MCV4U1
Using Limits to Develop the Derivative (or the slope of a tangent)
In general, if a curve has an equation y= f(x) and we want to find
the tangent to the curve at a point P(a, f(a)), then we consider a
nearby point Q(x, f(x)). We can draw a secant PQ and compute
the slope of the secant line PQ as:
(The slope of the secant PQ represents an average rate of change of y
with respect to x.)
If we let point Q approach point P (or x approaches a), in the limiting case the secants PQ become a
tangent at point P (i.e, P & Q coincide or the line only touches the curve in one place.) Thus we can
incorporate the concept of a limit to define the slope of the tangent at P as:
(The slope of the tangent represents an instantaneous rate of change of y
with respect to x at point P.)
This definition amounts to saying that the tangent line is the limiting
position of the sequence of secant lines PQ1, PQ2, PQ3, ... as the
points Q1, Q2, Q3, approach P along the curve.
Another expression for the slope of the secant line PQ (according
to the coordinates of P and Q shown in the figure at right) is:
Correspondingly, the expression for the slope of the tangent
becomes:
This concept of the slope of a tangent is so important that it is given a special name and many
notations. It is referred to as a derivative of a function. The process of taking a derivative is often
referred to as differentiation.
The derivative of a function f at a number a is:
Also...
.
Given a function f, the derivative of f is the function f’ defined by:
Interpretations of the Derivative:
1. As the slope of a tangent. The tangent line to the curve y=f(x) at the point P(a, f(a)) is the line
through P(a, f(a)) with slope f’(a).
2. As a rate of change. The instantaneous rate of change of y=f(x) with respect to x when x=a is
given by f’(a).
Other notations for the derivative of y=f(x):
The use of this limit approach to finding the derivative, because it is the based strictly on the
definition, is often referred to as a First Principles Approach.
Examples,
Use 1st Principles methods to find both
(i.e., Use both the definitions developed above.)
1.
and
for each of the following functions.
2.
3.
More Examples...
4.
5.
Even More... Just
7.
6.
for these...
8.
Oh no, I think I have a problem. I just can’t get enough of 1st Principles! (
9.
10.
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