The tower of Pizza is 55.86 meters tall and we start our stop watch right when, say, Galileo lets go of the ball-thus, at \(t=0\), accounting for Galileo's height, the ball's initial height of \(f(0)\) will be, roughly, \(f(0)=60m\) as shown in the graph in Figure 1. The function \(f(t)\) is represented by the red line (a parabola, of course) in Figure 1. For now the answer to, why a parabola?, will be a mystery which we'll address in a few separate lessons, each with a different perspective. If you are unfamiliar with physics, that is ok. Why the empirically correct function is a parabola was first answered by Newton. For now, I'll just tell you that the function \(f(t)\)-how far the ball moved away after \(t\) seconds-was experimentally determined by Galileo to be a parabola. It would be very difficult to determine how far the ball moved away from us and how fast it is going after \(t\) seconds with no knowledge of physics or without doing a tedious experiment. We know from everyday experience that if I say dropped a ball from the top of the tower of Pizza, as it descended towards the Earth's surface it would keep picking up speed as time flowed forward. The main point is that going from one function to another when \(f'(x)\) is changing (which, without differential and integral calculus, would be impossible) involves so many important problems which would be unsolvable without calculus. how fast someone grows when they hit a growth spurt from 12-14 years old) time, and distance and speed as functions of time time, and speed and acceleration as functions of time and all the other problems in your calculus textbook. height of a person when \(t\) years old), and the growth rate of that something as a function of time (i.e. Professor Strang (video below) explains, through many examples, how \(x\), \(f(x)\), and \(f'(x)\) are very general and can be 'many things': they can represent time, the height of something as a function of time (i.e. Single-variable calculus, which is basically a cook book or calculation procedure, is needed whenever \(f'(x)\) is changing. But if \(f'(x)\) is changing, then you have to do calculation procedures which we call differential calculus and integral calculus to go from \(f(x)\) to \(f'(x)\) and from \(f'(x)\) to \(f(x)\), respectively. You already know how to do this if \(f'(x)\) is a constant: just use algebra. Single-variable calculus is the study of how you go from \(f(x)\) to \(f'(x)\), and vice versa.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |