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(a) The graph of a position function of a car is shown, where $ s $ is measured in feet and $ t $ in seconds. Use it to graph the velocity and acceleration of the car. What is the acceleration at $ t = 10 $ seconds?

(b) Use the acceleration curve from part (a) to estimate the jerk at $ t = 10 $ seconds. What are the units for jerk?

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a) acceleration is 0 at $t=10$b) $\approx-10 \mathrm{ft} / \mathrm{sec}^{3}$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Missouri State University

Campbell University

Harvey Mudd College

Baylor University

Lectures

04:40

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.

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(a) The graph of a positio…

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The graph below shows the …

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Velocity of a car The grap…

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The position of a car as a…

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Car Acceleration The accel…

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The velocity of a car is g…

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Deceleration A car slows d…

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The changing velocity of a…

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The graph shows the positi…

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(a) Use the data in Proble…

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The velocity of a car, in …

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Acceleration The velocity …

this problem. Number fifty six of the Stuart Calculus, eighth edition, Section two point eight. Party, the graph of position function. The record has shown where s is measured in feet and tea in seconds. Use it to graph the velocity and acceleration of the car. What is acceleration At T equals ten seconds came. So we're going to look at this graph here That was given that we're going to pot. Ah, along the position functions of the position function is shown here in blue. We're also gonna plant velocity as well as the acceleration. And the way that we do that is by measuring certain tensions. So here we have a function position function, which is always increasing. So we're gonna have a Where we expect for the velocity function is that all of its residents are positive and we should be able to plot a function, Um, that is able to capture the behavior of this function. One notable point is to see what the slope of the tangent line is. Here At X equals ten aren t equals ten. So here we're just going to us to me best we can. A line that has a slope steep enough that it will be ten Shinde to dysfunction at exactly T equals ten. Maybe a little steeper. And we're just gonna estimate so it seems to be pretty tangent, and we want to find the slope of this. Well, we're just gonna bring it here and determine its appear. We see that it has an approximate rise of two hundred fifty over a run of about five seconds. Very rough estimate. Meaning that this slope, um, has meaning that this is, uh, two hundred and fifty two out of five, which is fifty our meters per second. So that would be the expected velocity. Um, for this function and t equals ten seconds. So we would expect a velocity function to reach this point here. Fifty meters for a second came and we're gonna take one more point before we let our function. Ah, we will expect the camera this this type of line here to be tension and we see that it is across this range here of of times. Uh, approximately time equals fifteen seconds. Twenty seconds. We see that this function are this tangent line is exactly tension at all those points. So this function. This position function increases its the slope of its tension lines up until ten. Part it where it is approximately the steepest. And we calculated that velocity at fifty, um, people second. And then over here we see that the velocity or the slope of the tension lines, I level out at a constant value. Proximately too. So we're taking the tangent line, and we're just going to measure the slope here at this point. We see there has arrived of about turning fifty feet and then a run of ten seconds. So turning fifty feet to every ten seconds gives us twenty five feet per second as the approximate velocity here at this region. Stone. With that information in mind, we're going to estimate the shape of the velocity graph and such. We want to reach speed of twenty five peeper. Second here at the end. We're going to test me just a little bit like this about the height that we want and that just ensure that no, there we meet this type of shape here. Okay? It's it's very subtle, but we're trying to achieve Is that the maximum way? Assume we see that this the maximum slope that we achieved here. Is that ten seconds. So this velocity function is increasing. What velocity? Until it reaches ten seconds where maxes out at fifty feet per second, which is what we calculated. And then afterwards it just decreases and speed The slopes of the tension lines over here decrease. And then it gets to be about twelve car, constant value of twenty five feet per second. So this is a rough sketch of the velocity graph the velocity function. Ah, and that seems pretty good test at the moment. Now we want to determine what acceleration function looks like, and we're going to be very approximate with this. We're going estimated, uh, quite a lunch. But what we want to capture is that this velocity function has a maximum here. T equals ten seconds, and then it also derivative. Ah, there's a local minimum or here because thie slopes level off one value. So the slopes of the tension lines for the velocity function should be their own meaning that acceleration should be zero. So if we were to use those points of reference and instead of green shooting, our Redford acceleration, right, I won't be able to apply it just like this, showing here a bit of her maximum for the acceleration right before ten is equal our ten seconds and then showing this tip here. This is more or less where we should expect again. It's very subtle, but this should be accurate given the magnitude, um of the acceleration. So what we're seeing here is that according to the velocity ground, the velocity graph is ah increasing up until it reaches maximum round here, a teak within seconds. Therefore, the derivative acceleration should be above the X axis. And then afterwards we see a slight decrease in the velocity graph, meaning that the function should be below the X axis. And then afterwards it levels off, meaning that they're too riveter. How were the acceleration levels off to around zero? It's very subtle, but the important part here is the acceleration function crosses the X axis at ten seconds. And this is exactly the information we need for party. We needed to graft the velocity, acceleration functions which we did and then answer What acceleration is a tense ten seconds. Well, here we are determined that it's in seconds. The position changes the position function has the largest change, which means that it has a large capacity. And if that's a large glass, it Ethan acceleration, it must have been zero at that point. So our answer here a A and R ten seconds is equal to zero. Parveen using acceleration curve in party to estimate the jerk at ten equals T equals ten seconds. What are the in its Frederick? So the jerk is the derivative of acceleration. So we need to use a, uh, slope the slip of attention. Really? So we're just going to come in is to me a tangent line and ten seconds for the acceleration. Raph going to be very, very not. Not super steep, But he cares what we're wanting to do. Maybe a little longer. And we have a tangent line, a line that we wanted to be tension right there for the ex celebration acceleration graph witches and ready. The tension line here has a slight negative slope. Um, we see that because it's the function is decreasing. The acceleration function is decreasing. So the jerk will have a negative value here at T equals seconds. And if we were to estimate exactly the slope of this crap of this line. It's very small across a Yeah. Ran, huh? Ten seconds. We see, like a difficult about Are you pretty good amounts? We'LL see a hundred. Therefore one hundred a decrease of one hundred feet To write in my changing Tisa T equals ten seconds. I think that our jerk will be ten a few per second. So we're going to answer that The jerk is ten feet Negative because it is a negative slope Negative ten feet per second cube. And this answers our second part. Water the units for trick. They are feet per second cube the unit sort position is feet that units for velocity is feet per second and the units for acceleration fee per second squared. Therefore the units of jerk which is the dirt of exploration and steeper second Cute. And this is our approximate answer for purple

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