Numerical Analysis - Know Before You Go

Notation

Due to the limitations of HTML and the awkwardness of embedding small images within text, certain mathematical notation will be displayed in a non-standard format.

NotationDescriptionExample(s)
Limit(f ,x) 'f ' is a function expression. 'x' is the variable and the limit such as 'x→5' meaning 'as X approaches 5'. Limit((x+2)/(x+5),x→-5)
Integral(f ,x,[y]) 'f ' is a function of one or two variables. 'x' is the first variable of integration with optional values of integration. 'y' is the second variable of integration with option values of integration. Integral(x*sin(x),x)
Integral(ey+x,y=1..5,x=0..3)
∑(f ,i=1..n) '∑(f(i),i=1..n)' is the sum of f(i) for all the integer values of i from 1 to n.

Calculus

Limits

A function f defined on a set R of real numbers has the limit L at x0, written Limit(f(x),x→x0)=L if, given any real number ε > 0, there exists a real number δ > 0 such that |f - L| < ε, whenever x ∈ X and 0 < |x - x0| < δ.

Continuous Functions

Let f be a function defined on a set X of real numbers and x0 ∈ X. Then f is continuous at x0 if Limit(f(x),x→ x0)=f(x). The function f is continuous on the set X if it is continuous at each number in X.

The notation for 'f is continuous on the open interval a, b' is f ∈ C[a,b].

Convergent Sequences

Let {xn}n=1∞ be an infinite sequence of real or complex numbers. The sequence {xn}n=1∞ has the limit x (converges to x) if, for any ε > 0 there exits a positive integer N(ε) |xn - x| < ε whenever n > N(ε). The notation Limit(xn),n → ∞)=x., or xn → x as n → ∞ means the sequence xn}n=1∞ converges to x.

Functions of Convergent Sequences

If f is a function defined on a set X of real numbers and x0 ∈ X, then the following statements are equivalent:

  1. f is continuous at x0;
  2. If {xn}n=1∞ is any sequence in X converging to x0, then Limit(f(xn),n→∞) = f(x0).

Differentiability

Let f be a function defined in an open interval containing x0. The function f is differentiable at x0 if f ′(x0) = Limit((f(x) - f(x0))/(x - x0),x→x0) exists. A function that has a derivative at each number in a set X is differentiable on X.

Continuity and Differentiability

If f is differentiable at x0, then f is continuous at x0.

Rolle's Theorem

Suppose f ∈ C[a.b] and f is differentiable on (a,b). If f(a) = f(b), then a number c in (a,b) exists with f ′(c) = 0.

Mean Value Theorem

if f ∈ C[a,b] and f is differentiable on (a,b), then a number c in (a,b) exists with f ′(c) = (f(b) - f(a)) / (b - a).

Extreme Value Theorem

If f ∈ C[a,b] then c1, c2 ∈ [a,b] exist with f(c1) ≤ f(x) &le f(c2), for all x ∈ [a,b]. In addition, if f is differentiable on (a,b), then the numbers c1 and c2 occur either at the endpoints of [a,b] or where f ′ is zero.

Riemann Integral

The Riemann integral of the function f on the interval [a,b] is the following limit, provided it exists: Integral( f(x), x ) = Limit( ∑(f(zi) Δx, i=1..n ), max Δxi→0), where the numbers x0, x1, ..., xn satisfy a = x0 &le x1 &le ... &le xn = b and where Δxi = xi - xi-1, for each i = 1, 2, ..., n, and zi is arbitrarily chosen in the interval [xi-1, xi].

Weighted Mean Value Theorem for Integrals

Suppose f ∈ C[a,b], the Reimann integral of g exists on [a,b], and g (x) does not change sign on [a,b]. Then there exists a number c in (a,b) such that Integral( f(x) g (x), x ) = f(c) Integral( g (x), x ).

Average Value