What is the significance of tangent lines in understanding curve slopes?

[aznHypnotix]

A tangent to a curve is a line that touches the curve but at what specific point where there is a slope to the curve. Equations like y = X^2 have many slopes because the curve is shaped differently at different points. We can choose 2 points on the graph and find the slope but it will different all the time. What is a true answer for tangent line?

 

[berkeman]

It's a limit concept. Are you familiar with limits?

 

[aznHypnotix]

Yes where x goes to zero or infinity or any value.

 

[Defennder]

A tangent to a curve is a line that touches the curve but at what specific point where there is a slope to the curve. Equations like y = X^2 have many slopes because the curve is shaped differently at different points. We can choose 2 points on the graph and find the slope but it will different all the time. What is a true answer for tangent line?

There isn't any "true" answer for tangent line. As you said, it all depends at which point on the curve you choose to evaluate the tangent line. Think about it this way: Suppose y=x+3. What is the "true" value of y here?

EDIT: Upon reading your OP for the 2nd time, I suppose you might be referring instead to secant lines. There are a number of animations you can view online:
http://www.math.umn.edu/~garrett/qy/Secant.html

And Wikipedia's page here:
http://en.wikipedia.org/wiki/Secant_line#Secant_approximation

Unfortunately the computer I'm using doesn't have java installed properly, so I can't verify if it works.

 

[aznHypnotix]

thanks defender, I'm beginning to understand. Those are some great links you provided. I forgot about the secant line. That makes more sense to me now.

 

[HallsofIvy]

Since the "slope" of the curve varies from point to point, the curve has a different tangent line at each point.

 

[chislam]

Furthermore, the slope of a tangent line touching at a specific x value is the result of the derivative of the equation.

For example, the derivative of y=x² is 2x. At x=2, the slope of the line tangent to your equation is 4. (2x = 2(2) = 4).

 

[aznHypnotix]

That is a good way of putting it chislam. I can see it now. I like derivatives and slopes. It is cool.