1. fundamental theorem of algebra
\(n\)-degree polynomial p(x) has \(n\) roots
i) roots of complex numbers including repeated zeros
ii) (\(x\)-root) is a factor of p(x)
iii) a zero of p(x) means a root of solution p(x)=0
2. complex conjugates theorem
paired roots \( a \pm bi \) for real coefficient polynormial
where \(a\) and \(b\) area real numbers and \( b \neq 0\)
3. decartes' law of signs
# of positive real zeros of \( p(x) \) is
(i) # of changes in signs of coefficients of \( p(x) \)
(ii) # of changes in signs minus 2 or 4 or 6...
# of negative real zeros of \( p(x) \) is
(i) # of changes in signs of coefficients of \( p(-x) \)
(ii) # of changes in signs minus 2 or 4 or 6...
4. newton's sum
let sm be the sum of mth power of roots of \( p(x) \)
\[p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0\]
then,
\[ a_ns_1+a_{n-1}=0 \]
\[ a_ns_2+a_{n-1}s_1+2a_{n-2}=0 \]
\[ a_ns_3+a_{n-1}s_2+a_{n-2}s_1+3a_{n-3}=0 \]
\[ a_ns_4+a_{n-1}s_3+a_{n-2}s_2+a_{n-3}s_1+4a_{n-4}=0 \]
...
1,2,3,4... after s1
5. vieta's formula
for \(p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0\)
kth symmetric sum is
\[ (-1)^k {a_{n-k} \over a_n} \]
6. remainder theorem
if polynormial \( p(x) \) is divided by \( x-r \)
i) remainder is \( p(r) \)
ii) \( p(x) \)=\( q(x) \)\( (x-r)\)+\( p(r) \)
where \( q(x) \) is one degree less (depressed polynomial)
7. factor theorem
\( x-r \) is a factor of \( p(x) \) if and only if \( p(r)=0\)
8. reciprocal transformation
for \(p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0\) with roots \( r_1,r_2,...,r_n \)9. rational zero theorem
for \(p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0\) with integral coefficients
its rational zero is
\[ a_0 \over a_n \]
10. location principle
for \( p(x) \) if \( p(a)<0 \) and \( p(b)>0 \)
\( p(x) \) has at least one real zero between \( a \) and \( b \)
where \( a \) and \( b \) are real numbers
11. synthetic/long division
12. derivatives of polynormial
for \( n \)-degree polynormial, its nth derivative is a constant
\[ a \times n!\]
where a is leading coefficient
1. infinite geometric \[ s = {a_1 \over 1-q} \]
2. finite geometric \[ s = {a_1(1-q^n) \over 1-q} \]
3. recursive vs explicit
recursion is key to generalize explicit equation for induction
4. binomial theorem \[ (x+y)^n= {n \choose 0} x^ny^0+ {n \choose 1} x^{n-1}y^1+...+ {n \choose n} x^0y^n\]
5. trinomial expansion \[ (x+y+z)^n= \sum_{i,j,k} {n \choose i,j,k} x^iy^jz^k \] where \( i+j+k=n \) and \[ {n \choose i,j,k} = {n! \over i!j!k!} \]
6. repeating decimal
geometric sequence
7. telescoping series \[ \sum_{i=1}^n {1 \over i(i+1)} = {n \over n+1} \]
8. sum of squares \[ \sum_{i=1}^n {k^2} = {n(n+1)(2n+1)\over 6} \]
9. sum of cubes \[ \sum_{i=1}^n {k^3} = {n^2(n+1)^2\over 4} \]
10. taylor's \[ \sum_{k=0}^{\infty} {{f^{(k)}(a) \over {k!}} (x-a)^k}\]
11. maclaurin's \[ \sum_{k=0}^{\infty} {{f^{(k)}(0) \over {k!}} x^k}\]
12. fibonacci
13. sum of two nth powers
i. special cases
\[a^{4n}+a^{-4n}=a^{2n}+a^{-2n}+2=a^{n}+a^{-n}+4\]
\[ a^3+b^3=(a+b)(a^2-ab+b^2)\]
ii. odd exponent
\[a^{n}+b^{n}=(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^{2}-\cdots -ab^{n-2}+b^{n-1}) \]
iii. even exponent
expression cannot be factorized without introducing complex numbers
14. difference of two nth powers
i. even exponent
\[a^{2n}-b^{2n}=(a^{n}+b^{n})(a^{n}-b^{n}) \]
ii. any exponent
\[a^{n}-b^{n}=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^{2}+\cdots +ab^{n-2}+b^{n-1}) \]
where the latter factor is a geometric sequence with ratio of \( b \over a \)
1. AM-GM \[ {a_1+a_2+...+a_n \over n} \ge \sqrt[n]{a_1a_2....a_n}\]
2. Cauchy's \[ {(a_1b_1+a_2b_2+...+a_nb_n)^2 \le (a_1^2+a_2^2+....+a_n^2)(b_1^2+b_2^2+....+b_n^2)}\]
3. Jensen's
secant line (weighted means) of a concave up function lies above function
\[ tf(x_1)+(1-t)f(x_2) \ge f(tx_1+(1-t)(x_2)) \]
where \( t \in [0,1] \)
secant line of a concave down function lies beneath function
\[ tf(x_1)+(1-t)f(x_2) \le f(tx_1+(1-t)(x_2)) \]
where \( t \in [0,1] \)
1. sum/difference \[ log(ab)=log(a) + log(b) \] \[ log({a \over b})=log(a) - log(b) \]
2. exponent \[ log(a^n)=nlog(a)\] \[ log_{a^n}(b)={1 \over n}log_a (b)\] \[ a^{log_a(b)}=b\] \[ a^{nlog_a(b)}=b^n\] \[ a^{n^2}=(a^n)^n\]
3. change base formula \[ log_a b={log_c b \over log_c a} \]
4. nested logarithmic
\[ y=log_a (log_b(log_c x)))\]
domain: \( x>0 \) and \( log_c x >0 \) and \( log_b(log_c x) >0 \)
5. exponential function
\[ f(x)=a^x\]
where a>0 and \(a \neq 1\)
i. growth: a>1
ii. decay: \(0 \lt a \lt 1\)
6. logarithmic function
\[ f(x)=log_a x\]
domain: \( x>0 \)
+ correlation: a > 1
- correlation: 0 < a < 1
1. complex conjugate
for \( z=a+bi \), \( \bar z=a-bi \)
for \( z=re^{i \theta} \), \( \bar z=re^{-i \theta} \)
2. complex power \[ (rcis \theta)^n=r^ncis(n \theta) \] \[ {1 \over cis(\theta)}=-cis(\theta)\]
3. demoivre's theroem \[ (rcis \theta)^{1/n}=r^{1 \over n} cis( {{2k \pi + \theta} \over n}) \] n roots of \( a^n=1 \) are \[ 1,cos({2 \pi \over n})+sin({2 \pi \over n}) ,cos({4 \pi \over n})+sin({4 \pi \over n}), ... \]
4. euler's formula \[ e^{i \pi} +1=0 \]
5. hyperbolic sine/cosine \[ sinh={{e^x - e^{-x}} \over 2} \] \[ cosh={{e^x + e^{-x}} \over 2} \]
1. circle
\[(x-a)^2+(y-b)^2=r^2 \]
2. ellipse
set of points whose distance from foci sum to a constant
\[ {(x-x_0)^2 \over a^2} + {(y-y_0)^2 \over b^2} =1 \]
major/minor axis \(2a/2b\)
foci \( \sqrt {a^2-b^2} \)
3. hyperbola
set of points whose distance from foci differ by a constant
\[ {(x-x_0)^2 \over a^2} - {(y-y_0)^2 \over b^2} =1 \]
transverse/conjugate axis \(2a/2b\)
foci \( \sqrt {a^2+b^2} \)
asymptote \( y=\pm { b \over a}x \)
4. parabola
set of points equidistant from focus and directrix
4.1. general form
\[ f(x)=ax^2+bx+c \]
where vertex is \( (-{b \over 2a}, -{{b^2-4ac} \over 4a}) \)
axis of symmetry is x= \( -{b \over 2a} \)
focus is \( (-{b \over 2a}, {1 \over 4a} -{{b^2-4ac} \over 4a}) \)
directrix is \( y=-{1 \over 4a} -{{b^2-4ac} \over 4a} \)
4.2. vertex form
\[ f(x)=a(x+h)^2+k \]
where vertex is \( (-h,k) \)
axis of symmetry is x= \( -h \)
focus is \( (-h, {1 \over 4a} + k) \)
directrix is \( y=-{1 \over 4a} +k \)
5. general form
\[ax^2+by^2+cx+dy+e=0\]
i. parabola: a=0 or b=0
ii. circle: \(a=b \neq 0 \)
iii. hyperbola: a and b have opposite signs
iv. ellipse: \(a \neq b\) and \(a \neq 0\) and \(b \neq 0\),
a and b have same sign
1. slope-intercept form
\[ y=kx+b \]
k: slope
b: y-intercept
2. point-slope form
\[ y-y_1=k(x-x_1) \]
k: slope
\( (x_1,y_1) \): a point on the line
3. two-point form \[ y-y_1=({y_2-y_1 \over x_2-x_1})(x-x_1) \] \( (x_1,y_1),(x_2,y_2) \): two points on the line
4. standard form \[ ax+by+c=0 \]
5. point to line formula \[ |ax_0+by_0+c| \over \sqrt {a^2+b^2} \] \( (x_0,y_0) \): a point in 2d
1. symmetric form
\[ {{x-x_0} \over a} = {{y-y_0} \over b} = {{z-z_0} \over c}\]
\( (x_0,y_0,z_0) \): a point
\( \langle a,b,c \rangle \): direction vector of line
1. standard form \[ ax+by+cz+d=0 \]
2. point to plane formula \[ |ax_0+by_0+cz_0+d| \over \sqrt {a^2+b^2+c^2} \] \( (x_0,y_0,z_0) \): a point in 3d
1. cyclic function
2. parametric function
express function with a third variable
3. piecewise function
different equations on different part of domain
4. rational function
\[ f(x)={p(x) \over q(x)} \]
i. vertical asymptote x=a when a is zero of q(x) but not p(x)
ii. hole is x when x is zero of both q(x) and p(x)
iii. horizontal asymtote
\[ y=0 \]
where degree of p < q
\[ y={p_{lcoeff} \over q_{lcoeff}}\]
where degree of p and q the same, lcoeff is leading coefficient
5. inverse function
1. rays and polygons
i. \( |x|+y=n \)
rays reflection of y where y \( \leq \) n
ii. \( x+|y|=n \)
rays reflection of y where x \( \leq \) n
iii. \( |x|+|y|=n \)
rhombus
2. absolute parabola
3. absolute circles
1. easy points
i. discontinuities
ii. endpoints
iii. easy points (x=0/y=0 or special values)
2. critical points
\[ f'(x)=0\]
3. inflection points
\[ f''(x)=0\]
between critical point and discontinuity interval
i. concave up
ii. concave down
4. cusp
1. continuous at a
2. \( {dy \over dx} \to \pm \infty \) at a
1. rotation by \( \theta \) counterclockwise of origin
$$
\begin{bmatrix}
cos \theta & -sin \theta \\
sin \theta & cos \theta \\
\end{bmatrix}
$$
2. rotation by \( \theta \) clockwise of origin
$$
\begin{bmatrix}
cos \theta & sin \theta \\
-sin \theta & cos \theta \\
\end{bmatrix}
$$
3. rotation by \( \pi \over 2 \) counterclockwise of origin
$$
\begin{bmatrix}
0 & -1 \\
1 & 0 \\
\end{bmatrix}
$$
4. rotation by \( \pi \) of origin
$$
\begin{bmatrix}
-1 & 0 \\
0 & -1 \\
\end{bmatrix}
$$
5. g(x) by rotation of f(x) \( \theta \) ccw of origin
(1) set (x,y) \( \in \) g(x)
(2) reverse rotation by \( \theta \) (or cw)
$$
\begin{bmatrix}
cos \theta & sin \theta \\
-sin \theta & cos \theta \\
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
\end{bmatrix}
$$
(3) plug in transformed x and y in f(x)
6. (x,y) rotation by \( \theta \) counterclockwise of (a,b)
$$
\begin{bmatrix}
cos \theta & -sin \theta \\
sin \theta & cos \theta \\
\end{bmatrix}
\begin{bmatrix}
x-a \\
y-b \\
\end{bmatrix}
+
\begin{bmatrix}
a \\
b \\
\end{bmatrix}
$$
7. g(x) by rotation of f(x) \( \theta \) counterclockwise by (a,b)
(1) set (x,y) \( \in \) g(x)
(2) reverse rotation by \( \theta \) (or cw)
$$
\begin{bmatrix}
cos \theta & sin \theta \\
-sin \theta & cos \theta \\
\end{bmatrix}
\begin{bmatrix}
x-a \\
y-b \\
\end{bmatrix}
+
\begin{bmatrix}
a \\
b \\
\end{bmatrix}
$$
(3) plug in transformed x and y in f(x)
8. reflection by x
$$
\begin{bmatrix}
1 & 0 \\
0 & -1 \\
\end{bmatrix}
$$
9. reflection by y
$$
\begin{bmatrix}
-1 & 0 \\
0 & 1 \\
\end{bmatrix}
$$
10. reflection by y=x
$$
\begin{bmatrix}
0 & 1 \\
1 & 0 \\
\end{bmatrix}
$$
11. reflection by y=-x
$$
\begin{bmatrix}
0 & -1 \\
-1 & 0 \\
\end{bmatrix}
$$
12. reflection by y=mx
$$
{1 \over {1+m^2}}
\begin{bmatrix}
1-m^2 & 2m \\
2m & m^2-1 \\
\end{bmatrix}
$$
this derives from:
$$
\begin{bmatrix}
cos \theta & -sin \theta \\
sin \theta & cos \theta \\
\end{bmatrix}
\begin{bmatrix}
1 & 0 \\
0 & -1 \\
\end{bmatrix}
\begin{bmatrix}
cos \theta & sin \theta \\
-sin \theta & cos \theta \\
\end{bmatrix}
$$
13. reflection by y=mx+b
$$
{1 \over {1+m^2}}
\begin{bmatrix}
1-m^2 & 2m \\
2m & m^2-1 \\
\end{bmatrix}
\begin{bmatrix}
x \\
y-b \\
\end{bmatrix}
+
\begin{bmatrix}
0 \\
b \\
\end{bmatrix}
$$
14. g(x) - reflection of f(x) by y=mx
(1) set (x,y) \( \in \) g(x)
(2) reflection by y=mx
$$
{1 \over {1+m^2}}
\begin{bmatrix}
1-m^2 & 2m \\
2m & m^2-1 \\
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
\end{bmatrix}
$$
(3) plug in transformed x and y in f(x)
15. g(x) - reflection of f(x) by y=mx+b
(1) set (x,y) \( \in \) g(x)
(2) reflection by y=mx
$$
{1 \over {1+m^2}}
\begin{bmatrix}
1-m^2 & 2m \\
2m & m^2-1 \\
\end{bmatrix}
\begin{bmatrix}
x \\
y-b \\
\end{bmatrix}
+
\begin{bmatrix}
0 \\
b \\
\end{bmatrix}
$$
(3) plug in transformed x and y in f(x)
16. reflection-rotation-reflection-rotation
$$
\begin{bmatrix}
cos \theta & -sin \theta \\
sin \theta & cos \theta \\
\end{bmatrix}
\begin{bmatrix}
cos 2\theta & sin 2\theta \\
sin 2\theta & -cos 2\theta \\
\end{bmatrix}
\begin{bmatrix}
cos \theta & -sin \theta \\
sin \theta & cos \theta \\
\end{bmatrix}
\begin{bmatrix}
cos 2\theta & sin 2\theta \\
sin 2\theta & -cos 2\theta \\
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 \\
0 & 1 \\
\end{bmatrix}
$$
1. commutative \[ a*b=b*a \]
2. associative \[ a*(b*c)=(a*b)*c \]
3. identity element (neutral)
\( e*a=a \) and \( a*e=a \)
4. inverse
\( a*a^{-1}=e \) and \( a^{-1}*a=e \)
5. distributive \[ a*(b@c)=(a*b)@(a*c) \]
1. definition
for set G with operation *
i) * is associative
ii) exact 1 identity element e in G
ii) for every a in G, \(a^{-1}\) in G
2. \( \mathbb Z \) or \( <\mathbb Z, +> \)
additive group of the integers
3. \( \mathbb Q \) or \( <\mathbb Q, +> \)
additive group of the rational numbers
4. \( \mathbb R \) or \( <\mathbb R, +> \)
additive group of the real numbers
5. \( \mathbb Q^* \) or \( <\mathbb Q^*, \bullet> \)
nonzero rational numbers/operation of multiplication
6. \( \mathbb R^* \) or \( <\mathbb R^*, \bullet> \)
nonzero real numbers/operation of multiplication
7. finite group
finite number of elements
8. \( \mathbb {Z_n} \)
addition modulo n
9. abelian group
cummutative group
10. cancellation law
for elements a,b,c in G
\[ ab=ac \Rightarrow b=c \]
\[ ba=ca \Rightarrow b=c \]
11. theorems of inverses
for elements a,b in G
\[ ab=e \Rightarrow a=b^{-1} \]
\[ ab=e \Rightarrow b=a^{-1} \]
\[ (ab)^{-1}=b^{-1}a^{-1} \]
\[ ((a)^{-1})^{-1}=a\]
\[ (a_1a_2...a_n)^{-1}=a_n^{-1}a_{n-1}^{-1}...a_1^{-1} \]
12. |G|
order of group G
1. definition
for subset S in G
i. closed with respect to operation
ii. closed with respect to inverses
S is a subgroup of G
2. trivial subgroup
e or whole group
3. proper subgroup
all except e or whole group
4. generators
for elements a,b,c in G
subgroup S contains all possible products of a,b,c
and their inverses
a,b,c are generators of S
5. cyclic group
subgroup generated by a single element a
a is generator \( \langle a \rangle \)
\[ \langle g \rangle = \{g^n, n \in \mathbb Z\} \]
\[ \langle g \rangle = \{ng, n \in \mathbb Z\} \]
6. cayley diagram
i) generators
ii) defining equations
1. injection
for function: \( f: A \rightarrow B \)
each element of B is image of \( \le 1 \) element of A
\[ f(x_1)=f(x_2) \Rightarrow x_1=x_2 \]
2. surjection
for function: \( f: A \rightarrow B \)
each element of B is image of \( \ge 1 \) element of A
3. bijection
for function: \( f: A \rightarrow B \)
both injective and surjective
4. composite functions
for functions: \( f: A \rightarrow B \) and \( g: B \rightarrow C \)
i) f(x) and g(x) are injective \( \Rightarrow g \circ f \) is injective
ii) f(x) and g(x) are surjective \( \Rightarrow g \circ f \) is surjective
iii) f(x) and g(x) are bijective \( \Rightarrow g \circ f \) is bijective
5. inverse
for function: \( f: A \rightarrow B \)
it has inverse iff it is bijective
1. symmetric group
for any set A
group of all permutations of A (\( S_A \))
2. dihedral group
for regular polygons (\( D_n \))
3. disjoint cycles
commutative
theorem: every permutation is either
i) identity
ii) a single cycle
iii) product of disjoint cycles
4. transposition
a cycle f length 2
5. even/odd permutation
for \( \pi \in S\)
\( \pi \) cannot be both even and odd
1. regular polygon vertex coloring
(n-gon, m-color, no flipping)
\[ {\sum_{i=0}^{n-1}} {m^{gcd(i,n)} \over n} \]
ii. 10-gon 2 color
\[ {{(2^{10}+2^1+2^2+2^1+2^2+2^5+2^2+2^1+2^2+2^1)} \over 10}=108 \]
2. regular polygon vertex coloring
(n-gon, m-color, flipping)
1. vertex n color (no flipping) \[ {(n^4+n^1+n^2+n^1)} \over 4 \]
2. vertex n color (4-rotation, 4-reflection) \[ {(n^4+n^1+n^2+n^1+n^2+n^2+n^3+n^3)} \over 8 \]
1. symmetry of regular tetrahedron
i. \( 1 \) identity element
\[ (1)(2)(3)(4)\]
ii. \( 8 \) \( \pm \)120-degree face/vertex rotations
\[(1)(234)\]
iii. edge center to edge center
\( 3 \) 180-degree rotations
(6 edges divided by 2)
\[(12)(34)\]
2. burnside of tetrahedron (face)
\[ {n^{4}+11n^{2}} \over 12 \]
3. burnside of tetrahedron (vertex)
\[ {n^{4}+11n^{2}} \over 12 \]
4. burnside of tetrahedron (edge)
\[ {n^{6}+3n^{4}+8n^{2}} \over 12 \]
5. regular tetrahedron face n-color
a) one color: \( n \choose 1 \)
b) two colors (2+2 faces): \( n \choose 2 \)
c) two colors (1+3 faces): \( {n \choose 2}{2 \choose 1} \)
d) three colors (2+1+1 faces): \( {n \choose 3}{3 \choose 1} \)
e) four colors (1+1+1+1 faces): \( {n \choose 4} \times 2 \)
2 colors \( 5/3 (all/only) \)
3 colors \( 15/3 \)
4 colors \( 36/2 \)
1. symmetry of cube
i. \( 1 \) identity element
\[(1)(2)(3)(4)(5)(6)\]
ii. \( 6 \) \( \pm \)90-degree face center to center rotations
\[(1234)(56)\]
iii. \( 3 \) 180-degree face center to center rotations
\[(13)(24)(5)(6)\]
iv. \( 8 \) \( \pm \)120-degree vertex to vertex rotations
\[(125)(346)\]
v. \( 6 \) 180-degree edge center to center rotations
(12 edges divided by 2)
\[(13)(52)(64)\]
2. burnside of cube (face)
\[ {n^{6}+3n^{4}+12n^{3}+8n^{2}} \over 24 \]
3. burnside of cube (vertex)
\[ {n^{8}+17n^{4}+6n^{2}} \over 24 \]
4. burnside of cube (edge)
\[ {n^{12}+6n^{7}+3n^{6}+8n^{4}+6n^{3}} \over 24 \]
5. cube face n-color breakdown
a) one color: \( n \choose 1 \)
b) two colors (1+5 faces): \( {n \choose 2}{2 \choose 1} \)
c) two colors (2+4 faces): \( {n \choose 2}{2 \choose 1} \times 2 \)
d) two colors (3+3 faces): \( {n \choose 2} \times 2 \)
e) three colors (4+1+1 faces): \( {n \choose 3}{3 \choose 1} \times 2 \)
f) three colors (3+2+1 faces): \( {n \choose 3}{3 \choose 1}{2 \choose 1} \times 3 \)
g) three colors (2+2+2 faces): \( {n \choose 3}(3+2+1) \)
h) four colors (3+1+1+1 faces): \( {n \choose 4}{4 \choose 1} \times 5 \)
i) four colors (2+2+1+1 faces): \( {n \choose 4}{4 \choose 1}{3 \choose 1} \times 4 \)
j) five colors (2+1+1+1+1 faces): \( {n \choose 5}{5 \choose 1} \times {(3+12)} \)
k) six colors (1+1+1+1+1+1 faces): \( {n \choose 6}{5 \choose 1} \times 3! \)
2 colors \( 10/8 (all/only) \)
3 colors \( 57/30 \)
4 colors \( 240/68 \)
5 colors \( 800/75 \)
6 colors \( 2226/30 \)
simply 6!/24=30
1. symmetry of octahedron
i. \( 1 \) identity element
\[ (1)(2)(3)(4)(5)(6)(7)(8)\]
ii. vertex to vertex
\( 6 \) \( \pm \)90-degree rotations
\[ (1234)(5678)\]
iii. vertex to vertex
\( 3 \) 180-degree rotations
\[ (13)(24)(57)(68)\]
iv. face center to face center
\( 8 \) \( \pm \)120-degree rotations
(8 faces divided by 2 times 2)
\[ (1)(7)(245)(638)\]
v. edge center to edge center
\( 6 \) 180-degree rotations
(12 edges divided by 2)
\[ (17)(26)(35)(48)\]
2. burnside of octahedron (face)
\[ {n^{8}+17n^{4}+6n^{2}} \over 24 \]
3. octahedron face n-color breakdown
2 colors \( 23/21 (all/only) \)
3 colors \( 333/267 \)
4 colors \( 2916/1718 \)
5 colors \( 16725/5250 \)
6 colors \( 70911/7980 \)
7 colors \( 241913/5880 \)
8 colors \( 701968/1680 \)
simply 8!/24=1680
1. symmetry of dodecahedron
i. \( 1 \) identity element
\[ (1)(2)(3)(4)(5)(6)(7)(8)(9)(a)(b)(c)\]
ii. face center to face center
\( 24 \) 72n-degree rotations (n=1,2,3,4)
(12 faces divided by 2 times 4)
\[ (1)(7)(23456)(89abc)\]
iii. edge center to edge center
\( 15 \) 180-degree rotations
(30 edges divided by 2)
\[ (12)(36)(4c)(48)(7a)(9b)\]
2. burnside of dodecahedron (face)
\[ {n^{12}+15n^{6}+44n^{4}} \over 60 \]
3. dodecahedron face n-color breakdown
2 colors \( 96/94 (all/only) \)
3 colors \( 9099/8814 \)
4 colors \( 280832/245008 \)
5 colors \( 4073375/2759250 \)
6 colors \( 36292320/15884004 \)
7 colors \( 230719293/52701264 \)
8 colors \( 1145393152/106866144 \)
9 colors \( 4707296613/134719200 \)
10 colors \( 16666924000/103118400 \)
11 colors \( 52307593239/43908480 \)
12 colors \( 148602435840/7983360 \)
simply 12!/60=7983360
1. symmetry of icosahedron
i. \( 1 \) identity element
\[ (1)(2)(3)(4)(5)(6)(7)(8)(9)(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)\]
ii. vertex to vertex
\( 24 \) 72n-degree rotations (n=1,2,3,4)
\[ (189a2)(357ik)(46hjc)(bdefg)\]
iii. edge center to edge center
\( 15 \) 180-degree rotations
\[ (12)(38)(49)(5a)(6k)(7d)(ch)(ei)(fj)(gb)\]
2. burnside of icosahedron (face)
\[ {n^{20}+15n^{10}+20n^{8}+24n^{4}} \over 60 \]
1. pythagorean theorem \[ a^2+b^2=c^2 \]
2. similar triangles
equiangular triangles (aaa) are similar
i. corresponding sides are proportional
ii. corresponding altitudes are proportional
iii. corresponding area are proportional sqaured
3. angle bisector theorem
\[ {ab \over ac} = {bd \over dc} \]
for angle bisector of a with cevian foot at d on bc
4. mass point
i. assign mass by mass balance over foot of a cevian
\[ mass_1 \times distance_1 = mass_2 \times distance_2 \]
ii. mass of cevian foot is mass addition of two vertices
iii. point of concurrency of cevians is mass sum vertex and foot
5. splitting masses
i. used for triangles with both cevians and transversals
ii. any vertex on both sides transversal crosses has split mass
6. heron's formula \[ \sqrt {s(s-a)(s-b)(s-c)} \] where \( s={a+b+c \over 2}\)
7. incircle
inenter is intersection of angle bisectors
\[ inradius = {area \over s} \]
where \( s={a+b+c \over 2}\)
8. circumcircle
circumcenter is intersection of perpendicular side bisector
\[ circumradius = {abc \over {4 \times area}} \]
where a,b,c are three sides
draw centroid to find center
9. concurrency of medians theorem
i. medians are concurrent at centroid
ii. \( {vertex \rightarrow centroid \over centroid \rightarrow median}=2 \)
10. stewart's theorem \[ ab^2 \times cd+ac^2 \times bd=bc(ad^2+bd \times cd) \] for d on bc
11. median formula \[ median^2 ={{2b^2+2c^2-a^2} \over 4} \] for median on bc
1. ptolemy's theorem
\( \sum \prod \) opposite sides = \( \prod \) diagonals
for cyclic quadrilaterals
2. law of cosine to solve diagnals
\[ ac^2=ab^2+bc^2-2 \times ab \times bc \times cos(b)\]
\[ ac^2=cd^2+ad^2-2 \times cd \times ad \times cos(d)\]
\[ cos(b)=-cos(d)\]
for cyclic quadrilateral abcd
3. varignon's theorem
a) midpoints form a parallelogram
b) area is half of quadrilateral
4. trapezoid
\[ area = {{a + b} \over 2} h \]
where \( {a+b} \over 2 \) is median or midsegment or midline
5. rhombus
equilateral parallelogram
6. rectangle
equiangular parallelogram
\[ area = \prod sides \]
7. square
regular quadrilateral
\[ area = side^2 \]
8. kite
i. two pairs of adjacent equal-length sides
ii. diagonal perpendicular
\[ area = {1 \over 2} \prod diagonals \]
9. brahmagupta's formula \[ area=\sqrt {(s-a)(s-b)(s-c)(s-d)} \] where \( s={a+b+c+d \over 2}\) for cyclic quadrilateral with sides a,b,c,d
10. cyclic quadrilateral with incircle \[ area=\sqrt {abcd} \]
1. equiangular vs equilateral
equal vertex angles vs equal sides
2. regular polygon
both equiangular and equilateral
interior angle
\[ {(n-2)*\pi \over n} \]
3. regular pentagon \[ area= {\sqrt {25+10 \sqrt 5} \over 4} side ^2 \]
4. regular hexagon \[ area={3 \sqrt 3 \over 2} side^2 \]
5. regular octogon \[ area=({2+ 2 \sqrt 2 }) side^2 \]
6. pick's theorem \[ area = i + {b \over 2} -1 \] where i and b are interior/boundary lattice points of a polygon
7. shoelace theorem \[ area = {{ \left|(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1) \right| } \over 2} \] for polygon with vertices \( (a_1, b_1), (a_2, b_2), ... , (a_n, b_n) \)
1. arc length \[ radian \times radius \]
2. chord bisector
i. radius bisect chord
ii. radius perpendicular to chord
3. inscribed angle
i. half of arc
ii. 90 deg if on diameter
4. intersecting chords
creating two similar triangles
\[ pa \times pb = pc \times pd\]
where two chords are ab and cd intersecting at p
5. tangent
perpendicular to radius at the point of trangency
6. power of a point
i) secant/secant
\[ pa \times pb = pc \times pd\]
where two secant lines from p intersecting circle at ab and cd
ii) tangent/secant
\[ pa^2 = pc \times pd \]
iii) tangent/tangent
\[ pa^2 = pc^2 \]
6. radical axis theorem
1. volume \[ v={4 \over 3} \pi r^3\]
2. surface area \[ a=4 \pi r^2\]
1. volume \[ v= \pi r^2 h\]
2. surface area \[ area= 2\pi r^2 + 2\pi r h \]
1. volume \[ v= {1 \over 3} \pi r^2 h\]
2. surface area \[ area= \pi r^2 + \pi r s \] where s is slant height
1. surface area \[ a=\sqrt 3 side^2\]
2. height \[ h={\sqrt 6 \over 3} side\]
3. volume \[ v={\sqrt 2 \over 12} side^3\]
1. surface area \[ a=2 \sqrt 3 side^2\]
2. height \[ h={\sqrt 2} side\]
3. volume \[ v={\sqrt 2 \over 3} side^3\]
1. volume \[ v= side^3 \]
1. volume \[ v= {15+7 \sqrt 5 \over 4} side^3 \]
1. edge|face|vertice
f+v-e=2
2. prism
i. two congruent polygons connected by parallelograms
ii. right prism vs oblique prism
\[ v=bh \]
3. pyramid \[ v={1 \over 3}a_{base}h \]
1. sphere inscribed in cube
\[ r={1 \over 2} side \]
2. cube inscribed in sphere
\[ r={\sqrt 3 \over 2} side \]
3. sphere inscribed in regular tetrahedron
\[ r={\sqrt 6 \over 12} side \]
4. regular tetrahedron inscribed in sphere \[ r={\sqrt 6 \over 4} side \]
5. sphere inscribed in regular octahedron
\[ r={\sqrt 6 \over 6} side \]
6. regular octahedron inscribed in sphere \[ r={\sqrt 2 \over 2} side \]
5. sphere stacking on tangent n-spheres
\[ height=\sqrt {(r_{top}+r_{bottom})^2- \bigl ({r_{bottom} \over sin({\pi \over n})} \bigr )^2} +r_{top}+r_{bottom} \]
1. pythagorean \[ sin^2 \theta + cos^2\theta =1 \] \[ tan^2 \theta + 1 = sec^2\theta \] \[ cot^2 \theta + 1 = csc^2\theta \]
2. complementary \[ sin({\pi \over 2}-\theta)=cos \theta \] \[ cos({\pi \over 2}-\theta)=sin \theta \] \[ tan({\pi \over 2}-\theta)=cot \theta \] \[ cot({\pi \over 2}-\theta)=tan \theta \]
3. sum/difference \[ sin (\alpha \pm \beta)=sin\alpha cos \beta \pm cos\alpha sin \beta \] \[ cos (\alpha \pm \beta)=cos\alpha cos \beta \mp sin\alpha sin \beta \] \[ tan (\alpha \pm \beta)= {tan\alpha \pm tan \beta \over 1 \mp tan\alpha tan \beta} \]
4. double \[ sin 2 \theta=2sin\theta cos\theta \] \[ cos 2 \theta=cos^2\theta-sin^2\theta=2cos^2\theta-1=1-2sin^2\theta\] \[ tan 2 \theta={2tan \theta \over {1 - tan^2 \theta}} \]
5. triple \[ sin 3 \theta=3sin \theta-4sin^3 \theta \] \[ cos 3 \theta=4cos^3 \theta-3cos\theta \] \[ tan 3 \theta={{3tan \theta-tan^3 \theta} \over {1-3tan^2 \theta}} \]
6. half \[ sin {\theta \over 2} = \pm \sqrt {1-cos \theta \over 2} \] \[ cos {\theta \over 2} = \pm \sqrt {1+cos \theta \over 2} \] \[ tan {\theta \over 2} = \pm \sqrt {1-cos \theta \over {1+cos \theta }} \]
7. product to sum \[ 2sin\alpha cos \beta =sin(\alpha + \beta) + sin(\alpha - \beta) \] \[ 2cos\alpha sin \beta =sin(\alpha + \beta) - sin(\alpha - \beta) \] \[ 2cos\alpha cos \beta =cos(\alpha + \beta) + cos(\alpha - \beta) \] \[ 2sin\alpha sin \beta =-cos(\alpha + \beta) +cos(\alpha - \beta) \]
8. sum to product \[ sin(\alpha) + sin(\beta) = 2sin({{\alpha + \beta} \over 2})cos({{\alpha - \beta} \over 2}) \] \[ sin(\alpha) - sin(\beta) =2cos({{\alpha + \beta} \over 2})sin({{\alpha - \beta} \over 2}) \] \[ cos(\alpha) + cos(\beta) = 2cos({{\alpha + \beta} \over 2})cos({{\alpha - \beta} \over 2}) \] \[ cos(\alpha) - cos(\beta) = -2sin({{\alpha + \beta} \over 2})sin({{\alpha - \beta} \over 2}) \]
9. special product \[ \prod_{k=1}^{n-1} {sin{k \pi \over n}} = {n \over {2^{n-1}}} \] \[ \prod_{k=1}^{n-1} {cos{k \pi \over n}} = {{sin{n \pi \over 2}} \over {2^{n-1}}} \] \[ \prod_{k=1}^{n-1} {tan{k \pi \over n}} = {n \over {sin{n \pi \over 2}}} \]
1. law of sine \[{a \over sin\alpha}={b \over sin\beta}={c \over sin\gamma}\]
2. extended law of sine \[{a \over sin\alpha}=2R\] where R is radius of circumcircle
3. law of cosine \[ a^2=b^2+c^2-2bc \times cos \alpha \] \[ b^2=a^2+c^2-2ac \times cos \beta \] \[ c^2=a^2+b^2-2ab \times cos \gamma \]
4. brocard angle \[ tan (\omega) = {{4area} \over {a^2+b^2+c^2}} \] \[ cot (\omega) = {{sin^2 \alpha + sin^2 \beta + sin^2 \gamma} \over {2sin \alpha sin \beta sin \gamma}} \] \[ csc^2 (\omega) = csc^2 \alpha + csc^2 \beta + csc^2 \gamma \]
1. \( sin^{-1}x \)
domain: [-1,1]
range: \( [-{\pi \over 2}, {\pi \over 2}] \)
2. \( cos^{-1}x \)
domain: [-1,1]
range: \( [0, \pi] \)
3. \( tan^{-1}x \)
domain: \( [- \infty, \infty] \)
range: \( [-{\pi \over 2}, {\pi \over 2}] \)
1. 15 deg \[ sin 15^o = {\sqrt 6 - \sqrt 2 \over 4} \]
2. 18 deg \[ sin 18^o = {\sqrt 5 -1 \over 4} \]
3. 36 deg \[ sin 36^o = {\sqrt {10 - 2 \sqrt 5 } \over 4} \]
4. 54 deg \[ sin 54^o = {\sqrt 5 +1 \over 4} \]
5. 72 deg \[ sin 72^o = {\sqrt {10 + 2 \sqrt 5 } \over 4} \]
6. 75 deg \[ sin 75^o = {\sqrt 6 + \sqrt 2 \over 4} \]
1. bayes' theorem \[ p(a|b)= {p(a)p(b|a) \over {p(b)}} \] for conditional probability
2. venn diagram \[ |a \cup b| = |a| + |b| - |a \cap b| \] \[ |a \cap b| = |a| + |b| - |a \cup b| \] \[ |a \cup b \cup c| = |a| + |b| + |c| - |a \cap b| -|a \cap c| - |b \cap c| + |a \cap b \cap c| \]
3. bijection
one-to-one correspondence between elements of two sets
4. law of alternatives \[ p(b)=p(a_1)p(b|a_1)+p(a_2)p(b|a_2)+\cdots+p(a_n)p(b|a_n)\] for disjoint events \(a_1, a_2, \ldots, a_n\)
5. law of successive conditioning \[ p(a1\cap a_2 \cap \ldots \cap a_n)= p(a_1)p(a_2|a_1)p(a_3|a1 \cap a_2) \cdots p(a_n|a_1 \cap a_2 \cap \cdots \cap a_{n-1}) \] for successive events
1. stars/bars \[ {s+b-1 \choose b-1} \]
2. blockwalk \[ {x+y \choose x} \]
3. mississippi rule
4. same size groups without order
\[ {(mn)!} \over {(n!)^m(m!)}\]
where \(m \times n \) objects into m groups
5. same size groups with order
\[ {(mn)!} \over {(n!)^m}\]
where \(m \times n \) objects into m groups
6. unequal size groups
\[ {(m+n+p)!} \over {(m!)(n!)(p!)}\]
where m+n+p objects into 3 groups of size of m,n,p ( \( m \neq n \neq p \) )
7. pascal's identity \[ {n \choose k} = {n-1 \choose k} + {n-1 \choose k-1}\] where \[ {n \choose k} = {n! \over {k!(n-k)!}} \]
8. extended pascal's identity \[ {n \choose k} = {n-1 \choose k} + {n-2 \choose k-1} + {n-3 \choose k-2} + \cdots + {n-k+1 \choose 0} \]
\[ {2n \choose n} = {n \choose 0}^2 + {n \choose 1}^2 + ... + {n \choose n}^2\]
1. addition/multiplication/exponent
\( a \pm x \equiv b \pm x \) (mod \(c\))
\( a \times x \equiv b \times x \) (mod \(c\))
\( a^x \equiv b^x \) (mod \(c\))
for \(a \equiv b \) (mod \(c\))
\( a \pm b \equiv x \pm y \) (mod \(c\))
\( a \times b \equiv x \times y \) (mod \(c\))
for \(a \equiv x \) (mod \(c\)) and \(b \equiv y \) (mod \(c\))
2. fermat's little theorem
\[ a^{p-1} \equiv 1 \pmod p \]
where \( p \) is prime and \( a \) is not a multiple of p
3. euler's totient theorem
\[ a^{\phi(n)} \equiv 1 \pmod n \]
where \( \phi(n)=n(1-{1 \over p_1})(1-{1 \over p_2})...(1-{1 \over p_n}) \)
and a is an integer relatively prime to n
4. chinese remainder theorem
\[ x \equiv a \pmod p \]
\[ x \equiv b \pmod q \]
then there is a unique solution for x modulo pq
where p and q are relatively prime
5. fermat's last theorem
xn+yn!=zn
where n>2 and x,y,z are positive integers
6. euclidean algorithm
\[ gcd(a,b)=gcd(a \pm b, b)=gcd(a,b \pm a) \]
1. base n to 10 \[ abc_n=an^2+bn+c\]
2. base 10 to n
3. decimals
1.sfft
2.bezout's lemma
ax+by=gcd(a,b)
where a,b,x,y are integers, with a,b>0
1.Lifting the Exponent Lemma
# of factors of:
p|(an-bn) = p|(a-b) + p|n.
where p is an odd prime, a and b relatively prime to p and p|(a-b).
n is a positive integer.
1. arithmatic mean
\[ \sum =am \times n\]
2. geometric mean
\[ rms \geq am \geq gm \geq hm \]
3. harmonic mean
reciprocal of mean of reciprocal
\[ hm={n \over {{1 \over a_1}+ {1 \over a_2}+ \cdots + {1 \over a_n}}} \]
4. root mean sqaure \[ rms=\sqrt {{a_1^2 + a_2^2 + \cdots + a_n^2} \over n} \]
1. geometric probability
1. continuity
\( f(x) \) continuous at x=a if
\( \exists f(a) \)
\( \exists \lim_{x\to a}\)
and \( \lim_{x\to a}\) = f(a)
2. discontinuity
"essential": vertical asymptote
"removeable": holes
"jump": \( x\to a^+ \neq x\to a^- \)
3. derivative \[ f'(x)=\lim_{h\to 0} f(x+h)-f(x) \]
4. l'hopital's rule
\[ \lim_{x\to a} {f(x) \over g(x)} = {f'(a) \over g'(a)}\]
for 0/0 or \( \infty / \infty \)
5. ratio test
\[ \lim_{n\to \infty} {a_{n+1} \over a_n} \]
converge if ratio < 1
diverge if ratio > 1
unknown if ratio = 1
6. p series
\[ \lim_{n\to \infty} {1 \over n^p} \]
converge if p < 1
diverge if p ≥ 1
7. radius of convergence (R)
\[ \lim_{n\to \infty} {c_n x^n} \]
\( x \in (R,R) \) when converge
1. power rule \[ {d \over dx} \left(x^n\right) = nx^{n-1} \] \[ \int {x^ndx}={x^{n+1} \over n+1} + c \]
2. product rule \[ {d \over dx} \left(f(x)g(x)\right) = g(x)f'(x)+f(x)g'(x) \]
3. quotient rule \[ {d \over dx}\left({f(x) \over g(x)}\right) = {g(x)f'(x)-f(x)g'(x) \over (g(x))^2}\]
4. chain rule \[ {d \over dx} \left( f(g(x) \right) = f'(g(x))g'(x) \]
5. exponential function \[ {d \over dx} \left( e^x \right) = e^x \] \[ {d \over dx} \left( a^x \right) = a^x ln(a)\]
6. logarithmic function \[ {d \over dx} \left( ln|x| \right) = {1 \over x} \] \[ {d \over dx} \left( log_ax \right) = {1 \over {xln(a)}}\]
7. trig function \[ {d \over dx} sin(x) = cos(x) \] \[ {d \over dx} cos(x) = -sin(x) \] \[ {d \over dx} tan(x) = sec^2(x) \] \[ {d \over dx} cot(x) = -csc^2(x) \] \[ {d \over dx} sec(x) = sec(x)tan(x) \] \[ {d \over dx} csc(x) = -csc(x)cot(x) \]
8. inverse trig function \[ {d \over dx} sin^{-1}x = {1 \over \sqrt {1-x^2}} \] \[ {d \over dx} cos^{-1}x = -{1 \over \sqrt {1-x^2}} \] \[ {d \over dx} tan^{-1}x = {1 \over {1+x^2}} \] \[ {d \over dx} cot^{-1}x = -{1 \over {1+x^2}} \] \[ {d \over dx} sec^{-1}x = {1 \over {|x| \sqrt {x^2-1}}} \] \[ {d \over dx} csc^{-1}x = -{1 \over {|x| \sqrt {x^2-1}}} \]
9. implicit differentiation
10. mean value theorem (MVT) - differentiation \[ f'(c) = {{f(b) - f(a)} \over {b-a}} \] for \( c \in [a,b] \)
11. rolle's theorem
\( \exists f'(c)=0 \) if f(a)=f(b)=0
for \( c \in [a,b] \)
12. related rate
1. fundamental theorem of calculus \[ \int_a^b f(x)dx=F(x)|_a^b=F(b)-F(a) \] where F(x) is antiderivative of f(x)
2. second fundamental theorem of calculus \[ {d \over dx} \int_a^x f(t)dt=f(x) \]
3. exponential function \[ \int {e^xdx}={e^x} + c \] \[ \int {a^xdx}={a^x \over ln(a)} + c \]
4. logarithmic function \[ \int {{1 \over x}dx}=ln|x| + c \] \[ \int ln(x)dx=xln(x)-x + c\]
5. trig function \[ \int {{sin(x)}dx}=-cos(x) + c \] \[ \int {{cos(x)}dx}=sin(x) + c \] \[ \int {{sec^2(x)}dx}=tan(x) + c \] \[ \int {{csc^2(x)}dx}=-cot(x) + c \] \[ \int {{sec(x)tan(x)}dx}=sec(x) + c \] \[ \int {{csc(x)cot(x)}dx}=-csc(x) + c \] \[ \int {tan(x)dx} =-ln|cos(x)| + c \] \[ \int {cot(x)dx} =-ln|sin(x)| + c \] \[ \int {sec(x)dx} =ln(sec(x)+tan(x)) + c \] \[ \int {csc(x)dx} =-ln(csc(x)+cot(x)) + c \]
6. inverse trig function \[ \int {{1 \over \sqrt {1-x^2}}dx}=sin^{-1}x + c \] \[ \int -{{1 \over \sqrt {1-x^2}}dx}=cos^{-1}x + c \] \[ \int {1 \over {1+x^2}}=tan^{-1}x + c \] \[ \int -{1 \over {1+x^2}}=cot^{-1}x + c \] \[ \int {1 \over {|x| \sqrt {x^2-1}}}=sec^{-1}x + c \] \[ \int -{1 \over {|x| \sqrt {x^2-1}}}=csc^{-1}x + c \]
7. u sub
\[ \int f(g(x))g'(x)dx=\int f(u)du\]
where u=g(x)
\[ \int_a^b f(g(x))g'(x)dx=\int_{g(a)}^{g(b)} f(u)du\]
where u=g(x) and g' is continuous on [a,b]
and f is continuous on range of u
8. partial fraction
i. factoring denominator
ii. set up numerator
iii. cover-up method, multiple by (x-a) and let x=a
iv. for repeated factors, one for each of the powers
\[{A\over {x-a}}+{B \over {(x-a)^2}}+ {C \over {(x-a)^3}}+\cdots \]
v. for quadratic factors, use numerator Bx+C
vi. if numerator has higher power use long divison
\[ {p(x) \over q(x)}=quotient+{r(x) \over q(x)}\]
9. \( \int sin^mxcos^nxdx \)
i)m=1 or n=1: u=cos(x) or sin(x)
ii)m or n is odd: pythagorean to reduce power
ii)m and n are even: use half angle formula
10. trig sub
\[ \int \sqrt {a^2-x^2} dx \]
let \( x=acos \theta \) or \( x=asin \theta \)
\[ \int \sqrt {a^2+x^2} dx \]
let \( x=atan \theta \)
\[ \int \sqrt {x^2-a^2} dx \]
let \( x=asec \theta \)
followed by undoing trig sub using "triangle"
11. integration by parts \[ \int {udv}=uv- \int vdu \]
12. arc length
\[ {ds \over dx}={\sqrt {1+({dy \over dx})^2}}\]
arc length=\({\int_a^b {\sqrt{1+f'(x)^2}dx}}\)
\[ \int_a^b {\sqrt{{\left({dx \over dt}\right)^2+\left({dy \over dt}\right)^2}}dt} \]
13. area between curves \[ A= \int_a^b(f(x)-g(x))dx\]
14. solids of revolution
(1) washer
\[ \pi \int_a^b[f(x)]^2dx \]
(2) disk
\[ \pi \int_a^b{[f(x)]^2-[g(x)]^2}dx \]
(2) shell
\[ 2\pi \int_a^b{x[f(x)-g(x)]}dx \]
15. surface area
16. parametric curve
17. polar coordinates
i. polar to rectangular
\[x=rcos\theta \]
\[y=rsin\theta\]
ii. rectangular to polar
\[r=\sqrt {x^2+y^2}\]
\[\theta=tan^{-1}({y\over x})\]
18. mean value theorem (MVT) - integration \[ f(c) = {1 \over {b-a}} \int_a^b f(x)dx\] for \( c \in [a,b] \)
1. linear approximation \[ f(x) \approx f(x_0)+f'(x_0)(x-x_0)\] \[ sin(x) \approx x \] \[ cox(x) \approx 1 \] \[ e^x \approx 1+x \] \[ (1+x)^r \approx 1+rx \] when x approaches 0
2. quadratic approximation \[ f(x) \approx f(x_0)+f'(x_0)(x-x_0)+{f''(x_0) \over 2}(x-x_0)^2\] \[ sin(x) \approx x \] \[ cox(x) \approx 1-{x^2 \over 2} \] \[ e^x \approx 1+x+{x^2 \over 2} \] \[ (1+x)^r \approx 1+rx+{{r(r-1) \over 2}x^2} \] when x approaches 0
3. maxima and minima
first derivative to find critical point
second derivative to determine concave up/down
4. newton's method of approximation \[ x_{n+1}=x_n-{f(x_n) \over f'(x_n)} \] \[ root= \lim_{n\to\infty} x_n \]
5. riemann's sums \[ left=(y_0+y_1+...+y_{n-1})\Delta x \] \[ right=(y_1+y_2+...+y_n)\Delta x \]
6. trapezoidal rule \[ area=({y_0 \over 2}+y_1+...+y_{n-1}+{y_n \over 2})\Delta x \]
7. simpson's rule
8. euler's method \[y_{n+1}=y_n+{dy_n \over dx_n}(x_{n+1}-x_n)\]
9. taylor's formula \[ f(x)=\sum_{n=0}^{\infty} {f^{(n)}(0)\over n!}x^n \]
1. linear \[ y'+p(x)y=q(x)\] use integration factor \[ u(x)=e^{\int p(x)dx} \]
2. bernoulli \[ y'=p(x)y+q(x)y^n\] use substitution \[ u={1 \over y^{n-1}}\]
3. homogenous \[ y'=F({y \over x})\] use substitution \[ u={y \over x}\] thus, y'=u'x+u
1. dot product of two vectors \[ \vec m \cdot \vec n = a_1b_1+a_2b_2+a_3b_3=\vert \vec m \vert \vert \vec n \vert cos\theta\] for \( \vec m = \langle a_1,a_2,a_3 \rangle \) and \( \vec n = \langle b_1,b_2,b_3 \rangle \)
2. orthogonality
when two vectors are perpendicular to each other
\( \vec m \bot \vec n \) means \( \vec m \cdot \vec n = 0\)
1. cross product of two vectors
is a vector perpendicular to both vectors
2. area of triangle enclosed by <a,b> and <c,d> is |ad-bc|/2
1. second-order determinint
$$ det \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} =ad-bc $$2. third-order determinint
$$ det \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} =a \cdot det \begin{bmatrix} e & f \\ h & i \\ \end{bmatrix} -b \cdot det \begin{bmatrix} d & f \\ g & i \\ \end{bmatrix} +c \cdot det \begin{bmatrix} d & e \\ g & h \\ \end{bmatrix} $$3. minors
4. matrix inverse
1. diagonal matrix
$$
det
\begin{pmatrix}
a_{11} & 0 & \cdots & 0 \\
0 & a_{22} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots & \\
0 & 0 & \cdots & a_{nn} \\
\end{pmatrix}
=a_{11} a_{22} \cdots a_{nn}
$$
2. triangular matrix
$$
det
\begin{pmatrix}
a_{11} & 0 & \cdots & 0 \\
a_{21} & a_{22} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots & \\
a_{n1} & a_{n2} & \cdots & a_{nn} \\
\end{pmatrix}
=a_{11} a_{22} \cdots a_{nn}
$$
1. read questions more than once
2. verify answers immediately
3. pythagorean triples
(3,4,5)
(5,12,13)
(7,24,25)
(8,15,17)
(9,40,41)
(11,60,61)
(12,35,37)
(20,21,29)
4. consecutive sum
\[ {(2n+p)(p+1)} \over 2\]
where \( {{2sum} \over {p+1}} \gt p \) or \( sum \gt {{p(p+1)} \over 2}\)
for n, n+1, n+2, ... n+p
2021 = [1010, 1011]
2021 = [26, 27,..., 68]
2021 = [20, 21,..., 66]
2020 = [402, 403, 404, 405, 406]
2020 = [249, 250, ..., 256]
2020 = [31, 32, ..., 70]
5. prime numbers
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499]
\[ 2001=3 \times 23 \times 29 \]
\[ 2002=11 \times 13 \times 14 \]
2003 is a prime
\[ 2004=2^2 \times 3 \times 167 \]
\[ 2005=5 \times 401 \]
\[ 2006=2 \times 17 \times 59 \]
\[ 2007=3^2 \times 223 \]
\[ 2008=2^3 \times 251 \]
\[ 2009=7^2 \times 41 \]
\[ 2010=2 \times 3 \times 5 \times 67 \]
2011 is a prime
\[ 2012=2^2 \times 503 \]
\[ 2013=3 \times 11 \times 61 \]
\[ 2014=2 \times 19 \times 53 \]
\[ 2015=5 \times 13 \times 31 \]
\[ 2016=2^5 \times 3^2 \times 7 \]
2017 is a prime
\[ 2018=2 \times 1009 \]
\[ 2019=3 \times 673 \]
\[ 2020=2^2 \times 5 \times 101 \]
\[ 2021=43 \times 47 \]
\[ 2022=2 \times 3 \times 337 \]
\[ 2023=7 \times 17^2 \]
\[ 2024=2^3 \times 11 \times 23 \]
\[ 2025=3^4 \times 5^2 \]
\[ 2026=2 \times 1013 \]
2027 is a prime
\[ 2028=2^2 \times 3 \times 13^2 \]
2029 is a prime
\[ 2030=2 \times 5 \times 7 \times 29 \]
\[ 2031=3 \times 677 \]
\[ 2032=2^4 \times 127 \]
\[ 2033=19 \times 107 \]
\[ 2034=2 \times 3^2 \times 113 \]
\[ 2035=5 \times 11 \times 37 \]
\[ 2036=2^2 \times 509 \]
\[ 2037=3 \times 7 \times 97 \]
\[ 2038=2 \times 1019 \]
2039 is a prime
\[ 2040=2^3 \times 3 \times 5 \times 17 \]
\[ 2041=13 \times 157 \]
\[ 2042=2 \times 1021 \]
\[ 2043=3^2 \times 227 \]
\[ 2044=2^2 \times 7 \times 73 \]
\[ 2045=5 \times 409 \]
\[ 2046=2 \times 3 \times 11 \times 31 \]
\[ 2047=23 \times 89 \]
\[ 2048=2^{11} \]
\[ 2049=3 \times 683 \]
6. be patient