1

15.Math-Review

Tuesday 8/15/00Tuesday 8/15/00

15.Math-Review 2

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

Convexity and Concavity

15.Math-Review 3

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.

Differentiation

x

xfxf

x

y

d

)(d)('

d

d

Notation: for a function y = f(x), the derivative of f with respect to

x can be written as:

15.Math-Review 4

This graphically:

Differentiation

t

f(t)

y= (x-t)f’(t)+f(t)

s

f(s)y= (x-s)f’(s)+f(s)

y

x

y=f(x)

15.Math-Review 5

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

15.Math-Review 6

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

Differentiation

g(x)f(x)

1

2g(x)

g'(x)f '(x)

Inverse rule as a special case of this:

Example:f(x) = 3x2

f(x) = 3x3 – 4 x2 + 6x – 20

f(x) = (3–7x)–3

15.Math-Review 7

More rules of differentiation:(g) f(x) = g(x)h(x) => f’(x) = g’(x)h(x)+ g(x)h’(x)

Differentiation

2g(x)

g'(x)f '(x)

g(x)f(x)

1

Inverse rule as a special case of this:

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)

2h(x)

g(x)h'(x)g'(x)h(x)- f '(x)

h(x)

g(x)f(x) (h)

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

15.Math-Review 8

Even more rules of differentiation:(j) f(x) = ax => f’(x) = ln(a)ax

(k) f(x) = ln(x) => f’(x) = 1/x

Differentiation

142 23 xx ef(x)

Example: f(x) = ex

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

f(x) = ln(x-3)

15.Math-Review 9

Example: logs, rates and ratios: For the following examples we will consider y a function of x, ( y(x) ). Compute:

Differentiation

)ln(

)ln(

)ln(

)ln(

xd

yddx

xddx

yd

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

15.Math-Review 10

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.

Differentiation

15.Math-Review 11

To differentiate is a trade….

Differentiation

xxf(x)

13 2

10

for expressionan get to)(' use )(i

i

i

i ixxfxxf

xx eef(x) 2221

))1)(1((log5 xxf(x)

15.Math-Review 12

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:

Differentiation

2

2

2

2

dx

f(x)d

dx

df'(x)f''(x)

dx

yd

n

n)(n-(n)

dx

f(x)d

dx

(x)df(x)f

1

Higher order derivatives are defined analogously.

Example: Second order derivative of f(x) = 3x2-12x +6f(x) = x3/4-x3/2 +5x

15.Math-Review 13

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.

Differentiation

y=f(x)

slope=f’(t +)

t

slope=f’(t)

t+

15.Math-Review 14

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:

Differentiation

x

yxf

),(

y

yxf

),(

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

15.Math-Review 15

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:

Stationary Points

Global Maximum

Local MaximumLocal Maximum

15.Math-Review 16

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:

Stationary Points

Local Minimum

Global Minimum

15.Math-Review 17

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?

Stationary Points

15.Math-Review 18

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.

Stationary Points

a1 a2 a3 a4

15.Math-Review 19

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?

Stationary Points

15.Math-Review 20

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!!

Stationary Points

Points of InflectionPoints of Inflection

15.Math-Review 21

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

Stationary Points

15.Math-Review 22

Application of derivative: L’Hopital rule.

Tough examples to kill time

)('

)('lim

)(

)(lim)(lim

thathave then we

)(lim and ,)(lim If

xh

xg

xh

xgxf

xhxg

xxx

xx

Use this rule to find a limit for f(x)=g(x)/h(x):

xexhxxg

xxhxxg

)( and 1,-4)( If

)( and ,)ln()( If2

0001.

15.Math-Review 23

Example: Let us consider the function

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

Tough examples to kill time

2

)2(

2

)(

x

exf

xexxf )3()( 2 Sketch the function

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