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15.Math-Review Tuesday 8/15/00 Convexity and Concavity Consider the function f(x)=x 2 over the interval [-1,1]. Is this function convex or concave? Prove it. Notation: for a function y = f(x), the derivative of f with respect to x can be written as: Differentiation The derivative

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### 15.Math-Review

Tuesday 8/15/00

• Consider the function f(x)=x2 over the interval [-1,1]. Is this function convex or concave? Prove it.

Differentiation

• The derivative

• The derivative of a function at a point is the instantaneous slope of the function at that point. This is, the slope of the tangent line to the function at that point.

f(t)

f(s)

y= (x-s)f’(s)+f(s)

s

t

Differentiation

• This graphically:

y

y=f(x)

x

• Rules of differentiation:

(a) f(x) = k => f’(x) = 0

(b) f(x) = ax => f’(x) = a

(c) f(x) = xn => f’(x) = nxn–1

• Example:

f(x) = x

f(x) = x5

f(x) = x2/3

f(x) = x–2/5

Differentiation

• Rules of differentiation:

(d) f(x) = g(x) + h(x) => f’(x) = g’(x) + h’(x)

(e) f(x) = kg(x) => f’(x) = kg’(x)

(f) f(x) = g(x)n => f’(x) = n g’(x)g(x)n–1

• Example:

f(x) = 3x2

f(x) = 3x3 – 4 x2 + 6x – 20

f(x) = (3–7x)–3

• Inverse rule as a special case of this:

Differentiation

• More rules of differentiation:

(g) f(x) = g(x)h(x) => f’(x) = g’(x)h(x)+ g(x)h’(x)

(i) f(x) = g(h(x)) => f’(x) = g’(h(x))h’(x)

• Example: product, quotient and chain for the following:

g(x) = x+2, h(x) = 3x2

g(x) = 3x2 + 2, h(x) = 2x – 5

g(x) = 6x2, h(x) = 2x + 1

g(x) = 3x, h(x) = 7x2 – 10

g(x) = 3x + 6, h(x) = (2x2 + 5).(3x – 2)

• Example:

f(x) = ex

f(x) = ln(3x3 + 2x+6)

f(x) = ln(x-3)

Differentiation

• Even more rules of differentiation:

(j) f(x) = ax => f’(x) = ln(a)ax

(k) f(x) = ln(x) => f’(x) = 1/x

• Example: logs, rates and ratios:

• For the following examples we will consider y a function of x, ( y(x) ).

• Compute:

• For this last example find an expression in terms of rates of changes of x and y.

• A non-linear model of the demand for door knobs, relating the quantity Q to the sales price P was estimated by our sales team as Q = e9.1 P-0.10

• Derive an expression for the rate of change in quantity to the rate of change in price.

• To differentiate is a trade….

Differentiation

• Higher order derivatives:

• The second derivative of f(x) is the derivative of f’(x). It is the rate of change of function f’(x).

• Notation, for a function y=f(x), the second order derivative with respect to x can be written as:

• Example: Second order derivative of f(x) = 3x2-12x +6

f(x) = x3/4-x3/2 +5x

slope=f’(t +)

slope=f’(t)

y=f(x)

t+

t

Differentiation

• Application of f’’(x)

• We have that f’(t)  f’(t+)

• This means that the rate of

change of f’(x) around t is

negative.

• f’’(t)  0

• We also note that around t,f is a concave function.

• Therefore:

• f’’(t) 0 is equivalent to f a concave function around t.

• f’’(t) 0 is equivalent to f a convex function around t.

• Partial derivatives:

• For functions of more than one variable, f(x,y), the rate of change with respect to one variable is given by the partial derivative.

• The derivative with respect to x is noted:

• The derivative with respect to y is noted:

• Example: Compute partial derivatives w/r to x and y.

f(x,y) = 2x + 4y2 + 3xy

f(x,y) = (3x – 7)(4x2 – 3y3)

f(x,y) = exy

Local Maximum

Local Maximum

Stationary Points

• Maximum

• A point x is a local maximum of f, if for every point y ‘close enough’ to x, f(x) > f(y).

• A point x is a global maximum of f, if f(x) > f(y) for any point y in the domain.

• In general, if x is a local maximum, we have that:

f’(x)=0, and f’’(x)<0.

• Graphically:

Global Minimum

Stationary Points

• Minimum

• A point x is a local minimum of f, if for every point y ‘close enough’ to x, f(x) < f(y).

• A point x is a global minimum of f, if f(x) < f(y) for any point y in the domain.

• In general, if x is a local minimum, we have that:

f’(x)=0, and f’’(x)>0.

• Graphically:

• Example:

• Consider the function defined over all x>0, f(x) = x - ln(x).

• Find any local or global minimum or maximum points. What type are they?

a3

a1

a2

a4

Stationary Points

• Consider the following example:

• The function is only defined in [a1, a4].

• Points a1 and a3 are maximums.

• Points a2 and a4 are minimums.

• And we have:

f’(a1)< 0 and f’’ (a1) ? 0

f’(a2)= 0 and f’’ (a2) 0

f’(a3)= 0 and f’’ (a3) 0

f’(a4)< 0 and f’’ (a4) ? 0

• The problem arises in points that are in the boundary of the domain.

• Example:

• Consider the function defined over all x[-3,3], f(x) = x3-3x+2.

• Find any local or global minimum or maximum points. What type are they?

Points of Inflection

Stationary Points

• Points of Inflection.

• Is where the slope of f shifts from increasing to decreasing or vice versa.

• Or where the function changes from convex to concave or v.v.

• In other words f’’(x) = 0!!

• Finding Stationary Points

• Given f(x), find f’(x) and f”(x).

• Solve for x in f’(x) = 0.

• Substitute the solution(s) into f”(x).

• If f”(x)  0, x is a local minimum.

• If f”(x)  0, x is a local maximum.

• If f”(x) = 0, x is likely a point of inflection.

• Example: f(x) = x2 – 8x + 26

f(x) = x3 + 4x2 + 4x

f(x) = 2/3 x3 – 10 x2 + 42x – 3

Tough examples to kill time

• Application of derivative: L’Hopital rule.

• Sketch the function

Hint: for this we will need to know that the ex‘beats’ any polynomial for very large and very small x.

Tough examples to kill time

• Example:

• Let us consider the function

Obtain a sketch of this function using all the information about stationary points you can obtain.