theory statements

Question 1

proposition 2.3 (sqrt(2) is irrational)

Answer 1

√2 is NOT rational

Question 2

de moivre’s theorem

Answer 2

Let z1, z2 be complex numbers with polar forms

Question 3

the fundamental theorem of algebra

Answer 3

Every polynomial equation of degree at least 1 has a root in C

Question 4

euler’s formula

Answer 4

For a convex polyhedron with V vertices, E edges and F faces, we have
V − E + F = 2

Question 5

the fundamental theorem of arthmetic

Answer 5

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 . ∈
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

Question 6

theorem 12.1 (there are infinitely many prime numbers)

Answer 6

n/(ln(n))

Question 7

fermat’s little theorem

Answer 7

Let p be a prime number, and let a be an integer that is not divisible by p . Then a^ p − 1 ≡ 1 mod p

Question 8

binomial and multinomial theorems

Answer 8

Let P be a process which consists of n stages, and suppose that for each r , the r^th stage can be carried out in a r ways. Then P can be carried out in a 1 a 2 . . . a (n) ways.

Question 9

inclusion-exclusion principle

Answer 9

Logically enough, we call a set S a finite set if it has only a finite number of elements. If S has n elements, we write Sn. If a set is not finite, it is said to be an infinite set.

For example, if S = {1,3,2), then S is finite and S = 3. And Z is an infinite set.

Question 10

real numbers are uncountable

Question 11

the powerset of a set has strictly larger cardinality

Answer 11

│N│ = │Q│= │N x N│ and │N│ < │R│

Question 12

the subgroup test

Answer 12

Let G be a group, and let H be a subset of G . The H is a subgroup of G if the following three conditions hold: ∈ (1) e (2) x , y (3) x ∈ ∈ H (where e is the identity element of G ), H xy H ⇒ ⇒ x − 1 ∈ H , ∈ H

Question 13

lagrange’s theorem

Question 14

squeeze theorem

Question 15

differentiation formulae: power rule

Answer 15

power rule is given by, d(xn)/dx OR (xn)’ = nx^n-1, where n is a real number.

d(x^n)/dx = n.x^n-1

Question 16

differentiation formulae: sum rule

Answer 16

If f and g are continuous at a point c , then the sum f + g is continuous at c .

Question 17

differentiation formulae: constant multiple rule

Answer 17

d(k f(x))/dx = k d(f((x))/dx

Question 18

differentiation formulae: product rule

Answer 18

If f and g are continuous at c , then the product function f . g (defined by ( f . g )( x ) = f ( x ) g ( x ) ) is continuous at c

Question 19

differentiation formulae: quotient rule

Answer 19

If f and g are continuous at c , and g ( x ) = 0 for all x , then the quotient f g is continuous at c

Question 20

differentiation formulae: chain rule

Answer 20

Dy/dx = dy/du × du/dx

Question 21

rolle’s theorem

Answer 21

“If a function f is defined in the closed interval [a, b] in such a way that it satisfies the following condition: i) f is continuous on [a, b], ii) f is differentiable on (a, b), and iii) f (a) = f (b), then there exists at least one value of x, let us assume this value to be c, which lies between a and b i.e. (a < c < b ) in such a way that f‘(c) = 0.”

In short: uwu
1) f(x) is continuous on [a,b],
2) f(x) is differentiable on (a,b), and
3)f(a) = f(b)

then there exisits at least on c in (a,b) such that f’(c) = 0

Question 22

mean value theorem

Answer 22

The function f(x) is continuous over the interval [a, b].
The function f(x) is differentiable over the interval (a, b).
There exists a point c in (a, b) such that f’(c) = [ f(b) - f(a) ] / (b - a)

Question 23

→a and →b are orthogonal if and only if →a*→b = 0

Question 24

→a x →b is orthogonal to both →a and →b

Question 25

the fundamental theorem of calculus

Answer 25

For a continuous function f(x) on an interval [a, b] with the integral: F(x) = ∫xa f(t) dt then the derivative of the integral F(x) gets us the original function f(x) back again: F'(x) = f(x)

Question 26

general solution of the nonhomogeneous 2nd order differential equations as a sum of the complementary function and particular solution

Question 27

l'hospital's rule

Answer 27

lim f(x)/ lim f’(x)/ = x→c g(x) x→c g’(x) The limit as x approaches c of "f-of−x over g-of−x" equals the the limit as x approaches c of "f-dash-of−x over g-dash-of−x"

Question 28

matrix addition (theorem 3.2)

Question 29

matrix multiplication (theorem 3.3)

Question 30

matrix transpse (theorem 3.4)

Question 31

matrix symmetry (theorem 3.5)