Chapter 02 differentiation

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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

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LIMIT AND CONTINUITY

LIMIT OF A FUNCTION

Limits of a Function

• 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

Left-side and Right-side limits

• 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.

Properties of Limits

Example

To avoid indetermination (0/0) we can multiply both terms by

CONTINUITY OF A FUNCTION

Continuity of a Function

Continuity of a function

• 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

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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

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.

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.

Example: removable discontinuity

, since

Example (cont.)

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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

• 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

• 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

Properties of Continuous Functions

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Properties of Continuous Functions

DIFFERENCE QUOTIENT AND THE DERIVATIVE

Difference Quotients and Derivatives

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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

Derivatives of Elementary Functions- Differentiation Rules

Derivatives of composite and Inverse Functions

DIFFERENTIAL; RATE OF CHANGE AND ELASTICITY

Differential, Rate of Change & Elasticity

The differential dy

Example-continued

GRAPHING FUNCTIONS

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

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Graphing Functions

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Taylor Polynomials

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Chapter 02 differentiation

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