RATES OF CHANGE TANGENTS A tangent can be more precisely defined as follows:
Given a curve y = f(x), and a point P(a, f(a)), then the tangent line to the curve is the line through P with slope
m
=
lim x→a
f(x) – f(a)
1
x–a
(provided this limit exists)
In the earlier section Introduction to Calculus – The Limit Of A Function, we tried to guess the slopes of tangents by picking nearby points to P (a, f(a)) and compute the slope of the secant line PQ, which is given by
mPQ
=
f(x) – f(a) x–a
Then we let Q → P by letting x → a. By doing so, we discovered that the slope of the secant line approached a certain value m. Thus, in terms of limits, we say that ➔ the tangent line is the limiting position of the secant line PQ as Q → P, OR ➔ the slope m of the tangent line PQ is the limiting value of the slope of the secant lines PQ as Q → P. which leads to the mathematical expression
m
=
lim x→a
f(x) – f(a) x–a
Q(x, f(a))
f(x) – f(a) P(a, f(a)) x–a
a
The formula for m can be expressed in another way: Let h = x – a which gives m
=
lim x→a
f(x) – f(a) h
x