Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College average rate of change f of z minus f of x over z minus x or f of x plus h minus f of x all over h average rate of change is really just the slope we could think of it as the change of y's over change of x's 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 minus f of x over z minus x or the limit as h goes to zero of f of x plus h minus f of x over 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 gonna 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 gonna 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 ds/dt or sometimes that notation is d/dt of the function in terms of time or we could think of it as the limit as time changes going to zero of f of t plus delta t minus f of t all over delta t you could think of that h really being your delta t in this case we want out time interval to get smaller and smaller and smaller the average velocity is gonna 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 of t plus delta t minus f of 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 that it's talking about the slope so the velocity tells the direction of motion along with how fast the object is moving moving backwards it's gonna have a negative velocity moving forwards it's gonna have a positive velocity sometimes we wanna 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 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 squared s over dt squared sometimes it's written as d squared 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 and average velocity for a given time interval so it's gonna give us a given time interval and it wants us to find the displacement if we're talking about the displacement we wanna know how far it's change in the time so we're really talking about the delta s so where was the distance at s 3 minus the distance at s zero so the delta s if we stick in three we get 81 fourths minus 27 plus nine if we stick in zero we get zero plus zero minus zero or the displacement was nine-fourths in this case we're referring to it in meters so nine-fourths meters if we wanted the average velocity the average velocity is just the displacement over the change of time so nine-fourths divided by three or three-fourths meters per second the next part of this says find the body's 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 gonna use the power rule bring down the four so the fours cancel one less power t cubed bring down the three so negative three t squared bring down the two and the exponents to one less power so the velocity is t cubed minus three t squared plus two t but we're asking to find the speed so the speed is the absolute value of the velocity so v of three 27 minus 27 plus six we get six meters per second v of zero when we stick that in we're gonna see that we really just get zero meters per second now from there we're being asked to find the acceleration at the endpoints 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 squared minus six t plus two at the endpoints we're gonna have a of three so literally we just stick three in and we get 11 meters per second squared we stick zero in and we get two 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 is gonna change direction whenever the velocity is zero 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 zero we must be changing our direction so here we're gonna take that velocity formula up here and we're gonna set it equal to zero we're gonna then factor so if I pull out a t I get t times t squared minus three t plus two t times t minus one time t minus two this is a cubic equation so if we think about what a cubic equation's gonna look like we know we're gonna have a zero two at one and zero the leading coefficient was positive so our graph would look like this and if we wanna know where it's gonna be moving forwards and backwards if the y is positive in this graph we're gonna be moving forward so we're gonna be moving forward from zero to one where it's positive and also from two on in this case we're only looking at the interval zero to three so from two to three we're moving forward when the y was negative we're gonna be moving backwards so we're moving backwards from one to two Thank you and have a wonderful day