Statements Of Definitions Flashcards
The Principle of Mathematical Induction
Suppose that for each positive integer n we have a statement P(n).
If we prove the following two things:
(a) P(1) is true;
(b) for all n, if P(n) is true then P(n + 1) is also true; then P(n) is true for all positive integers n.
The Principle of Mathematical Induction II
Let k be an integer.
Suppose that for each integer n ≥ k we have a statement P(n).
If we prove the following two things:
(a) P(k) is true;
(b) for all n ≥ k, if P(n) is true then P(n+1) is also true; then P(n) is true for all integers n ≥ k.
The Strong Principle of Mathematical Induction
Suppose that for each integer n ≥ k we have a statement P(n). If we prove the following two things: (a) P(k) is true; (b) for all n, if P(k),P(k+1),...,P(n) are all true, then P(n+1) is also true; then P(n) is true for all n ≥ k.
Prime number
A prime number is a positive integer p such that p ≥ 2 and the only positive integers dividing p are 1 and p.
Plane graph
A plane graph is a figure in the plane consisting of a collection of points (vertices), and some edges joining various pairs of these points, with no two edges crossing each other.
Division of integers
Let a,b ∈ Z. We say a divides b (or a is a factor of
b) if b = ac for some integer c. When a divides b, we write a|b.
Highest Common Factor
The highest common factor of a and b, written hcf(a,b), is the largest positive integer that divides both a and b.
Coprimality
If a,b ∈ Z and hcf(a,b) = 1, we say that a and b are
coprime to each other
Congruence of integers
Let m be a positive integer. For a, b ∈ Z, if m divides b−a we write a≡b mod m and say a is congruent to b modulo m.
Derivative (First Principles)
The derivative of a function f at a number a, denoted by f’(a) is
f’(a) = lim(h->0) f(a+h)-f(a) / h
if this limit exists.
Critical Number
A critical number of a function f is a number c in the domain of f such that either f’(c) = 0 or f’(c) does not exist.
Concavity
If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I.
If the graph of f lies below all of its tangents on I, it is called concave downward on I
Inflection Point
A point P on a curve y = f(x) is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P
Dot Product
If a = and b = , then the dot product of a and b is the number a.b given by
a.b = a1b1 + a2b2 + a3b3
Cross Product
If a = and b = , then the cross product of a and b is the vector
axb =
Squeeze Theorem (statement)
It says that if g(x) is squeezed between f(x) and h(x) near a, and if f and h have the same limit L at a, then g is forced to have the same limit L at a.
Differentiation Formulae: Power Rule (statement)
If n is any real number, then
d/dx(x^n) = nx^(n-1)
Differentiation Formulae: Power Rule (proof)
Hhhh
The formula
x^n - a^n = (x-a)[x^(n-1) + (x^(n-2)a + ∙∙∙ + xa^(n-2) + a^(n-1)]
can be verified simply by multiplying out the right-hand side (or by summing the second factor as a geometric series).
If fsxd − xn, we can use Equation 2.1.5 for f9sad and the equation above to write
Second prooF
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fsxd 2 fsad xn 2 an − lim
f 9sad − lim
xla x2a xla x2a
− lim sxn21 1 xn22a 1 ∙∙∙ 1 xan22 1 an21d xla
−an21 1an22a1∙∙∙1aan22 1an21 − nan21
Differentiation Formulae: Sum Rules (statement)
A
Differentiation Formulae: Sum Rule (proof)
A
Differentiation Formulae Constant Multiple Rule (statement)
A
Differentiation Formulae: Constant Multiple Rule (proof)
A
Differentiation Formulae Product Rule (statement)
A
Differentiation Formulae: Product Rules (proof)
A
Differentiation Formulae Quotient Rule (statement)
A
Differentiation Formulae Quotient Rule (proof)
A
Differentiation Formulae: Chain Rule: statement
A
Mean Value Theorem: statement
A
Rolle’s Theorem: statement
A
2 vector statements that I can’t format 😬😬
A
connectedness of a plane graph
A plane graph is connected if we can get from any vertex of the graph to any other vertex by going along a path of edges in the graph.
