num sol midterm

Continuity

at a point if its graph has no breaks, holes, or jumps at that point. Mathematically, a function f(x) is continuous at x=c if:

Differentiability

at a point if it has a well-defined tangent (slope) at that point (means that the derivative is there: Left Hand and Right Hand Derivative) In simple words: The graph must be smooth at that point — no sharp corners or vertical tangents

Georg Friedrich Bernhard Riemann

was a German mathematician who made profound contributions to analysis, number theory, and differential geometry . In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series

Riemann sum

n approximation of a region's area, obtained by adding up the areas of multiple simplified slices of the region. It is applied in calculus to formalize the method of exhaustion, used to determine the area of a region. This process yields the integral, which computes the value of the area exactly

Taylor series

is of great

value in the study of numerical methods. It provides a means to predict a

function value at one point in terms of the function value and its derivatives

at another point. In particular, the theorem states that any smooth function

can be approximated as a polynomia

matrix

is any rectangular array of numbers or functions:

entries or elements

The number or functions in the array

square matrix

An n × n matrix

main diagonal entries

For an n × n square matrix, the entries aa11, a22, a33,

column vector

An n × 1

row vecto

A 1 × n matrix

different

Matrices that are not equal

MATRIX TRANSPOSITION

We obtain the transpose of a matric by writing its rows as columns (or

equivalently its columns as rows).

Symmetric

matrices are square matrices whose transpose equals the matrix

itsel

Skew-symmetric

matrices are square matrices whose transpose equals

minus the matrix

Upper triangular matrices

are square matrices that can have nonzero entries

only on and above the main diagonal, whereas any entry below the diagonal

must be zero

Lower triangular matrices

are square matrices that can have nonzero entries

only on and below the main diagonal, whereas any entry above the diagonal

must be zero

Diagonal Matrices

These are square matrices that can have nonzero entries only on the main

diagonal. Any entry above or below the main diagonal must be zero

Scalar Matrice

f all the diagonal entries of a diagonal matrix s are equal, we call s

unit matrix

In particular, a scalar matrix, whose entries on the main diagonal are all 1, is denoted by I