This section deal with the problem of finding a straight line L that is tangent to a curve C at a point P. Before we go to learning that, let us make a few assumptions to start with (so as to avoid mathematical errors before we come to the end result). C is the graph of a function y = f(x) and this function is continuous over the interval (a,b). The co-ordinates of the point P are (x0,y0) and the point P lies on the curve C so that f(x0) = y0. Also we assume that P is not the end point of C. That means x0 is not equal to a or b. Therefore C extends to some distance on either sides of P.
A reasonable definition of tangency can be stated in terms of limits. If Q is a point on C different from P, then the line through PQ is called a secant line to the curve. This line rotates around P as Q moves along the curve. If L is a line through P, whose slope is the limit of the slopes of these secant lines PQ as Q approaches P along C (see picture below), then we say that L is tangent to C at P.
Since C is the graph of the function y = (x), then vertical lines can meet C only once. Since P = (x0,f(x0)), a different point Q on the graph must have a different x-co ordinate, say x0+h, where h ? 0. Thus the co-ordinates of Q would be (x0+h, f(x0+h)) and the slope of the line PQ would be:
= (f(x0+h) – f(x0))/h, based on the formula for slope of line joining two points with co-ordinates (x1,y1) and (x2,y2). Slope = m = (y2-y1)/(x2-x1)
So for our line PQ, that would be:
m = (f(x0+h) – f(x0))/(x+h – x) = (f(x0+h) – f(x0))/h
The above expression is called the difference quotient formula or simply the difference quotient of a function f. Note that h can be positive or negative based on whether Q is to the right or to the left of P.
This method of finding the difference quotient is further applied in finding the derivative of functions in calculus. The basic limit definition of derivatives stems from this difference quotient.
Given the function y = f(x) and a point x = x0, or given a table of values of f(x) for various values of x, to evaluate the difference quotient and subsequently to simplify the difference quotient, is simple.
A reasonable definition of tangency can be stated in terms of limits. If Q is a point on C different from P, then the line through PQ is called a secant line to the curve. This line rotates around P as Q moves along the curve. If L is a line through P, whose slope is the limit of the slopes of these secant lines PQ as Q approaches P along C (see picture below), then we say that L is tangent to C at P.
= (f(x0+h) – f(x0))/h, based on the formula for slope of line joining two points with co-ordinates (x1,y1) and (x2,y2). Slope = m = (y2-y1)/(x2-x1)
So for our line PQ, that would be:
m = (f(x0+h) – f(x0))/(x+h – x) = (f(x0+h) – f(x0))/h
The above expression is called the difference quotient formula or simply the difference quotient of a function f. Note that h can be positive or negative based on whether Q is to the right or to the left of P.
This method of finding the difference quotient is further applied in finding the derivative of functions in calculus. The basic limit definition of derivatives stems from this difference quotient.
Given the function y = f(x) and a point x = x0, or given a table of values of f(x) for various values of x, to evaluate the difference quotient and subsequently to simplify the difference quotient, is simple.
