Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Average rate of change F of Z -, F of X / Z -, X or F of X + H -, F of X all over H Average rate of change is really just the slope. We could think of it as the change of YS over change of X as if we want. The instantaneous rate of change is really just the first derivative. So the limit as Z approaches X of F of Z -, F of X / Z -, X. Or the limit is H goes to 0 of F of X + H -, F of X / H really just the first derivative. The velocity is the instantaneous velocity. The word instantaneous is important. So the instantaneous rate of change. Now if we're just referring to rate of change from here on out, we're going to assume it's an instantaneous rate of change. If they want it to be an average rate of change, we'll specify average. So the velocity, the same way it's going to be considered instantaneous velocity unless specified otherwise. It's the derivative of the position with respect to time. So if a body's position at time T is S, typical variable for distance S equals some function in terms of time. When the body's velocity at time T is, then the body's velocity at time T is V of T equals the derivative in respect to the distance or DSDT. Or sometimes that notation is DDT of the function in terms of time. Or we could think of it as the limit as time changes going to 0 of F of T plus delta t -, F of T all over delta T, you could think of that H really being your delta T. In this case, we want our time interval to get smaller and smaller and smaller. The average velocity is going to be the displacement in the distance over the travel time. So average velocity is basically a slope. Again, it's the change of the distance over the change of the time. Or F at T plus delta t -, F at T over delta T Remember, if it just says velocity, we're assuming instantaneous. It has to have the word average velocity for us to understand it's. I'm talking about the slope. So the velocity tells the direction of motion along with how fast the object is moving. Moving backwards, it's going to have a negative velocity. Moving forwards, it's going to have a positive velocity. Sometimes we want to think of the velocity as only being positive. So if we took the absolute value of velocity, we have a new thing called speed. Speed is always going to be the absolute value of velocity. So that equals the absolute value of the velocity in terms of time, which is also the first derivative of the distance in terms of time. Rate at which a body's velocity changes is the body's acceleration. It it measures how quickly the body picks up or loses picks up or loses speed. So the acceleration is just the derivative of the velocity in terms of time, or the derivative of the derivative of the derivative of the derivative of S written as d ^2 s over DT squared. Sometimes it's written d ^2 DT squared of S. So the acceleration is the derivative of the velocity, second derivative of distance. Jerk is a sudden change of acceleration. If you think about being in your car and all of a sudden it jerks forward, it's because there was a sudden change of acceleration or even jerks to a stop. So jerk is the derivative of acceleration in terms of time, which is the second derivative of velocity in terms of time, which is the third derivative of distance in terms of time. If we look at an example where we're given some equation for the distance and we're asked to find the body's displacement in the end average velocity for a given time interval. So it's going to give us a given time interval and it wants us to find the displacement. For talking about the displacement, we want to know how far it's changed in the time. So we're really talking about the delta S. So where was the distance at S3 minus the distance at S0? So the delta S, if we stick in three, we get 81 fourths -27 + 9. If we stick in zero, we get 0 + 0 - 0, or the displacement was 9 fourths. In this case, we're referring to it in meters, so 9 fourths meters if we wanted the average velocity. The average velocity is just the displacement over the change of time. So 9 fourths divided by three or three fourths meters per second. The next part of this says find the body speed and acceleration at the end points of the interval. So to find the velocity, it's just the first derivative. If we take this original distance formula, we're going to use the power rule. Bring down the four, so the fours cancel 1 less power t ^3. Bring down the three, so -3 T squared, bring down the two and the exponents to 1 less power. So the velocity is t ^3 -, 3 T squared plus 2T. But we're asking to find the speed. So the speed is the absolute value of the velocity. So at V of 327 -, 27 + 6, we get 6 meters per second, V of 0. When we stick that in, we're going to see that we really just get 0 meters per second. Now from there, we're being asked to find the acceleration at the end points. Well, the acceleration is the derivative of the velocity. So we come up here to the velocity equation and we take the derivative. We bring down the three, three t ^2 -, 6 T +2. At the end points we're going to have a of three. So literally we just stick 3IN and we get 11 meters per second squared. We stick 0IN and we get 2 meters per second squared. Then we're asked if ever, when, if ever, does the interval? When, if ever, during the interval, does the body change direction? Well, the body's going to change direction whenever the velocity is 0. Because remember, if the velocity is negative, we're going backwards, and if the velocity is positive, we're going forwards. So if the velocity is 0, we must be changing our direction. So here we're going to take that velocity formula up here, and we're going to set it equal to 0. We're going to then factor. So if I pull out ATI get t * t ^2 - 3 T plus two t * t - 1 * t - 2. This is a cubic equation. So if we think about what a cubic equation's going to look like, we know we're going to have a zero at 2 at one and 0 the leading coefficient was positive. So our graph would look like this. And if we want to know where it's going to be moving forwards and backwards, if the Y is positive in this graph, we're going to be moving forward. So we're going to be moving forward from zero to 1 where it's positive and also from 2 on in this case, we're only looking at the interval zero to three. So from 2:00 to 3:00, we're moving forward. When the Y was negative, we're going to be moving backwards. So we're moving backwards from 1:00 to 2:00. Thank you and have a wonderful day.