U1S2V03 Introduction to tangent lines.txt

#
# File:   content-mit-18-01-1x-captions/U1S2V03 Introduction to tangent lines.txt
#
# Captions for MITx 18.01.1x module [XvuHw1a1F6o]
#
# This file has 36 caption lines.
#
# Do not add or delete any lines.  If there is text missing at the end, please add it to the last line.
#
#----------------------------------------

Today we're going to start with a geometry problem.
So at its outset, it doesn't seem to have anything
to do with derivatives.
I'm just going to start with the graph of some function y
equals f of x.
And what I want to know is, what is this tangent line
that I somehow intuitively know how
to draw through my graph above a point x equals a?
Let's be more specific about what
it is that I'm asking in this geometry problem.
So in order to understand a tangent line, first of all this
is a line.
And in order to understand a line,
it's enough to know a point on that line
and the slope of that line.
Now we already know a point on this line
because we specified that it went through our graph
above a point x equals a.
That means a point on this line is the point a, f of a.
Then we can write the equation for this tangent line
as y minus f of a is equal to m times the quantity x minus a.
Now we have a really simple problem.
What we want to find is m, the slope of this line.
And this is exactly where calculus
is going to come into play.
What we're going to discover in this sequence is
that m is equal to f prime of a, the derivative of this function
at the point a.
How did this happen?
How did our simple geometry problem
become a calculus problem?
Well, that is what we're going to figure out.
So first I want you to go ahead and do a few problems,
thinking about the tangent line and we'll be right back.