ميةس محمد بن سعود اامممعة ا جاداريةعلوم اد والقتصا كلية استثمارتمويل وا قسم الAl-Imam Muhammad Ibn Saud Islamic University College of Economics and Administration Sciences Department of Finance and Investment Financial Mathematics Course FIN 118 Unit course 3 Number Unit Derivability Critical points Inflection point Unit Subject Dr. Lotfi Ben Jedidia Dr. Imed Medhioub 1
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
جامعة اإلمام محمد بن سعود اإلسالمية
كلية االقتصاد والعلوم اإلدارية
قسم التمويل واالستثمار
Al-Imam Muhammad Ibn Saud Islamic University College of Economics and Administration Sciences
Department of Finance and Investment
Financial Mathematics Course
FIN 118 Unit course
3 Number Unit
Derivability Critical points Inflection point
Unit Subject
Dr. Lotfi Ben Jedidia Dr. Imed Medhioub
1
1. Derivative of function
2. How to compute derivative?
3. Some rules of differentiation
4. Second and higher derivatives
5. Interpretation of the derivative
6. Critical points
7. Inflection point
2
We will see in this unit
Learning Outcomes
3
At the end of this chapter, you should be able to:
1. Understand what is meant by “Differentiation and Derivative
of function”.
2. Compute these derivatives.
3. Apply some rules of differential calculus that are especially
useful for decision making.
4. Find the critical & inflection points of a function.
5. Apply derivatives to real world situations in order to optimize
unconstrained problems, especially economic and finance.
Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y (the function) with respect to x. The derivative gives the value of the slope of the tangent line to a curve at a point (rate of change). The slope of the tangent line is very close to the slope of the line through (a, f(a)) and a nearby point on the graph, for example (a + h, f(a + h)).
Derivative of Function
4
h
xfhxf
dx
dfxf
h
0
' lim
Definition1:
Definition2:
Definition3:
How to compute derivative ?
To compute the derivative of a function, we can use four
steps.
Step 1 : compute
Step2 : compute
Step 3 : compute
Step 4 : compute
5
hxf
xfhxf
h
xfhxf
h
xfhxfxf
h
0
' lim
Find the derivative of
Step 1 : Step2 : Step 3 : Step 4 :
6
536353 222 hxhxhxhxf
xhhxfhxf 63 2
xh
h
xhh
h
xfhxf63
63 2
xxhh
xfhxfxf
hh663limlim
00
'
53 2 xxf
xxfxxf 6 53 '2
Example1:
How to compute derivative ?
7
Some rules of differentiation
To simplify the determination of derivatives we can use the following rules. 1/ 2/ 3/ 4/ 5/ 6/ 7/ 8/
0kdx
dccx
dx
d
1 nn nxxdx
d 1 nn ncxcxdx
d
xx
dx
d
2
1
xx eedx
d
xx
dx
d 1ln
2
11
xxdx
d
8
Some rules of differentiation
9/ 10/ 11/ 12/
13/
14/
xgxfxgxf '''
xgxfxgxfxgxf '''
2
'''
xg
xgxfxgxf
xg
xf
xnfxnfxnf 1''
xf
xfxf
''
ln
xfxf exfe ''
Find the derivatives of the functions
1/
2/
3/
4/
5/
6/
9
12 xxf
1073253 xxxxf
3
1072
xxxf
1
1)(
2
2
x
xxf
1ln)( 2 xxf
12
)( xexf
Examples:
Derivative of Function
Second and Higher Derivatives
Given a function f we defined first derivative of the
function as:
The second derivative is obtained by:
And the n-th derivative is obtained by:
10
h
xfhxf
hxnf
nn 11
0lim
h
xfhxf
hxfxf
112
0lim''
h
xfhxf
hxfxf
0lim'1
Example:
1/ the function:
2/ first derivative :
3/ second derivative :
4/ third derivative :
5/ fourth derivative :
6/ fifth derivative :
11
10234 25 xxxxf
2620 4'1 xxxff
680 3''2 xxff
2'''3 240xxff
xxff 480''''4
480'''''5 xff
Second and Higher Derivatives
12
Interpretation of the derivative
First derivative:
• If in a particular interval, then the
function is increasing in that particular interval.
• If in a particular interval, then the
function is decreasing in that particular interval.
Second derivative:
• If in a particular interval, then the graph of the function is concave upward or strictly convex in that particular interval.
• If in a particular interval, then the graph of the function is concave downward or strictly concave in that particular interval.
0 xf
0 xf
0'' xf
" "
" "
0'' xf
13
Example1:
Find where the function
is increasing and where it is decreasing.
Solution:
Thus the function is increasing on [-1;2] and it is decreasing on ]-, -1[ and ]2, +[ .
965.1 23 xxxxf
2or 10
213633 2
xxxf
xxxxxf
Interpretation of the derivative
14
Example2:
Find where the function
is strictly concave and where it is strictly convex.
Solution:
Thus the function is convex on ]-, 0.5[ and concave on ]0.5, +[ .
965.1 23 xxxxf
210;36 '''' xxfxxf
Interpretation of the derivative
15
Critical points: local extrema
A critical point of a function of a single real variable , is a
value x0 in the domain of ƒ where either the function is not
differentiable or its derivative is 0, .
The point is a local minimum if
The point is a local maximum if
.
0' xf
xf
00 , xfxM
0 0 0''
0' xfandxf
00 , xfxM
0 0 0''
0' xfandxf
Definition 1:
Inflection point
The point is an inflection point if the
second derivative of the function changes signs
from to or to , i.e. the curve changes from
concave upward to concave downward or from
concave downward to concave upward at M.
16
00 , xfxM
Definition2 :
Inflection point
17
Inflection point
18
Example1: maximum
Find the critical point of the following quadratic
function:
Step 1 : and
Step 2 :
Step 3 : we find the sign chart table.
322 xxxf
22' xxf 2'' xf
1 ,022 0' xthenxxf
4
- -
Critical points
19
Graph of the function
• M correspond to the
vertex point (see the
chapter 2)
• X = -1 and x = 3 are the
roots of the equation.
• The function is increasing
in ]-, 1[ and decreasing in
]1, +[
M(1,4) is an absolute
maximum
Critical points
20
Example2: minimum
Find the critical point of the following quadratic
function:
Step 1: and
Step 2 :
Step 3 : we find the sign chart table
322 xxxf
22' xxf 2'' xf
1 ,022 0' xthenxxf
2
Critical points
21
Graph of the function
• M(-1,2) is an absolute
minimum.
• It correspond to the vertex
point (see chapter 2)
• There is no roots for the
quadratic equation.
• The function is decreasing in
]-, -1[ and increasing in
]-1, +[.
Critical points
M(-1,2) is an absolute
minimum
22
Example3: inflection point
Find the critical and inflection points of the
following cubic function:
Step 1: and
Step 2 :
Step 3 : we find the sign chart table
133 xxxf
33 2' xxf xxf 6''
1 ,013 0 2' xthenxxf
0 ,06 0'' xthenxxf
Critical & Inflection points
23
Graph of the function
•The function is
increasing on ]-, -1[ and
]1, +[. But decreasing
on ]-1,1[
•It is concave on ]-, 0[
and convex on ]0, +[.
M(-1,3) is a
local maximum
M(1,-1) is a
local minimum
M(0,1) is an
inflection point
Critical & Inflection points
Example 4:
Find the critical and the inflection points of the
following function:
24
12 23 xxxxf
Critical & Inflection points
1. The derivative is the rate of change of a dependent variable with respect to an independent variable 2. The derivative of the function f at the point x = a is defined by: Provided the limit exists.
3. First and Second derivatives of a function are useful
tool to determine critical points and inflection point.
h
xfhxf
hxf
)()(
0
lim)('
Time to Review !
25
f • Increasing • Decreasing • Maximum or minimum
value (when slope = 0)
f’ • Positive (above x axis) • Negative (below x axis) • Zero
f’’ equals zero in x0 and changes of signs, M (x0 , f(x0)) is an inflection point
Time to Review !
26
we will see in the next unit
1. The inverse process of differentiation:
Integration.
2. The connection between integration and summation.