Derivatives - what are they - falling sphere example

Mark Misin
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Aerospace & Robotics Engineer
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INTUITION MATTERS! - Applied Calculus for Engineers-Complete

Calculus + Engineering + PID: Functions, Limits, Derivatives, Vectors, differential equations, integrals: BEST CALCULUS

34:53:37 of on-demand video • Updated October 2020

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English [Auto] Welcome back in this lecture we are finally going to start discussing derivatives. What are derivatives then what are they useful for in this lecture. I'm going to make the concept of derivatives as intuitive as it possibly can for you. I will really pay a lot of attention to details and and just go with the lecture watch my videos listen to what I'm saying and pay attention to the examples and I promise you after this lecture you will really truly understand what derivatives are. OK. So I will start with this example. Let's say that this is a corner of your house here and. And you have some kind of beam that is put here on the corner of the house and then from that beam you have some kind of spheres some kind of object that has some kind of mass or some kind of metallic sphere. And now here you have a chain that holds the sphere in place. But then something happens and that the chain breaks down. And then this sphere will starts falling down and it falls down faster and faster until it reaches the ground. OK. So we know that the initial height of the sphere is 45 meters. So this is this distance here. And the final height of the sphere is when it reaches the ground. So how what would be the relationship of height with respect to time. The function would be the height as a function of time equals the initial height. Which is a constant which 45 metres minus five times square miles five times Times Square and it's negative because you can see from this picture that as the sphere falls down the altitude of the sphere decreases. So the function has to decrease. So I can rewrite it like this. Age is a function of t equals 45 minus five times the square. OK. So how would we find when the when the sphere touches the ground in how much time the sphere would touch the ground. Well we know that the final height would be zero meters right. So the final height and time final and we can just say that the final height happens at time. Final equals zero. So we can take this equation and equate it to zero. So it's 45 minus five t squared equals zero. This would be T.F. and then we would have five T.F. square equals 45 T.F. square it equals 45 divided by five would be nine seconds squared because it's Times Square right. The units would be second squared and then T.F. would be square root of 9 would be three seconds. So in three seconds the sphere would reach the ground. But look let's grab this information. So I have the access here and on the X-axis I have time. And then the y axis I have the altitude and the sphere. Now what would be the height of the sphere at time because one second or we just take this function and do we say altitude at time because one would be 45 minus five times one squared would be 40 meters. And then at two the altitude at that time equals two seconds it would be 45 minus five times to square it would be 45 minus 20 25 meters. OK let's go that. Time equals one we would have we were 40 and time equals two we would have 25 and now we can graph like this. OK. Well you can see that essentially what we are having here. We are having an upside down parabola a square root function something like this. And we are considering this part here and this the square function or the parable function would be y equals a x squared plus B and B here would be 45 and a would be minus 5. So you can see 45 with B B and minus 5 would be a. In our airplane example in our very first airplane example that we had we had a function that the map distance and it showed its relationship to time. So we we had a function like this one time here and we had distances here and then we had a straight line function. Distance equals a hundred times time and we measured how much this has changed with respect to time by first of all taking this into a whole new mine is old. Remember Delta the new mine is old. And then we divided by the delta T which was new and is old. That's how we got the change of function. Now what can we do here though. Well we could do the same thing. We can just take one interval which would be Delta H and we could take the other interval which would be delta time. Right. And then or we could just write it down like this. We could say like Delta h over delta time equals new minus old. So the new height would be zero. Minus the old height which would be forty five. And we can divide it by by the new time which is three and the all time which is zero. And you will get minus 15 meters per second and you can graph it like this. So you would have one point here and another point here and then you could just Craford like this. So it would be a straight line graph. Now according to this red graph this red straight line function the sphere flies down at the at the speed of 15 meters per second. However is it really true. Is it really what happens. Does it really fall down with 50 meters per second at the concentrate. Well no it's not because the function here. The blackline it's not a straight line function. So in fact this red line represents the average change over the interval of three seconds. So we take an average US average change of of the height with respect to time over the interval of three seconds. However you can have as many average changes as you want because you can take any interval you want. Let me give you an example I could take the interval from zero to 1. So I would have something like this would have this then my Delta h over that t would be where it would be Delta h over delta T. And then I'm just going to continue riding it over here. OK. So Delta edge over delta-T over over the time interval of one second from zero to 1 and these are connected it would be new. Mine is old so it would be 40 minus 45 and the time interval would be 1 minus zero. And it would be minus five meters per second. We can also take the other interval which is one over to take this interval and we can take this change in altitude. So here the Delta h over delta T. Would be 25 minus 40 to minus one would be minus 15 meters per second. And notice this is the same thing like the initial one. So here the average change corresponds. So if I draw a line here then in fact this line the straight line here would be parallel to the other straight line. So they see this straight line here is parallel to this one here. In this case the straight line would be different. It would be more flat and you can see from the numbers that change is smaller it's closer to zero. And of course I can take another interval which is this one. And here I can also take the interval Delta H. And then I can divide it by Delta T. So now I'm taking the interval from two to three and it would be zero minus 25 over three minus two and it would be minus 25 meters per second which is a greater change. And you can see it from this straight line. If I grab the straight line it would be let's say more vertical. So you can see that I can take whatever interval I want and depending on the interval that I take in time I can take for example from 1.5 to 1.6 so whatever interval I take the average change of altitude with respect to time or the average speed. If I had different variables here like the volume and mass then it would be the average change of mass with respect to volume and then it wouldn't be speed it would be the change of mass with respect to volume. But in this case since we have distances and times then the change would be speed. Naturally I could have an infinite amount of Avars change depending on the interval that I take.