Newton's Method , Differentials, Related Rates

Hi guys
we have Quiz on Nov 6th, Tuesday.

Newton's Method
The general method

More generally, we can try to generate approximate solutions to the equation using the same idea. Suppose that is some point which we suspect is near a solution. We can form the linear approximation at and solve the linear equation instead.
That is, we will call the solution to . In other words,

If our first guess was a good one, the approximate solution should be an even better approximation to the solution of . Once we have , we can repeat the process to obtain , the solution to the linear equation

Solving in the same way, we see that

Maybe now you see that we can repeat this process indefinitely: from , we generate and so on. If, after n steps, we have an approximate solution , then the next step is

Provided we have started with a good value for , this will produce approximate solutions to any degree of accuracy.


Dfferentials

Given a function we call dy and dx differentials and the relationship between them is given by

Note that if we are just given then the differentials are the df and dx and we compute them the same manner

Related Rates

There is 13ft of string beteen two people. The people holding the string.

One person(x) is moving left and another person (Y)is moving forward. (keeping the distance between people 13ft)

speed of Y person is 4ft/s

We can know that the length of x side is 5ft and length of y side is 12ft to use pythagorean theaorem.

2 2 2

x + y = 13

2x (dx/dt) + 2y (dy/dt) = 0

x(dx/dt) + y(dy/dt) = 0

x(dx/dt) + y*4 = 0

5(dx/dt) + 12*4 = 0

dx/dt = - 9.6