chapter 02 differentiation
TRANSCRIPT
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TunisTunis bbususiinneessss scschhooooLL
MATHEMATICS FOR BUSINESS BCOR 120
DIFFERENTIATION
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Content
• Limits of a function
• Continuity of a function
• Derivatives
• Differential
• Local/ Global Optimum
• Convexity and Concavity
• Taylor Polynomials
201/02/12
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LIMIT AND CONTINUITY
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LIMIT OF A FUNCTION
1.1
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Limits of a Function
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• 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
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Left-side and Right-side limits
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• 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.
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• 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.
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Properties of Limits
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Example
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Example
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To avoid indetermination (0/0) we can multiply both terms by
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CONTINUITY OF A FUNCTION
1.2
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Continuity of a Function
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Continuity of a function
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• 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.
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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)
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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)
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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
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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
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Example: removable discontinuity
22BCOR-120 Automn 2011
, since
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Example (cont.)
23BCOR-120 Autumn 2011
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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
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• 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
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• 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.
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One-sided Continuity
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Properties of Continuous Functions
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Properties of Continuous Functions
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Properties of Continuous Functions
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Properties of Continuous Functions
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Properties of Continuous Functions
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Properties of Continuous Functions
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DIFFERENCE QUOTIENT AND THE DERIVATIVE
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Difference Quotients and Derivatives
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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.
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DERIVATIVES OF ELEMENTARY FUNCTIONS; DIFFERENTIATIONRULES
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Derivatives of Elementary Functions- Differentiation Rules
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Derivatives of composite and Inverse Functions
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DIFFERENTIAL; RATE OF CHANGE AND ELASTICITY
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Differential, Rate of Change & Elasticity
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The differential dy
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Example-continued
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GRAPHING FUNCTIONS
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Graphing Functions
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Monotonicity
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Extreme Points
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Convexity & Concavity
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Limits-revisited
11801/02/12
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Graphing Functions
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Taylor Polynomials
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