can anyone give me a function(curve) that has a tangent that crosses the curve more than once?

Question

I know the tangent of y=sinx touches the curve more than once but I need one that crosses to the other side of the curve. understand ?I know i'm making it very complicated!


I would als+o like the function of a curve with a tangent that bisects the curve at the point of tangency.

if anyone can help me even slightly with either of these problems please respond.

Answer 1

I think I know what you mean by the first question, but I also think you're confused by the idea of a tangent. Curves have more than one tangent. In fact, any curve has an infinite number of tangents -- one at every point along the curve. If you look at the sin wave (y=sinx) you can almost visualize the different tangent lines as you move from one point to another. If you start at a peak, the tangent line is perfectly horizontal (slope=0). As you look at points further down the curve (in other words as x increases) the tangent slope decreases (slope<0) until you reach y=0. Then the slope begins to increase. It keeps increasing, all the way through the bottom peak (when it is once again horizontal, slope=0) and continues increasing (slope>0)until you hit y=0 again.

So start at the peak again. The perfectly horizontal tangent line is tangent again at the next peak -- exactly one wavelength away. As you increase x, the tangent line is not tangent to the next peak; it forms a secant to that peak. More importantly, it forms a secant to the next "valley" as well (the one after the next peak). Keep increasing x, and notice what happens to this secant -- the two points where it instersect the valley are converging. Eventually, they will converge to the exact same point, and viola! the line will be tangent at that point!

So the answer to your first question is y=sinx! (There are other possibilities, too.)

As for your second question, I don't think that's possible, but I can't prove why not.

I hope this helps!

Answer 2

A sin curve....I am sorry my my is rusty but a sin curve will help.