Fin118 Unit 3

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


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


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


• 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


•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


Example 4:

Find the critical and the inflection points of the

following function:

24

12 23 xxxxf


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.

3. How to calculate area under a curve.

4. How to calculate area between two Curves.

27

Fin118 Unit 3

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