THE TANGENT & SLOPE OF SECANT LINES

Tangents are found on every continuous curve. A parabola, like a circle, is a continuous curve.

In a simple sense your teacher is trying to incorporate several things that you already know. You are familiar with the slope intercept equation of a line right? y=mx+b.

If I look at an analytic form of the tangent trigonometric function then tan is the rise over the run of a triangle, which interestingly enough, is the same as the slope of a line. Connecting these factoids then a line that touches a continuous curve at a single point is the tangent line.

Now how do you find this tangent line on the parabola?

Think about how to find a line from two points. One point you know, its (1,1) and it's on the parabola y=x^2. Now look for another point that is slightly off of (1,1) sat x=1+0.1, and y=(1.1)^2. This gives (1.1, 1.21).

The slope of this line is then m=(1.21-1)/(1.1-1)

Now have the difference between x=1 and 1.1. X=1.05 and y=1.1025 & m=(1.1025-1)/(1.05-1)

Now think about a tangent, it's the line at a single point. So if I want to find the slope, I'd pick two points that are incredibly close to each other* on the parabola.

The first point is given (1,1) the second? x=1+del where del is almost 0. Then y=(1+del)^2=1+2del+del^2

So the slope is

m=(1+2del+del^2-1)/(1+del-1)=(2del+del^2)/del

Now since del is almost zero then del^2 is incredibly smaller than almost zero, so compared to del, del^2=0 and

m=2del/del=2

Now I have the slope of the tangent line to the parabola y=x^2 at x=1.

*at some point you have, or will be, introduced to the concept of a limit. This is an application of that concept.

As for your teacher--yes, it is a good teaching method. He is simply trying to get you to think about the conception of what you already have learned, and since all concepts of mathematics is cumulative he is effectively guiding you subtly toward the next concepts.

Rap

See here for the definition of the secant line:
http://mathworld.wolfram.com/SecantLine.html

This probem will be easier since you just have to plug x values into the function to determine y values. Then you can easily calculate the slope between the two points.

okay
I want to thank both of you for your tips and great help.

raprap
One question for raprap:

What do you mean by the letters del in your explanation of my question?

It's an abbreviation for the Greek letter delta. Delta traditionally signifies difference between two independent variable, that is, if I have an x and a want a nearby x a specific distance (del) from x (I'll call this second x---x0) then x0=x+del.

Rap

ok
Got it now.