Mathematics - Formulario

BAg f : A C, g f(x) = g(f(x))∘ g(f(x))f(x)(x)iB non commutativeInvective (one-to-one) : each element in BA .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)) = xClass 3 : derivaitves - linear approximationThe first derivative is the slope of the tangent line to the graph of f at (x ,f(x ))p pDenoted by f (x ) : or secantp Approximation error :tangent ...My curve is decreasing when m is negative " The linear approximation of : Df ℝ⊆My curve is increasing when m is positive at D isa the functionℝ : :∈ È!Monotonicity theorem : f (x) > 0 for all x in the interval I f increasing: " 1. compute f (x).2. change x with aA strictly increasing errorapproximationf (x) < 0 for all x in the interval I f decreasing its graph is the tangent line to(resp., decreasing) maynot have a strictly f (x) = 0 for all x in the interval I f constant the graph of'

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

Rules:

• Product: f'(x) = f(x) * g'(x) - f'(x) * g(x)
• Quotient: f'(x) = (f(x) * g'(x) - f'(x) * g(x)) / (g(x))^2
• Chain: f'(x) = f'(g(x)) * g'(x)
• Sum & Difference: (f(x) + g(x))' = f'(x) + g'(x) and (f(x) - g(x))' = f'(x) - g'(x)
• Product Rule: (f(x) * g(x))' = f'(x) * g(x) + f(x) * g'(x)
• Quotient Rule: (f(x) / g(x))' = (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2
• Power Rule: (x^n)' = n * x^(n-1)
• Exponential Rule: (a^x)' = a^x * ln(a)
• Logarithmic Rule: (log_a(x))' = 1 / (x * ln(a))
• Root Rule: (sqrt(x))' = 1 / (2 * sqrt(x))

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'(x)

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

f(a)_Class 4 - 5 : limitsDefinition : 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 toA than ε.

Limits computation : ∞withLimitsNeither f(x) nor g(x) can

be eliminated numberscomposea number ifI can undo the number, but not the denominator ifand ifLeft and right limits may simplify ifhave different results = limitdoes not esxistI can undo only the denominator Complex cases :Logarithms & Exponentials[for a > 1, b > 0, c > 0]+ (right) [>x₀]- (left) [<x₀] does not existifI can undo the nominator and the denominator② ifscompose 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∈ padreIf 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 differentiabilityConsider :Df 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 Dcontinuous ∈| ' -Lemmapolynomials : ℝIdentifying continuous functions :continuous ⅔powers : | x [0, +∞) A function is continuous at aℝIf f and g are continuous functions at a then : even root : [0, +∞); point a of the domain if andf + g and f - g are continuous at a odd root : ℝi. only if:exponentials : ℝf · 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) = Adiscontinuous x a+if f is one-to-one on interval [b,c], f:[b,c] then f⁻¹ is continuousℝ 3. f(a)=A.Continuity vs Differentiability : Kink : a point where the? tangents from the left and thedifferentiable continuous [NOT viceversa]f f•• right exist but are differentf(x) = |x| is continuous at but not differentiable at ( )a a••Differentiable functions are smooth: they do not have kinksLeft 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 valuef fbetween 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 point 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 nSteps:

1. Take into consideration the mid point: c = |
2. If f(c) < ε stop here otherwise |
3. If f(a) · f(c) < 0 set: ₊₁ ₊₁) ₊₁)
4. 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]

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

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

D ⊆ ℝ

Epigraph of f: Epi(f) = {(x,y) | x ∈ D, y ≥ f(x)} ∈ ℝ²

Hypograph of f: Hyp(f) = {(x,y) | x ∈ D, y ≤ f(x)} ∈ ℝ²

Let f : [a, b] → ℝ be convex or concave

f continuous in (a, b)

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 ∈ ℝ

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) → ℝ², 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 existµit'1 lmax f = f(c) min f = f(d). minimum value of f in Dmaximum 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 convex1. f increasing in (a, c) 1. f decreasing in (a, c)if concave↳ Then c is a minimum point2. 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 isℝ 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

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