Lecture Note Differentiation
d. The actual change in the worker’s rate of production between 11:00 and
11:10 A.M. is the difference between the values of the rate when
and when
Qt′()
t= (^316)
19
3
6
t==. That is,
()
Actual change in 19
3 1.08 units per hour
rate of production 6
⎛⎞⎛⎞
⎜⎟=−=−′′⎜⎟
⎝⎠⎝⎠
6.2 The nth Derivative ...................................................................................
For any positive integer n, the nth derivative of a function is obtained from the
function by differentiating successively n times. If the original function isy=fx(),
the nth derivative is denoted by
or ()()
n
n
n
dy
f x
dx
Example 3
Find the 5th derivative of each of the following functions:
a. fx( )=++− 52 x^642 x x 3 b.
1
y
x
=.
7 Concavity and the Second Derivative Test
Concavity
Suppose that f is differentiable on the interval (a,b).
a. Iff′is increasing on (a,b),then the graph of f is concave upward
on (a,b).
b. If f′ is decreasing on (a,b), then the graph of f is concave downward on
(a,b).
To Determine Concavity
Suppose that f is a function and f′ and f′′ both exist on the interval (a,b).
a. If fx′′()> 0 for all x in(ab, ), then f′ is increasing and f is concave
upward on (a,b).
b. If fx′′() 0< for all x in (a,b), then f′ is decreasing and f is concave
downward on (a,b).
Critical Points
A critical point of a function is a point on its graph where either:
+ The derivative is zero, or
+ The derivative is undefined
The relative maxima and minima of the function can occur only at critical points.
To Determine Points of Inflection
A point on the graph of a function at which the concavity of the function changes is
called an inflection point.
- Slope
is negative
- Slope
Slope
is 0
Slope
is positive
Slope
is negative
Slope
is 0
()
Concave upward
holds water
()
Concave downward
spills water