zero slope of your position graph. So um we see two different intervals for that Between one and 2. It's a nice horizontal line In between three and 4 and wonderful. So there we did it. So I'll kind of put a box around that's just kind of figuring out moving right, moving left and standing still. Okay, so finally let's figure out our velocity graph. And remember velocity is the slope of our position graph. So between zero and one notice we can do rise over run, we get four up over one, so we'll be at four and we're at four during the whole interval, 1-2, sorry, from 0 to 1 and 1 to 2, we have zero velocity, right? We're standing still. So this is going to go all the way down to zero, make that look a little bit better. They'll use little straight line marks, let's see if they can make little straight lines. Okay, so zero uh from 0 to 1, we're at four, and then from 1 to 2 we have zero, let me change colors, You can see some kind of contrast. There we go. There, there there. Okay, I guess you can kind of see it. Okay, There we go. Okay. And then so 1-2 at zero velocity. Then we have a negative velocity. It looks like we're going down 3/1, so -3. So we're going to go down to -3 here and then back down to zero. So we have kind of like that. Um and then uh they were going positive soap again um looks like we're going up to over two, so so per one, so like this and it might have been better. I think I'm just going to draw freehand because that's kind of looking bad. Alright, so let's fix that. So -3 there, zero slope and the slope of positive one rise over run. Okay, It looks like we did it and accomplished our task. So um hopefully that all made sense and have a wonderful day. See you next time.
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