Hi, I want to get a solid understanding of the actual definition of a tangent line. I believe a tangent line is a line which touches the graph of another equation at one point? But is it always perpendicular to something at that point? I mean, if two perpendicular linear equations intersect with each other at 1 point, does that mean the lines are tangent to each other? I know that when finding tangent lines to circles, the line is always perpendicular to the radius, right? Is it possible to find a tangent line to a point on an equation like y=x^2 without using calculus?
I'm very new to calculus, and I'm still studying pre-calc, but when you say the tangent is the rage of change at any given point P, this is also the derivative, correct? I was just thinking that there's probably points on the equation of y=x^2 where you cannot even use a linear equation as a tangent line because the line would cross the parabola more than once. The only areas I can think of where it wouldn't cross it twice is near the vertex. For example: find the tangent line equation to the point (-2,4) on the equation y=x^2.
Not exactly. A tangent line to a curve at a point on the curve is a line that goes through that line and is going in the same direction as the curve at that point.[/quote]
Any line is perpendicular to something! If you mean "perpendicular to the curve it is tangent to", absolutely not! If two lines are perpendicular they are NOT tangent! In fact, two lines are only "tangent" if they are, in fact, the same line.But is it always perpendicular to something at that point? I mean, if two perpendicular linear equations intersect with each other at 1 point, does that mean the lines are tangent to each other?
Well, yes, but you have to use "almost" calculus. For example, Fermat, just before Newton developed the calculus used this method: (Almost) any straight line can be written y= ax+ b. If that line is tangent to y= x^2, at, say, then it must touch the graph: x=1 must satisfy ax+ b= x^2 or x^2- ax-b= 0. Further- and this is the critical point- if they are tangent that x must be a double root. That is, we must have x^2- ax- b= 0 has 1 as a double root if and only x^2- ax- b= (x-1)^2= x^2- 2x+ 1. That is, we must have a= 2 and b= -1. The tangent line to y= x^2 at x= 1 is y= 2x- 1.I know that when finding tangent lines to circles, the line is always perpendicular to the radius, right? Is it possible to find a tangent line to a point on an equation like y=x^2 without using calculus?
Of course, if you had given me a more complicated function, that would be harder but the same idea works.