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The table shows the position of a motorcyclist after accelerating from rest.
(a) Find the average velocity for each time period: (i) $ [2, 4] $ (ii) $ [3, 4] $ (iii) $ [4, 5] $ (iv) $ [4, 6] $(b) Use the graph of $ s $ as a function of $ t $ to estimate the instantaneous velocity when $ t = 3 $.
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Calculus 1 / AB
Limits and Derivatives
The Tangent and Velocity Problems
University of Michigan - Ann Arbor
University of Nottingham
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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All right here we have a motorcyclist with some data on for each second shown in the chart with the position of the motorcyclist. And so our goal is to find the average velocity while actually instantaneous velocity. Um That's our goal. What's the instantaneous velocity of the motorcyclists at three seconds. However, we're going to use the data and find average velocities for different time periods and then use that to predict are instantaneous philosophy At three circles, basically approximately The velocity at three seconds. Make that a little bit more even. Alright, that's our goal. Okay, so notice we have different time periods. We have 2-4, 3 to 4, 4-5 and 4-6. And we're going to find the average velocity. So let's figure out what's the formula we're going to use, average velocity is displacement over time. So it's basically going to be S at T two minus s at T. Once. That's the displacement between two times And then T 2 -11. So I'll show you for one of these. Then let's do the time two and 4. So the average then Would be s at four -S at two Over for -2. So we plug into our calculator. Well let's first actually use the data and then we can plug in our calculator. Okay, so SF four is 79.2 minus. Uh Oops, I put a support both times. I got to fix that. It's s a four minus s a two. There we go. Okay so it's a delta position over delta time. So we'll get sf 4 79.2, we'll subtract S. A two which is 20.6 and then we divide by two and that will be in meters per second. And when we plug that in our calculator we get 29.3 m/s. So this will be then our average from 2-4 is 29.3 m/s. Okay, so from 3 to 4 it would be the same process or should at least one more, So V. F. For the second one will change color. So let's do one more and then you get the idea I think. So let's do um the F for the second interval. So that's s of four minus s. Of three. All over 4 -3. So that would be 79.2 -46.5. All over one. So that comes out to be 32.7 m/s, so we can put that in our chart. Um Okay, so the same process plugging into the average value formula for the different intervals um will give us then let's get a different color here. um so if we went from 4 to 5 then we get 45.6 meters per second. And if we go 4-6 it's um Fort Wolf, let's get one more color. Just for a variety of one more, let's try. How about brown. Okay, so that gives us, When we use the formula we get 48 75 m/s. Alright, so now we're ready to look at um approximating VF three. And let's take a look at the data here Since I want via three. Um one way to do a best approximation would be To approximate the average from 2 to 3 and then from 3 to 4 and then take the average of those two average velocities. So we're going to do that. But I noticed that though I have The 3-4 interval already, I don't have the 2-3. So let's do that one real quick 2-3. Then I'll just show the work down here. It'll be rise over run. So 46 .5 -20.6 all over 3 -2 meters per second. And that gives us a value of 25 0.9 meters per second. So then to get via three, I'm going to take the average of the VF from 2 to 3. And the VF From 3 to 4 scenarios. They went a little bit to the right and a little bit to the left and I'm going to average those two. So that will give us then um Let's see. V 2 to 3 was 25.9 m/s Plus the 32 .7 m/s divide by two. And that gives us 29.3 meters per second. And I'll still put squiggles because really we are approximating. But that's about the best we can do with this data Is approximate that the instantaneous velocity at three seconds is 29.3. Alright, I hope that helps have a wonderful day.
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