Mathematics - Formulario

Class 2 : Functions of a real variable denoted

number,

real

unique

a

D

in

x

element

every

to

associating

rule

a

is

variable

real

one

of

function

real

a

D

Given ℝ

⊆ f(x)

x

D

:

f

f(x)

with ↦

ℝ, Range : subset of R values attained by f(x) (= dependent variable)

Domain : set of all possible values of x (= independent variable)

il Vertical line text : used in

*

* order to understand whether 1

DOMAIN RANGE :

Ratios 0

≠

x

domain

it is a function or not x

2n 0

≥

x

domain

:

roots

Even x

not a function

a function inverse function f ⁻¹ 0

>

x

x domain

log

:

Logarithms

Let A, B and C , f : A B and g : f(B) C, the composite function g f (g after f) is :

ℝ ∘

⊆ #÷

C

B

A

g f : A C, g f(x) = g(f(x))

∘ g(f(x))

f(x)

(x)

i

B non commutative

Invective (one-to-one) : each element in B

A .

0¥ is the image of at most one element in A

: Surjective (many-to one) Inverse function is denoted with f (x) : From the definition f(g(x)) = g(f(x)) = x

Class 3 : derivaitves - linear approximation

The first derivative is the slope of the tangent line to the graph of f at (x ,f(x ))

p p

Denoted by f (x ) : or secant

p Approximation error :

tangent ..

.

My curve is decreasing when m is negative " The linear approximation of : D

f ℝ

⊆

My curve is increasing when m is positive at D is

a the function

ℝ : :

∈ È

!

Monotonicity theorem : f (x) > 0 for all x in the interval I f increasing

: " 1. compute f (x)

.

2. change x with a

A strictly increasing error

approximation

f (x) < 0 for all x in the interval I f decreasing its graph is the tangent line to

(resp., decreasing) may

not have a strictly f (x) = 0 for all x in the interval I f constant the graph of

' f

positive derivative.

If the limit exists, we say that f is differentiable at x. The process of finding the derivative of a functions called differentiation

-

Rules

.at?..o..!Ea

- Differentiation

l'

1- f (x) · g(x) - f(x) · g (x)

f(x) | a > 0 | a ≠ 1

Quotient : in

= f (x) = 1

f(x) = log (x)

f (x) = 0

.

f(x) = k a

g(x) (g(x)) x · ln (a)

f (x) = 1

f(x) = x g - 1 f(x) = ln (x)

(f (g(x)) · g (x)

(f (g(x) ) f (x) = 1

: g

' f (x) = g · x

f(x) = x

÷

Product : x

x

x f (x) = a · ln (a)

(f(x) · g(x)) f (x) · g(x) + f(x) · g (x) f(x) = a

. f(x) = |x| f (x) = |x|

x - x

x [-e ]

f (x) = e

f(x) = e [with -x]

Sum & Diff : (a f(x) b g(x)) af (x) bg (x) x

∓ ∓ z = g(x) f(x) = x

f(x) = g(x)

Roots : f (x) = 1

f (x) = g · z

solution 2 x

=

g = z z

Rates of change of a certain economic variable (meaning of derivatives) :

Change of f over the interval [a, a+h] : f(a+h) - f(a)

Average rate of change : f(a+h) - f(a)

" h

-

Instantaneous rate of change = f’(a)

/ Relative rate of change : f’(a) [percentage per unit of time]

' f(a)

_

Class 4 - 5 : limits

Definition : the expression lim f(x) = A means we can make f(x) as close to A as we want for all x sufficiently close to a

Neighbourhoods : '

Rules for limits example

-

?

|x−a|<δ x∈(a−δ,a+δ) n

/ .

(a − δ, a + δ) : symmetric neighborhood of a

(−∞, a), (a, +∞) : neighborhoods of −∞ and +∞ [f continuous]

If the function is defined in a given interval [a, b] : [B≠0] Memorise

[a, +δ): is a right neighborhood of a [A≠0; B≠0]

lim f(x) = A if: > 0, > 0 : |x − a| < δ |f (x) − A| < ε.

x a ∀ε ∃δ

Given : threshold of error ε, we can find a corresponding threshold δ such that, if x is closer to a than δ, then f(x) is closer to

A than ε.

Limits computation : ∞

with

Limits

Neither f(x) nor g(x) can be eliminated number

scompose

a number if

I can undo the number, but not the denominator if

and if

Left and right limits may simplify if

have different results = limit

does not esxist

I can undo only the denominator Complex cases :

Logarithms & Exponentials

[for a > 1, b > 0, c > 0]

+ (right) [>x₀]

- (left) [<x₀] does not exist

if

I can undo the nominator and the denominator

② if

scompose and [for a > 1, for b > 0]

simplify Important limits :

L’Hopital’s Rule Used in order to solve limits such as : !

Let and be differentiable functions in (α , β) and α (α, β):

f g TED

∈ padre

If lim g(x) = 0, g’(x) ≠ 0 for all x (α, β)\{a}, and there exists then:

∈

x a

• pp

: ÷

If lim f(x) = ±∞, and lim g(x) = ±∞, and there exists then:

x a x a

. .

!! The same hold for x ±∞ !!

-

Class 6 - 7 : continuity and differentiability

Consider :D

f ℝ ℝ

⊆

1. Function f is at a D if lim f (x) = f (a)

continuous x a

∈

2. Function f is in D, if it is continuous at each a D

continuous ∈

| ' -

Lemma

polynomials : ℝ

Identifying continuous functions :

continuous ⅔

powers : | x [0, +∞) A function is continuous at a

ℝ

If f and g are continuous functions at a then : even root : [0, +∞); point a of the domain if and

f + g and f - g are continuous at a odd root : ℝ

i. only if:

exponentials : ℝ

f · g and f ÷ g (g(a)≠0 are continuous at a logarithms: (0, A +∞) 1. lim = A.

f(x)

↳ f g is continuous at x a-

a

∘ 2. lim f(x) = A

discontinuous x a+

if f is one-to-one on interval [b,c], f:[b,c] then f⁻¹ is continuous

ℝ 3. f(a)=A.

Continuity vs Differentiability : Kink : a point where the

? tangents from the left and the

differentiable continuous [NOT viceversa]

f f

•• right exist but are different

f(x) = |x| is continuous at but not differentiable at ( )

a a

••

Differentiable functions are smooth: they do not have kinks

Left and right derivatives

· The left derivative (right derivative) of f at a point a is the limit (if existing and finite) :

Check differentiability : '

1. Check whether f is continuous at a; if not it’s not differentiable at a;

2. Compute the limits L⁻ = lim f′(x) | L⁺ = lim f′(x);

x a+

x a−

3. If L⁺ and L⁻ are finite :

· L⁺ = L⁻ differentiable at a; L⁺≠ L⁻ NOT differentiable at a;

÷ f′ exists and is continuous at a = function of class C¹

.

If f’(x) > 0 on I f strictly increasing on I f one to one on I f invertibile on Rf Lg

"

we can compute g=f⁻¹ : " lim x a f(x) = A |x−a|<δ |f(x)−f(a)|<ε

∀ε>0,∃δ>0:

Intermediate value theorem [intuitive]

Suppose that the function is continuous on a closed interval [a,b]; will take on every value

f f

between f(a) and f(b) over the interval (no need to take up pencil)

If f(a) · f(b) < 0 there exists at least one poin c (a,b) such that f(c)= 0

∈

For any y between the values f(a) and f(b), there exists a number c in [a,b] for which f(c) = y

* If f(x) is strictly monotone then there will only be one zero

The Bisection algorithm is used for searching zeros

Let f(a) < f(b) [f(a) · f(b) < 0] and set tolerance level ε > 0. Let I₀ = [a₀, b₀] = [a, b]. For every n

Steps: 1. Take into consideration the mid point : c = | 2. f(c < ε stop here otherwise | 3. If f(a )·f(c < 0 set :

₊₁ ₊₁) ₊₁)

n n n n

I = [a , c a ] = [a b and go to (1) || I = [b , c b ] = [a b and go to (1.)

₊₁ ₊₁]∪[c ₊₁, ₊₁, ₊₁] ₊₁ ₊₁]∪[c ₊₁, ₊₁, ₊₁]

n n n n n n n n n n n n n n

As n goes larger the c gets closer to a solution of f(c) = 0

n

Class 8 : convexity and concavity

A subset C is convex when the segment joining any two points in C is contained in C [convex - concave ]

ℝⁿ

⊆

Convex subsets of : intervals | : circle, square, rectangle, line, segment | : sphere cube, cone

ℝ ℝ² ℝ³

-

Consider function f : D R R:

⊆ convex

Epigraph of f: Epi(f) = {(x,y) | x D,y ≥ f(x)}

∈

Hypograph of f: Hyp(f) = {(x,y) | x D,y ≤ f(x)}

∈

Let f : [a, b] R be convex or concave f continuous in (a, b)

te

The slopes of tangent

Let I be an interval and let f : I be a twice differentiable function, then:

ℝ lines increase

f on I f′′(x) ≥ 0 for all x I f on I f′′(x) ≤ 0 for all x I

convex concave

∈ ∈

An inflection point is a point of a curve at which the curve changes from being concave to convex, or vice versa

Consider f : (a, b) R R, with a continuous second derivative

⊂

1. c inflection point f′′(c) = 0, the vice versa is not true in general [condizione non necessaria e non sufficente]

2. c inflection point f′′ changes sign at c.

Class 9 - 10 : optimization

(Used in profit maximization) } May fail

Let f : D : c is a maximum point for f if f(c)≥f(x) for all x in D |d is a minimum point f(d)≤f(x) for all x in D

ℝ ℝ

⊂ to exist

µ

it

'

1 l

max f = f(c) min f = f(d)

. minimum value of f in D

maximum value of f in D

Lemma D D

-

Let f : (a, b) be continuous. If : Let f : (a, b) be continuous. If :

ℝ ℝ

¥

Ì Then c is a maximum point f convex

1. f increasing in (a, c) 1. f decreasing in (a, c)

i

f concave

↳ Then c is a minimum point

2. f decreasing in (c, b) 2. f increasing in (c, b)

If f not continuous, anything can happen

!! study the sign of the derivatives !!

Bounded or unbounded : A function f : D is

ℝ ℝ

⊂ ti

· bounded above if f(x) ≤ M for all x D, for a suitable choice of a constant M

∈ The range of any bounded function is

· bounded below if f(x) ≥ M for all x D, for a suitable choice of a constant M

∈ contained in a bounded real interval of ℝ

· bounded if it is both bounded above and below

Facts :

A function having a min is also bounded below | A function having a max is also bounded above.

The vice versa is false: a function may be bounded below (above) but have no min (max).

(Useful for exercises) If we prove that a function is unbounded below (above) then it has no min (no max).

Local extreme points : A subset U of R is a neighborhood of a if it contains a as an interior point. global maximum

In particular (a − ε, a + ε), where ε > 0, is a neighborhood of a local maximum

f : D ℝ ℝ

⊂

· if f(c) ≥ f(x) for all x in a neighborhood of c, then c is a local maximum point for f local minimum

· if f(d) ≤ f(x) for all x in a neighborhood of d, then d is a local minimum point for f global minimum

If f is differentiable in (a, b), a stationary point is a point c (a, b) such that f′(c) = 0; c can be : local (/global) max(/min) point

∈

↳ first order condition

:

Candidates To spot max and min employ :

· Interior stationary points of f [f’(x₀)=0] · The definition of max/min points

· Boundary points of D [ex. D= [a,b] check {a,b} • · Second order condition *

-

· Points of D where f is NOT differentiable · Compactness of the domain

* If f is twice differentiable on (a,b) : Remark : if f(c)>0 then f(x) > for all x in a

1. f’’(x) ≥ 0 for all x (a,b) c is a minimum point [> local] | f convex neighbourhood of c

∈ Knowing that a point is of local extrema is rarely

2. f’’(x) ≤ 0 for all x (a,b) c is a maximum point [< local] f concave

∈ conclusive

3. f’’(x) = 0 anything

The extreme value (Weierstrass) theorem

Let f : [a,b] be continuous. Then f has a minimum and a maximum poi