chapter 02 differentiation
TRANSCRIPT
TunisTunis bbususiinneessss scschhooooLL
MATHEMATICS FOR BUSINESS BCOR 120
DIFFERENTIATION
Content
• Limits of a function
• Continuity of a function
• Derivatives
• Differential
• Local/ Global Optimum
• Convexity and Concavity
• Taylor Polynomials
201/02/12
LIMIT AND CONTINUITY
1
3
LIMIT OF A FUNCTION
1.1
4
Limits of a Function
5
• The limit L has to be (a finite) number.
• Otherwise we say that the limit does not exist.– If the limit does not exist:
• f is said to be definitely divergent if L = ±∞; otherwise
• f is said to be indefinitely divergent
6
Left-side and Right-side limits
7
8
• We also write
for the right-side and left-side limits, respectively.
• A relationship between one-sided limits and the limit as introduced in Definition 4.1 is given by the following theorem.
• We note that it is not necessary for the existence of a limit of function f as x tends to x0 that the function value f (x0) at point x0 be defined.
9
10
11
Properties of Limits
12
Example
13
Example
14
To avoid indetermination (0/0) we can multiply both terms by
CONTINUITY OF A FUNCTION
1.2
15
Continuity of a Function
16
Continuity of a function
17
• or, using the notation
• continuity of a function at some point x0 D∈ f means that small changes in the independent variable x lead to small changes in the dependent variable y.
Continuous Function
18BCOR-120 Automn 2011
for all x from the open interval (x0 −δ, x0 +δ)
the function values f (x) are within the open interval (f (x0) − ε, f (x0) + ε)
Discontinuous Function
19BCOR-120 Automn 2011
There exist a least an x from the open interval (x0 −δ, x0 +δ)
For which the function values f (x) is outside the open interval (f (x0) − ε, f (x0) + ε)
When f is discontinuous at x0?
• The function f is discontinuous at x0
– If the one-sided limits of a function f as x tends to x0 are different; or
– If one or both of the one-sided limits do not exist; or
– if the one-sided limits are identical but the function value f (x0) is not defined; or
– if value f (x0) is defined but not equal to both one-sided limits.
20BCOR-120 Automn 2011
Types of discontinuities
A. Removable discontinuity: – f the limit of function f as x tends to x0 exists but
• (4) the function value f (x0) is different or
• (3) the function f is not defined at point x0. In this case we also say that function f has a gap at x0.
21BCOR-120 Automn 2011
Example: removable discontinuity
22BCOR-120 Automn 2011
, since
Example (cont.)
23BCOR-120 Autumn 2011
A. Irremovable dicontinuities
• finite jump at x0:
– (1) if both one-sided limits of function f as x tends to x0 exist and they are different.
• Infinite jump at x0:
– (2) if one of the one-sided limits as x tends to x0 exists and from the other side function f tends to (+ or -)infinity
24BCOR-120 Automn 2011
• Pole at point x0
A rational function f = P/Q has a Pole at point x0 if Q(x0)=0 but P(x0)≠ 0 – As a consequence, the function values at x0
+ or to x0
- tend to either ∞- or +∞.• The multiplicity of zero x0 of polynomial Q defines the
order of the pole:– In the case of a pole of even order, the sign of the function f
does not change at point x0;
– In the case of a pole of odd order, the sign of the function f changes at point x0
25BCOR-120 Automn 2011
• Oscillation point at x0 .
A function f has an oscillation point at x0 it he function is indefinitely divergent as x tends to x0.
• This means that neither the limit of function f as x tends to x0 exist not function f tends to ±∞ as x tends to x0.
26BCOR-120 Automn 2011
2701/02/12
One-sided Continuity
28
Properties of Continuous Functions
29
Properties of Continuous Functions
30
Properties of Continuous Functions
31
Properties of Continuous Functions
32
Properties of Continuous Functions
3301/02/12
Properties of Continuous Functions
34
DIFFERENCE QUOTIENT AND THE DERIVATIVE
2
35
Difference Quotients and Derivatives
3601/02/12
3701/02/12
3801/02/12
3901/02/12
4001/02/12
4101/02/12
4201/02/12
4301/02/12
4401/02/12
4501/02/12
A
B
A: the set of all points where function f is continuous
B: the set of all points where function f is Defferentiable
B included A.
DERIVATIVES OF ELEMENTARY FUNCTIONS; DIFFERENTIATIONRULES
3
46
Derivatives of Elementary Functions- Differentiation Rules
Derivatives of composite and Inverse Functions
DIFFERENTIAL; RATE OF CHANGE AND ELASTICITY
4
67
Differential, Rate of Change & Elasticity
The differential dy
Example-continued
GRAPHING FUNCTIONS
5
83
Graphing Functions
8401/02/12
8501/02/12
Monotonicity
8601/02/12
8701/02/12
8801/02/12
Extreme Points
8901/02/12
9001/02/12
9101/02/12
9201/02/12
01/02/1294
01/02/1295
01/02/12106
Convexity & Concavity
10701/02/12
01/02/12108
01/02/12109
01/02/12110
01/02/12111
01/02/12112
01/02/12113
01/02/12114
01/02/12115
01/02/12116
01/02/12117
Limits-revisited
11801/02/12
01/02/12119
01/02/12120
01/02/12121
01/02/12122
01/02/12123
01/02/12124
01/02/12125
01/02/12126
01/02/12127
01/02/12128
01/02/12129
01/02/12130
Graphing Functions
13101/02/12
01/02/12132
01/02/12133
01/02/12134
01/02/12135
01/02/12136
01/02/12137
01/02/12138
01/02/12139
Taylor Polynomials
14001/02/12
01/02/12141
01/02/12142
01/02/12143
01/02/12144
01/02/12145
01/02/12146
01/02/12147
01/02/12148
01/02/12149
01/02/12150
01/02/12151
01/02/12152
01/02/12153