1518 words - 7 pages

Game Theory

Math Review

1

1.1

Function

Deﬁnition

If we write down the relation of x and y as follows, y = f (x) this means y is related to x under the rule f . Also we say that the value of y depends on the value of x. This relation y = f (x) is called as a function if 1) the rule f assigns a single x value to single y value or 2) assigns multiple x values to single y value.

2

2.1

Shape of function

When f (x) = ax + b

Suppose that the function is given as follows. y = f (x) = ax + b ´ 1) Slope: f (x) = a and Y −intercept: b Y − axis is b..

b −a

2) Intersection with Y − axis: This is the case when x = 0. So from f (0) = b, the Intersection with 3) Intersection ...view middle of the document...

If x ≤ 0, function y = ln x can’t be deﬁned. Example : y = ln x

2

y

20

10

0 0 5 10 15 x -10 20

-20

Also the basic rules of lug function are as follows.

ln 1 = 0 ln(ab) a ln( ) b ln(ab ) 1 ln( ) a = ln a + ln b = ln a − ln b = b ln a = ln 1 − ln a = − ln a = ln a−1

2.4

Function y = x

The basic rules are as follows.

x0 = 1 xm xn = xm+n xm = xm−n xn 1 x−m = xm m n (x ) = xmn xm ym = (xy)m

3

Diﬀerentiation (One variable)

Let f (x) be a continuous function. Then the ﬁrst derivative of a function at x = x0 is

Deﬁnition

the function deﬁned by following equation. f (x0 ) = lim .

0

h→0

f (x0 + h) − f (x0 ) f (x0 + h) − f (x0 ) = lim h→0 (x0 + h) − x0 h

3

The ﬁrst derivative of a function at point x = x0 can be interpreted as a slope at that point. So f (x0 ) > 0 =⇒ Value of function is increasing at x =⇒ Positive slope at x f (x0 ) = 0 =⇒ Value of function is constant at x =⇒ Constant slope at x f (x0 ) < 0 =⇒ Value of function is decreasing at x =⇒ Negative slope at x Also, for every point x, f (x) > 0 =⇒ Value of function is increasing for all x f (x) = 0 =⇒ Constant for all x f (x) < 0 =⇒ Value of function is decreasing for all x Deﬁnition (Second derivative) f ” (x) = h 0 i ∂ f (x) ∂x

0 0 0 0 0 0

The second derivative of a function can be interpreted as a change of slope. That is, it gives the information about the curvature of function.

f ” (x) = 0 ⇒ the slope of function has no change =⇒ linear function f ” (x) < 0 ⇒ the slope of function is decreasing Example ´ 1) What is the shape of function, f (x) if f (x) > 0 and f ” (x) > 0 for all x? ´ 2) What is the shape of function, f (x) if f (x) > 0 and f ” (x) < 0 for all x? ´ 3) What is the shape of function, f (x) if f (x) < 0 and f ” (x) = 0 for all x?

f ” (x) > 0 ⇒ the slope of function is increasing

3.1

General rules

Let f and g are both diﬀerential function of x.

y = f (x) ± g(x) =⇒ y = f (x) ± g (x)

0 0

0

0

0

y = f (x) · g(x) =⇒ y = f (x) · g(x) + f (x) · g (x) y = f (x) · g(x) − f (x) · g (x) f (x) 0 =⇒ y = g(x) [g (x)]2

0 0 0 0 0

0

y = f [g (x)] =⇒ y = f [g(x)] · g (x)

4

3.2

Special rules

y = c =⇒ y = 0 (if c is constant) y = xa =⇒ y = a · x(a−1) (if a is constant) √ 1 1 1 1 0 x = x 2 =⇒ y = · x(− 2 ) = √ y = 2 2 x y = eax (exponential function) =⇒ y = a · eax (if a is constant) 1 0 (for x > 0 always) y = ln(x) =⇒ y = x

0 0 0

4

Partial Derivatives (Multivariable)

Deﬁnition 1 (Two variable case) if f = f (x1 , x2 ) ∂f ∂x1 ∂f ∂x2 For example,

∂f ∂x1

f (x1 + h, x2 ) − f (x1 , x2 ) h f (x1 , x2 + h) − f (x1 ,...

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