Math 142 Lecture Notes for Section 3.3
Section 3.3 -
1
The Derivative
Definition 3.3.1:
The derivative of a function f at a number a, denoted f 0 (a) is
lim
h−→0
f (a + h) − f (a)
h
provided that this limit exists.
Interpretations of the Derivative There are many ways to view the derivative of a function,
(1) Slope of the tangent line: f 0 (a) is the slope of the line tangent to the graph of f at the point
(a, f (a)).
(2) Instantaneous rate of change: f 0 (a) is the instantaneous rate of change of the value y = f (x)
at the value x = a.
(3) Velocity: If f (t) is the change in position of a moving object at time t, then the velocity of the
object at time t is the value f 0 (t).
(4) Acceleration: If f (t) is the change in velocity of a moving object at time t, then the acceleration
of the object at time t is the value f 0 (t).
Math 142 Lecture Notes for Section 3.3
Example 3.3.2:
Given the function f (x) = x2 − 3x + 5, find the following:
(a) f 0 (3).
(b) The equation of the line tangent to f (x) at the point x = 3.
2
Math 142 Lecture Notes for Section 3.3
3
Definition 3.3.3:
If we allow a to be a variable instead of a constant number, then we can find a new function from f (x),
f 0 (x) which is defined as
f (x + h) − f (x)
lim
.
h
h−→0
Other notations for the derivative may be f 0 (x), y 0 ,
dy df
d
dx , dx , dx f (x).
A note on the properties of the derivative f 0 (x) of the function y = f (x):
The following facts hold true for the derivative of f (x):
(i) If f 0 (x) < 0 on an interval I, then f (x) is decreasing on I.
(ii) If f 0 (x) = 0 on an interval I, then f (x) is constant on I.
(iii) If f 0 (x) > 0 on an interval I, then f (x) is increasing on I.
Example 3.3.4:
Find the derivative f 0 (x) for f (x) =
1−x
:
2+x
Math 142 Lecture Notes for Section 3.3
4
Nonexistance of the Derivative:
If f 0 (a) does not exist, then we say f (x) is nondifferentiable at x = a. This occurs when the graph:
(1) has a discontinuity.
(2) has a sharp turn or corner.
(3) has a vertical tangent line.