Math formulas

Def of a real variable function

A= domain

R= codomain

Rule associating every element of A with a real number

Range vs Codomain

Range is the set of values taken by the function while the Codomain consists of all real numbers

Definition of Surjectivity

f is surjective when the range=the codomain

Multiple inputs may correspond to the same output here

Every x value corresponds to at least one y value in the codomain

Definition of Injectivitiy

f is injective when: for every x₁ and x₂ included in A (where x₁ does not equal x₂), f(x₁) does not equal f(x₂)

Injectivity can be seen by using the horizontal line test

These functions are also invertible

Definition of Bijectivity

A function is bijective if it is both surjective and injective

A function is increasing if…

(see image for general definiton)

f is strictly increasing if f(x₁) < f(x₂)

A function is decreasing if…

(see image for general definiton)

f is strictly increasing if f(x₁) > f(x₂)

Bounded Below

If K belongs to R s.t. f(x) is greater than or equal to k for all values of x which are less than A

the values of the function will always be greater than the k value

Bounded Above

If k belongs to R s.t. f(x) is less than or equal to k for all values of x which are less than A

the values of the function will always be less than the k value

Definition of a bounded function

a function is bounded if it is bounded both above and below

Definition of a sequence

A sequence is a function which assigns every natural number with a real number

a_n : N —> R

The values taken by a sequence are called the terms of a sequence

a_n= {a₀, a₁, a₂, … a_n}

The graph of a sequence

The natural number and its corresponding pair

= {(n, a_n) : n belonging to N}

Properties of sequences

a_n is positive if a_n is greater than or equal to n belonging to N

a_n is negative if a_n is less than or equal to n belonging to N

Boundedness of a Sequence

a_n is unbounded if it is unbounded

Monotonicity of a sequence

Definition of a geometric sequence

A sequence of real numbers s.t. the ratio between the consecutive terms is constant

i.e. it has a multiplier

Properties of a geometric sequence

q>1 increasing exponential behavior, unbounded

q=1 a_n= 1, constant

0<q<1 decreasing exponential behavior, bounded

-1<q<1 oscillates between positive and negative, bounded by K belonging to R; where R is greater than or equal to 1

Formula for the sum of terms in a geometric sequence

Definition of an Arithmetic sequence

A sequence of real numbers s.t. the difference between the two numbers is constant

Definition of a sequence defined by Recursion

A sequence is defined by recursion if an initial value is assigned and a law is used to compute a_n+1 from a_n (or a_n from a_n-1)

Definition of a Global Extreme

f(x₀) is the maximum/minimum value

The point must belong to the domain

The same definition applies for a strict max/min but the sign changes to < or >

Max/Min with a neighborhood of points

The same definition applies for a strict max/min but the sign changes to < or >

Global extrema = local extrema

Definition of Convex Functions

For a function to be concave or convex it must be defined on that interval

Image is the definition for strict convexity. A function is just convex if there is a less than or equal to symbol

Definition of Concave Funtions

Constant functions (f(x)=k) are both concave and convex

Image is the definition for strict concavity. A function is just concave if there is a less than or equal to symbol

Definition of an Inflection Point

X₀ belonging to I is an inflection point if there exists N_r(x₀) where f changes from concave to convex or convex to concave

Definition of a convergent sequence

A sequence converges to the real number L if for any y, e is greater than 0, the distance of a_n from L is less than e when n is large enough (when n —> infinity)

Definition of a divergent sequence

A sequence diverges to ± infinity for all real numbers N, we have a_n>M (a_n<M) when H is large enough a_n>M (a_n<M), for all n > n₀

Definition of the limit of a real variable function

Uniqueness of Limits

If a_n has a limit (finite or infinite) then it is unique

Existence of limit of a monotonic sequence

A monotonic sequence is always regular

If a_n is increasing bounded above or decreasing bounded below a_n converges

If a_n is increasing unbounded above it diverges to + infinity

If a_n is decreasing unbounded below it diverges to - infinity

Existence of a limit for monotonic functions

Let a<c<b and let f be monotonic on the interval (ab)

a) the limits lim as x—>c⁺ of f(x) and lim as x—> c^- of f(x) both exist

b) the limits lim as x—> a⁺ of f(x) and lim as x—>b^- of f(x) both exist

Zero value theorem

Let f be defined on [a,b], continuous on [a,b] s.t f(a)*f(b)<0 so there exists c included in (a,b) s.t. f(c)= 0

If f is continuous on [a,b] and strictly monotonic the zero is unique

Weiastrass theorem

If a function is defined on a continuous, bounded, and closed interval f has a global max and global min in the interval

Difference quotient

Let x move to the x₀+h, with h>0 or h<0

When f values f(x₀+h), the variation of f: Δf=f(x₀+h)-f(x₀)

Ratio: Δf/h = Δf/Δx = (f(x₀+h)-f(x₀))/h

Definition of a derivative with h

Equation of a tangent line at point (x₀, f(x₀))

x₀= starting point

y= f(x₀) + f^’(x₀)(x-x₀)

Theorem of derivability and continuity at a point

Definition of a stationary point

Fermat’s Theorem

Definition of differentiability

Def of 1st derivative

The linear term m(x₀)*h is called the first differential of f at x₀ with the increment h

Theorem of derivability and differentiability

Taylor’s formula of the first order with Peano’s remainder

Taylor’s formula of the first order WITHOUT Peano’s remainder

Limit definition of a second order derivative

Taylors formula of the second degree

Note: this is also called Maclauren’s formula

Maclauren’s formula is just Taylor’s formula of the second degree in the neighborhood of zero

Definition of inflection points without derivatives

x₀ ∈ Domf is an inflection point if f is concave (convex) in a left neighborhood and convex (concave) in a right neighborhood

Definition of orthogonal vectors

Definition of a linear combination of vectors

Definition of a convex linear combination

Definition of linearly DEPENDENT vectors

a set of vectors is linearly dependent if at least one of them can be written as a linear combination of the others

Definition of linearly INDEPENDENT vectors

A set of vectors is linearly independent if none of them can be written as a linear combination of the other

Theorem of Linear DEPENDENCE

Theorem of Linear INDEPENDENCE

Definition of a matrix

A matrix (mxn) is a table of m rows and n columns of real numbers

Definition of a square matrix

A square matrix A(nxn) is symetric if the elements with are opposite with respect to the main diagonal are equal

Definition of Transpositon

Given A(mxn), the transpose of A, A^T is a matrix n/m whose rows are the columns of A and whose columns are the rows of A

Definition of Addition

A(mxn) + B(mxn)= C(mxn) s.t. C_ij = a_ij + b_ij

Product by a scalar

K*A(mxn)= C(mxn) s.t. C_ij= K*a_ij

Determinant of a SQUARE matrix

The determinant of A(nxn), detA or |A|, is a real number associated to the matrix calculated as:

if n=1 A=[a₁₁] detA= a₁₁

if n=2 detA = a₁₁*a₂₂ - a₁₂*a₂₁

Leplace rule

if n is greater than or equal to 3

The determinant is the sum of the product of each element of any row or column of the matrix, but the corresponding cofactor or algebraic complement

Binet Theorem

If A, B are matrices of order n then |A*B| = |A|*|B|

Definition of a singular square matrix

A(nxn) is said to be singular if |A|=0

Theorem of linearly DEPENDENT vectors with square matrix

n vectors of Rⁿ are linearly dependent iff the determinant of the square matrix containing the vectors is equal to zero

Theorem of linearly INDEPENDENT vectors with square matrix

n vectors of Rⁿ are linearly independent iff the determinant of the square matrix containing the vectors is different from zero

Definition of an inverse matrix

Given A(nxn), the inverse matrix of A, if it exists, is a matrix B(nxn) s.t. A*B=B*A=I_n

I_n is the identity matrix

Notation A^-1

Definition of the Rank of a Matrix

the rank of A(mxn) is the maximum number of linearly independent vectors (either row or column vectors)

alternative definition: the rank of A(mxn) is the maximum order of a submatrix (square) with a determinant that is not equal to zero

Kroneker algorithm

A(mxn) has rank r iff all the minors of order r+1 which are obtained augmenting (adding rows/columns) the submatrix of order r, are equal to zero

Cramer’s theorem

Given Ax=b where A(nxn) with |A| not equal to zero

Ax=b has 1 and only 1 solution calculated as x= A^-1*b

Cramer’s rule

Rouche’- Cappelli Theorem