Question

@all I am finding a problem interpreting the meaning of a derivative professor David Jerrison in his first lecture explained the meaning of a derivative using the difference between a tangent line and a secant line but let us suppose the curve y= f(x) is wiggly and the secant line intersects the curve at more than 1 point then how do we interpret the meaning of the derivative.This may be an easy proposition for most of you but I am getting a little confused solving this question.

Answer 1

The derivative at a point can be interpretated as being the slope of the tangent to a function at that spesific point. The derivative at a point x (that is any point on the graph of y=f(x)) will then be interpretated as being a general expression for the slope of the tangent at x. In order to do this rigourosly one need to find an expression for the secant line through a point P and Q on the graph, and then compute the limit of this expression as the change of x approaches zero.

The resulting expression will, provided the limit exists, be the slope of the tangent at that point.

Answer 2

So, the derivative has 3 intepretations really:
1) The slope of the graph of y=f(x) at x
2) The slope of the tangent to the y=f(x) at x
3) The rate of change of f(x) with respect to x

Answer 3

even if the secant line intersects the curve at n number of points I think what we are doing is taking all those different \[\delta\]x's as one and making them approach to 0 am i right to generalize the equation

Answer 4

Yes. The secant line actually has to cross the curve at least once, because it wouldn't be a secant line unless it does this. It can cross as many times as it wishes, but so long as we select two points, write out an expression for the slope of this line, take the limit as delta x moves to zero, we will have the derivative. What actually happens geometrically when you take this limit, is that, say you have a point P on the curve and another point Q (see the drawing)

ance you let the expression approach zero, the point Q will progressively move down along the curve towards point P, until it is "infinitely close" to point P. Then point Q actually just sits right on top of point P, and the secant line has turned into a tangent to the curve at point P.