How Do Tangent Lines Relate to Calculus Problems?

[ACLerok]

how does tangent lines play into this problem?

http://www.eden.rutgers.edu/~cjjacob/images/calc.jpg

 

[gnome]

Do you understand that f(7) is the value of y when x=7?

 

[ACLerok]

yeah. i don't understand F'(7).

 

[gnome]

dy/dx = f'(x) is the slope of the curve y = f(x), so what is f'(7)?

 

[ACLerok]

am i supposed to plug 7 in for x and then take the derivative? how? doesn't the equation end up canceling out?

 

[gnome]

Why would you take the derivative?

The slope of the tangent line = the slope of the curve at the point where the line touches the curve.

 

[ACLerok]

oh, i didnt realize the d's stood for the difference.
so to find F(7), I isolate y on one side of the equation and then plugin 7 for x and solve for y? what do i do for the F'(7)? slope of the line when x=7?

 

[HallsofIvy]

Originally posted by ACLerok
oh, i didnt realize the d's stood for the difference.
so to find F(7), I isolate y on one side of the equation and then plugin 7 for x and solve for y? what do i do for the F'(7)? slope of the line when x=7?

These are all things that should be elementary (believe me, the problems are going to get a lot harder!).

In the first place, the "d's" do not stand for difference!
They are simply the notation for derivative. Yes, one method of finding the derivative of a function is to take the limit of the "difference quotient" but I don't think you should think "the d's stand for the difference".

In any case, the first thing you should have learned about the derivative of a function is that "the derivative IS the slope of the tangent line".

In this case, yes, solve for y. The value of y when x= 7 is the value of the function F when x= 7, F(7). The slope of that line is the derivative of F when x= 7 dF/dx(7) or F'(7).

By the way, you do not find the derivative of a function "when x= 7" by substituting 7 for x and then differntiating. The value of any function for a specific x is a number (a constant) and the derivative of a constant is always 0. Do it the other way around: first differentiate and then substitute.

 

[ACLerok]

so correct me if wrong. to find F(7), i just plug 7 for x in the equation of the tangent line and to find F'(7), I just differentiate the equation and then plug in 7 for x and solve for y?

I forget, what does differentiate mean?