Monday, November 19, 2007

Tangent Approximation and Newton`s Method

Hi everyone! Its Mark here for today`s scribe.

We started today`s class by finding an approximate value of the square root of 37, without using our calculators to get the answer. The burning question is , how are we going to do that? We all know that its some where between the range of 6-6.1 since the square root of 36 is 6, but that is not a good approximation. Here comes the tangent approximation to the rescue. Our plan is to use the tangent line approximation of f(x)=square root of 36 @ x=36. First we need to graph the function, to see whether our approximate will be over or under the real value. If the function is concave up at that given point then our approximation will be over.If the function is concave down at that given point then our approximation will be under the real value. The graph of the function can be seen on the first slide. Second, find the derivative of the function @ x=36. After that create the tangent line at that given point. Then substitute x=37. We can see that our answer is approximately 0.006 larger than the actual value. The operations involve can be seen on the second slide.

We can therefore make a general rule that is true to every function to approximate any given value. The rule can be seen on the third slide.

The second thing we talked about in class is Newton's Method. We started our discussion about this method by going on a side story about Newton and Leibniz feud. After that, we went on describe Newton's method. His method is used for approximating zeros or roots of differentiable functions. Well, the big question is, how does it work? First, it starts with an initial guess, which is reasonably close to the true root, then the function is approximated by using the tangent line , and one computes the x-intercept of this tangent line . This x-intercept will typically be a better approximation to the function's root than the original guess. This process can be done over and over again until the root is found, this process is called iteration. In general, we can apply the formula that can be seen on the fifth slide to find a better approximation.

The third thing we talked about are the scenarios where Newton's method may fail. His method may fail if it produces a derivative of 0 or undefined. Also, it may fail if the steps just continues with the same set of values. Lastly, it wont work if the results diverge away from the root that you are interested in.

If you are feeling a little confused by this new concept, you can check this animation out.

That's it for today's scribe. Our home work for tonight is Exercise 4.7 (all the odd numbered questions). The next scribe will be Van.

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