Total distance traveled with derivatives

the position of a particle moving along a number line is given by s of T is equal to two-thirds T to the third minus 6t squared plus 10t for T is greater than or equal to 0 where T is time in seconds the particle moves both left and right in the first six seconds what is the total distance traveled by the particle four zero is less than or equal to T is less than or equal to six so let's just remind ourselves what they mean by total distance if I were to say start there and if I were to move three units to the right and then I were to move four units to the left and four units to the left I don't say negative four to show that I'm moving to the left then my total distance right over here is the total distance right over here is seven three to the right and four to the left even though my position right over here is going to be negative one so my position right over here is negative one or you could say my net my net distance or you could say my displacement is negative one I'm one to the left of where I started the total distance is seven so now that we've clarified that I encourage you to now pause this video and try to answer the question what is the total distance traveled by the particle in these first six seconds so we the the easiest way I can think of addressing this is to think about well what is this thing moving to the right and when is it moving to the left it's going to be moving to the right when the velocity is positive and it's going to be moving to the left when the velocity is negative so this really boils down to thinking about when is the velocity positive or negative and to think about that let's actually graph the velocity function or make a rough sketch of it so this is the position function the velocity function is going to be the derivative of the position function with respect to time so derivative of two thirds t to the third is going to be two T squared and then we have minus 12t plus ten and so let's just try to graph this let's just try to graph this so it's going to be an upward-opening parabola this is clearly a quadratic and the coefficient on the second degree term on the t squared term is a positive number so it's going to be an upward-opening parabola is going to look something like this and we're assuming that it switches directions so it's going to be positive some of the time a negative for some of the time so it should intersect the T it should intersect the T axis where it's negative the function is going to be negative in that interval and it's going to be positive outside of that interval so the easiest thing I can think of doing is let's try to find what the zeros are and then we could draw this upward opening parabola so to find its zeros let's just set this thing equal to 0 so we get 2t squared minus 12t plus 10 is equal to zero divide both sides by two just to get rid of this to make this leading coefficient of one we get T squared minus 6t plus five is equal to zero that made a lot easier to factor this can be factored into t minus one times t minus 5 negative 1 times negative 5 is 5 negative 1 plus negative 5 is negative 6 this is equal to 0 so this left-hand side of the expression is going to be equal to 0 if either one of these things is equal to 0 take the product of two things equaling 0 well you get 0 if either one of them is 0 so either T is equal to 1 or T is equal to 5 so now let's graph it so if I let's draw our axes so that's my I could say that's my velocity axis and let me draw them we only care for positive values of time so let's draw it something like this positive time and let's see let's take that 1 2 3 4 5 keep going so this is T equals 1 this is T is equal to 5 this is our T axis and let's graph it so it's going to be an upward-opening parabola it's going to intersect both of these points and so its vertex is going to be when T is equal to 3 right in between those points so it's going to look something like this it's going to look something like this that's the only way to make an upward-opening parabola that intersects the T axis at both of these points at both of those points so it'll go like that and it'll go like this it'll intersect when T equals 0 we actually can figure out when T equals 0 our velocity is 10 so this the V intercept we could say is 10 right over here so that's what it looks like so we see that the velocity is positive we see that the velocity is positive for time between zero and one and it's also positive for time is greater than five seconds and we see that our velocity is negative or that we're moving to the left we're moving to the left between one and five seconds the velocity is below the T axis right over here it is a negative so let's just think about what our position is at each of these points at time zero at time one at time five and what we care about times six and two times six and then think about how what the distance it would have had to travel between those times so let's think about it so let's make a little table here let's see what make it look make a little table so this is time and this is our position at that time so we care about time zero time one time five seconds and times six seconds so at zero seconds we know that our position is zero s of zero is a zero at one second this is going to be 2/3 minus six plus 10 so it's going to be four and two thirds so I'll write down four and two thirds at five seconds let's see it's two thirds times I'm gonna have to write this one down so this is going to be two thirds times 125 that's the same thing as 250 over three that's the same thing as 250 over three which is the same thing let's see 250 over three that's the same eighty-three times three is 249 so this is two four this is 83 and 1/3 so this is 83 and 1/3 that's this first term minus six times -6 times 25 minus 6 times 25 so that's going to be minus 150 - 150 plus 10 times 5 so plus 50 so this is going to simplify minus 150 plus 50 that's going to be minus 100 minus 180 3 and 1/3 mine 100 that's going to be negative 16 and 2/3 so negative 16 and 2/3 is this position after 5 seconds and then at the 6 at 6 seconds it's going to be 2/3 times 6 to the 3rd let's write this one down 2/3 times 6 to the 3rd 2/3 times 6 to the 3rd minus 6 times 6 squared well that's just going to be that's going to be that's going to be minus 6 to the 3rd again 6 times 6 squared plus 60 and let's see how can we simplify this right over here well we could this part right over here we can rewrite as we could factor out a 6 to the third this is 6 to the 3rd times 2/3 minus 1 plus 60 scroll down a little bit get some more space so there's going to be 6 to the 3rd times negative 1/3 plus 60 and let's see this is 6 let's write it this way this is going to be 6 squared times 6 times negative 1/3 plus 60 this right over here is negative 2 so it's negative 2 times 36 this is negative 72 plus 60 so this is going to be negative 12 so this is going to be negative 12 right over there so now we just think how far did it travel well it starts traveling to the right it's going to travel to the right 4 and 2/3 so let's write this down so we're going to have 4 and 2/3 and then it's going to travel to the left let's see to go from 4 and 2/3 to negative 16 and 2/3 that means you traveled 4 and 2/3 again you traveled 4 and 2/3 to the left and then you travel another 16 and 2/3 to the left another 16 and 2/3 left just a reminder we're 4 and 2/3 to the right now we have to go four and 2/3 to the left get it back to the origin then we have to go 16 and 2/3 again to the left so that's why this move from here to here is going to be 4 and 2/3 to the left followed by 16 and 2/3 to the left another way to think about it the difference between these two points is is what it's going to be 4 and 2/3 plus 16 and 2/3 if you do 4 and 2/3 minus negative 16 and 2/3 you're going to add that's the same thing as 4 and 2/3 plus 16 and 2/3 and then and then to go from negative 16 and 2/3 to negative 12 that means you went another 4 and 2/3 now to the right so now this is 4 and 2/3 now you're moving 4 and 2/3 to the right and so we just have to add up all of these we just have to add up all of these values so what is this going to be so this is going to be 2/3 times 4 which is so the this part of it right over here the fraction part of it 2/3 times 4 is 8 over 3 and let's see 4 plus 4 plus 16 plus 4 is 28 so 28 and 8/3 it's a very strange way to write it because 8/3 is the same thing as 8/3 is the same thing as 2 and 2/3 so 28 plus 2 and 2/3 is 30 and 2/3 so the total distance traveled over those six seconds is 30 and 2 30 and 2/3 units