Mathematics - Formulario

Class 2: Functions of a real variable

Functions of a real variable denotednumber,realuniqueaDinxelementeverytoassociatingruleaisvariablerealoneoffunctionrealaDGiven ℝ⊆ f(x)xD:ff(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≠xdomain it is a function or not x2n 0≥xdomain: roots even xnot a functiona function inverse function f ⁻¹ 0>xx domainlog: logarithms.

Let A, B and C, f : A B and g : f(B) C, the composite function g f (g after f) is: ℝ ∘⊆ #÷CBAg f : A C, g f(x) = g(f(x))∘ g(f(x))f(x)(x)iB non commutative.

Injective (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)) = x.

Class 3: Derivatives - linear approximation

The 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' fpositive 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- Differentiationl'1- f (x) · g(x) - f(x) · g (x)f(x) | a > 0 | a ≠ 1Quotient: in= f (x) = 1f(x) = log (x)f (x) = 0.f(x) = k ag(x) (g(x)) x · ln (a)f (x) = 1f(x) = x g - 1 f(x) = ln (x)(f (g(x)) · g (x)(f (g(x) ) f (x) = 1: g' f (x) = g · xf(x) = x÷Product: xxx 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 - xx [-e ]f (x) = ef(x) = e [with -x]Sum & Diff: (a f(x) b g(x)) af (x) bg (x) x∓ ∓ z = g(x) f(x) = xf(x) = g(x)Roots: f (x) = 1f (x) = g · zsolution 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

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 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 existI 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 differentiability

Consider: Df ℝ ℝ⊆.

• Function f is at a D if lim f (x) = f (a)continuous x a∈.
• Function f is in D, if it is continuous at each a Dcontinuous ∈| ' -Lemmapolynomials: ℝ.

Identifying continuous functions: continuous ⅔powers: | x [0, +∞) A function is continuous at aℝ.

If 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 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

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. f(c < ε stop here otherwise | 3. If f(a )·f(c < 0 set: ₊₁ ₊₁) ₊₁)n n n nI = [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 nAs n goes larger the c gets closer to a solution of f(c) = 0n.

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, coneℝ ℝ² ℝ³-

Consider function f : D R R:⊆ convexEpigraph 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)teThe slopes of tangentLet I be an interval and let f : I be a twice differentiable function, then:ℝ lines increasef on I f′′(x) ≥ 0 for all x I f on I f′′(x) ≤ 0 for all x Iconvex 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 sufficiente]2. c inflection point f′′ changes sign at c.

Class 9 - 10: Optimization

Optimization (Used in profit maximization) } May failLet 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 Dℝ ℝ⊂ to existµit'1 lmax f = f(c) min f = f(d). minimum value of f in Dmaximum value of f in DLemma 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 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 maximumIn particular (a − ε, a + ε), where ε > 0, is a neighborhood of a local maximumf : 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 a1. 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 rarely2. f’’(x) ≤ 0 for all x (a,b) c is a maximum point [< local] f concave∈ conclusive3. f’’(x) = 0 anything.

The extreme value (Weierstrass) theoremLet f : [a,b] be continuous. Then f has a minimum and a maximum point.