Newton's Method of Approximation

WARNING: these blogs puts the last blog on top of the page. I strongly suggest you start at the bottom with the first post on Calculus. This one is the fourth post on the subject and least friendly....lol. (Maybe my son or Uncle Mel can show me how to change the order of the posting ? That would be helpful)

Newton's Method of Approximation

This really does not have much value today because of the graphing calculator, but you will quickly learn that Calculus as we know it (the last 200 years) is pretty much obsolete. The graphing calculator does it ALL (except the reading of the problem and the setting up of the equations). Why we still do the Calculus by hand, i do not know. That is why there are FOUR different types of Calculus offered in universities today.

There is the Calculus for Math Majors (I would avoid this one unless you plan on being a mathematicians). This course starts off getting all bogged down with limits and proofs.

There is Calculus for the engineering majors. Again they do not waste too much time with proofs and go right into USING the derivative for solving physics problems.

There is Business Calculus which is my favorite. Again they do not waste too much time with proofs and go right into USING the derivative for solving business problems.

And there is Calculus for the Liberal Arts major. Usually called "Descriptive Calculus". It gives students an overview of Calculus and its uses for max and mins in graphs and word problems.

I strongly suggest that you take at least one of these Calculus. In today's world, many students are going on to graduate school. If you plan to apply to Law school or Medical school or a top notch MBA program, the admissions committee wants to see Calculus on your college transcript. It separated you from the "rest of the applicants". Even Wall Street financial institutes like to see Calculus on the transcript (by the way, they love hiring engineers).

Now the questions is why do we need to study Calculus ?

Well it started with Sputnik, the Russian satellite launched in the 1950's. The government asked the leading scientist why American was falling behind. The answer was we are not producing enough engineers. And why not ? Because most freshman fail Calculus when they come to college. Well what can we do about that ? Lets let the high school teachers teach them Calculus. And Calculus at the high school level was introduced. We have the students 5 days a week. And we can actually teach to a class of 30 or 40 students. At the university, they have 200 or 300 students in a lecture hall. Plus the college professors did not have the patients to show the algebra steps plus they got bogged down with limits and proofs.

Always remember learn math from a teacher who knows math, NOT from a mathematician they stuck in a classroom (sorry mathematicians....you guys are just too smart...both Einstein and Newton taught at the university, but had no students).

So you have the Russians to blame for having to take Calculus in high school or college. You might as well enjoy it while you are there and learn some neat stuff.
Who knows what the Chinese will force us to learn. Maybe Chinese !!

Ok getting back to Newton's Method of Approximation.

Lets say we want to find the square root of 2. (yea, yea i know we can just use our graping calculator...but assume we are stranded on some island without our calculator and our life depends on us finding an approximate value for the square root of 2). By the way the square root of 2 was very important to Pythagoras back in 800 BC. He was trying to find the length of the diameter of a square 1 by 1. Using his famous formula a^2 + b^2 = c^2, he was able to get the square root of 2. Of course back then they did not have algebra (which would not be invented until 1,000 AD by the Arabs and Hindu's). So since Pythagoras could not find the exact value of the square root of 2, he called it an unmeasurable number.

So we want to approximate the square root of 2.

We first must look at the function y = square root of 2...or y = x^(1/2).

We do know that the square root of 1 is equal to one...so we have at least one point (1,1) on the graph y = x^(1/2)

now what we want to do is find the ETL (equation of the tangent line....i told you to read the other blogs first....bottom of the page up !!)

the slope of the tangent line which is the derivative is y' = (1/2) x^(-1/2)

(remember the rule for derivative y = x^n, then y' = n x^(n-1) )

so if we plug in x = 1, we get slope = m = y' = (1/2) 1^(-1/2) = (1/2)

and remember the equation of the line y = mx + b

now we find b....by plugging in (1,1).... 1 = (1/2)(1) + b...therefore b = (1/2)

so the ETL at (1,1) on the graph is y = (1/2)x + (1/2)

now we are ready to use the ETL to approximate the square root of 2. we plug in 2 !!

y = (1/2) (2) + (1/2)...or the approx value of 2^(1/2) = 1.5

the calculator tells us the square root of 2 is 1.4....we are very close...(neat huh? yea, if you love math...lol)

Lets try another one...try square root of 3...and this time we have to go to the point (4,2) because we know that the square root of 4 is equal to 2.

remember the slope = m = y' = (1/2) x^(-1/2)...plugging in 4 we get m = (1/4)

and then we get the ETL y = (1/4) x + 1

therefore when we plug in 3, we get y = 1.75 which is our aprox. for 3^(1/2)

the calculator says square root of 3 is 1.73...again very close

WAY TO GO NEWTON !!! and now YOU can do it !!

COMMENTS
(Thanks for sharing your blogs with me but you have gotten over my head but I still appreciate you sending them....30 year algebra teacher and friend.)

(I really enjoy playing with these numbers. I have not done Calculus since college. I notice that if you go further out your approximations get better. Using pencil and paper really helped me follow your blog much easier. Keep them coming...20 year algebra teacher and friend.)