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Tangent lines passing through points.

Posted: Mon Dec 07, 2009 8:11 am UTC
by tzar1990
I'm just coming up to my calculus final now, and I have realized, to my discontent, that I am still incapable of figuring out a certain type of question in which we are given an function, and told to find a point on said function where the tangent line will pass through a given point. (Example: given y=e^x find where the tangent line passes through 2,7.) I tried reviewing my notes, but it just looks like a variety of gibberish today (especially because my writing is quite messy when I have math first thing in the morning...) Could one of you wise forumites give me a hint or two as to how to begin something something like that?

Re: Tangent lines passing through points.

Posted: Mon Dec 07, 2009 8:28 am UTC
by Sabatini
If you want a few hints, try these.

(1) You know how to find the slope of a tangent line at x, right? You learned that fairly early on in calculus.

(2) Think back to algebra classes. Remember something called the "point-slope" equation of a line? Back then, you were given a slope and a point and asked for the equation of a line that (a) passed through the point, and (b) had the given slope.

(3) Now you are given information that's a little different from what you saw back in your algebra classes. You are given a point that the line must pass through, just as before, but now instead of being told its slope you are being told what function y = f(x) it is tangent to. What happens when you use the point-slope equation with this information?

I think you can figure it out from those three starters. Try it, and if you're really stuck look at the final hint below.

Spoiler:
(4) Because the slope of a tangent line to y = f(x) at (x,f(x)) is f'(x), the slope of the line you're considering is a function of x, which is what you're trying to figure out.

Re: Tangent lines passing through points.

Posted: Mon Dec 07, 2009 8:28 am UTC
by jaap
Let (p,q) be the point you are looking for.
You know two things about that point:
1. It lies on the given function.
2. The tangent line at (p,q) goes through the given point (or equivalently, the line from (p,q) to the given point is tangent to the function).

Each of these facts gives you some equation in p and q. Solve these two equations simultaneously to give you the values of p and q.

Re: Tangent lines passing through points.

Posted: Mon Dec 07, 2009 9:21 am UTC
by tzar1990
Hmm...

I get that the tangent line passing through point (a,b,) and tangent to the function f(x) will have an equation of y=f`(c)(x-a)+b, where c is the the x-value of the point where the tangent line touches the function, but I don't get how to use that plus my ability to calculate what f`(x) is for any given point to find out what specific value of c sends off a tangent line passing through (a,b).

Oh well, it'll probably make more sense in the morning, when I can keep my eyes open and focus for more than a few minutes. Thanks for the help, guys.

Re: Tangent lines passing through points.

Posted: Mon Dec 07, 2009 11:41 am UTC
by mr-mitch
but I don't get how to use that plus my ability to calculate what f`(x) is for any given point to find out what specific value of c sends off a tangent line passing through (a,b).


What is the value at x at the point on the curve? In fact, it might make more sense to ignore the point-gradient equation for now, and just work with the definition of the gradient,
hint:
Spoiler:
m = f'(x) = (y2 - y1)/ (x2-x1)

If you let (x2,y2) be the pair you're given, what is (x1,y1)?

Re: Tangent lines passing through points.

Posted: Mon Dec 07, 2009 9:51 pm UTC
by tzar1990
Okay, I just got it. Thank you very much!