ETL - Equation of the Tangent Line

So what is the first thing we are going to do with this new concept, the derivative. Well since the derivative is the slope of the tangent line, why don't we find the equation of this tangent line (ETL).

A historical note first. Calculus was discovered or refined by two mean. One was British (the apple fell on his head), Newton and the other was German, Leibniz.
(A funny note, if Germany had won WWII, we would probably be studying Leibniz's Calculus, but since the British and the Americans won WWII, we studying Newton's calculus.) Children in Germany study Leibniz's calculus. I am sure other civilizations like the Chinese and American Indians (South and North) were developing their own mathematical ideas, but once Christopher Columbus sailed the ocean blues, the world got a lot smaller...or like they like to say these days, Flatter.

Another interest historical note about Newton. At the time of many of his discoveries, Europe was experiencing a plague and in those days people left the city and went out to their estates in the country. The university was closed for 6 months. It just happened that Newton's father had an extensive library and Newton had time to kill. In that time he developed some of his greatest ideas, including the new math called Calculus.

This phenomenon occurs many time throughout history. Today our greatest mind is Steven Hawkins who is paralyzed from the neck down. All he can move is his finger. But thanks to computers, he has been able to write several books. The most famous being the Short History of Time (which everyone has on their book shelves, but no one understands including yours truly...i have tried three or four times). But the idea is that Hawkins has his mind free from every day duties to just think. One of his big ideas is the Big Bang theory of the Universe. And his final conclusion is that there is a God out there.

Another person in history who was confined was Galileo. He was put under house arrest (not for his ideas of the sun being the center which was a concept accepted long ago by most learned men of Europe including the Pope) because he published a book after being ORDERED by the Church NOT to publish.

So Galileo had a lot of time on his hands to observe the moons of Jupiter. And he was visited by other great thinkers of the time and they developed many of the ideas of the heavens that we follow today.

The Athenian were the first (we know of) who understood that men need leisure time in order to think. They needed shelter and food provide. Today we have great univeristies like Harvard where half of their faculty do not teach anymore in the classrooms, but instead do research with grad students. MIT is that was too. As is Stanford and other great universities. It helps to have great endowments from generous people to help pay for these men and women, shelter and food.

Now getting back to my ETL (Equation of the Tangent Line).

We know the equation of the line is y = mx + b.

And now we know that m = slope = the derivative !!

Going back to our original equation y = x^2, remember that the derivative is y' = 2x.

Therefore the slope = m = y' = 2x.

(By the way, Newton used y' for the derivative. Leibniz used dy/dx. The dy/dx is has more practical uses in the long run, but here we will use y'. In your calculator TI-84 the notation under 2nd function/Calc/ is dy/dx.)

Now we want to know at WHAT POINT are we looking for this tangent line. Lets used the point (1,1).

Since the slope = y' = 2x, when we plug in one for x, we get m = 2 !!

We know the equation of any line is y = mx + b. So now we find b, by plugging in the point (1,1).

So 1 = (2)(1) + b...or 1 = 2 + b...subtract 2 from both sides and we get b = -1.

And therefore the ETL is y = 2x - 1.

Time for pencil and paper. Draw the graph y = x^2 and draw the line y = 2x - 1.

(or use your child's graphing calculator.)

Now you see the parabola y = x^2 and the ETL y = 2x - 1.

And that is CALCULUS !!