Formulas: Flashcards
1
Q
Complex Analysis: Basic Formula
A
Z=x+yi
2
Q
Complex Analysis: Polar formula
A
z=rcos@+rsin@i
z=r<@
3
Q
Complex Analysis: Euler’s
A
z=re^@i
4
Q
Complex Analysis: r (modulus)
A
|z|=sqr root of (x^2 + y^2)
5
Q
Complex Analysis: Theta @ (Argument)
A
@ = arctan (y/x)
6
Q
Complex Analysis: De moivre’s
A
Caltech: CMPLX mode
|z|^n <(n x arg (z)) =
7
Q
Number System: Binary, Bin
A
0 and 1
8
Q
Number System: Decimal, Dec
A
0 to 9
9
Q
Number System: Octagecimal, Oct
A
0 to 7
10
Q
Number System: Hexadecimal, Hex
A
0 to 9 and A to F
11
Q
Linear Equation
A
ax+b
12
Q
Vieta’s Formula
A
If X1 and X2 are roots of quadratic equation
13
Q
Vieta’s Formula: Sum of Roots
A
X1 + X2 = -B/A
14
Q
Vieta’s Formula: Product of Roots
A
X1X2=C/A
15
Q
Binomial Theorem: (x+y)^n
Formula for rth term
A
nCn-r+1(x^(n-r+1))(y^(r-1))
16
Q
Binomial Theorem: (x+y)^n
Formula for Sum of coefficient
A
f(1,1)-f(0,0)
17
Q
Binomial Theorem: (x+y)^n
Formula for Sum of Exponents
A
(SOEi)(n+1C2)
18
Q
Binomial Theorem: (x+y)^n
Formula for Number of Terms
A
n+1
19
Q
Binomial Theorem: (x+y)^n
Formula for middle term
A
r=(n/2) +1
20
Q
Multinomial Theorem:
(x^alpha + y^beta + z^ gamma) ^n
Formula for rth term
A
[n!/(a! x b! x c!) ] (x^a y^b z ^c)
21
Q
Multinomial Theorem:
(x^alpha + y^beta + z^ gamma) ^n
Formula for Sum of Exponents
A
(SOEi) (n+t-1Ct)
22
Q
Multinomial Theorem:
(x^alpha + y^beta + z^ gamma) ^n
Formula for number of terms
A
n+t-1Ct-1 (t no. Of terms)
23
Q
Arithmetic Sequence: Stat 2
An
A
An=A1+(n-1)d
24
Q
Arithmetic Sequence: Stat 2
Sn
A
Sn=(n/2)(2A1+ (n-1)d)
25
Geometric Sequence: Stat 6
| An
An=A1(r^(n-1))
26
Geometric Sequence: Stat 6
| Sn
Sn=A1(1-r^n)/1-r
27
Fibonacci Sequence: add 2 numbers before it
| Golden ratio, Phi Formula
Phi=(1+sqr root of 5)/2
28
Fibonacci Sequence: add 2 numbers before it
| Silver ratio, Gamma Y Formula
Gamma, Y=(1- sqr root of 5)/2
29
Fibonacci Sequence: add 2 numbers before it
| Rth Term formula
(Phi^n - Y^n)/sqr root of 5
30
Lucas Sequence, L
| Rth term:
Phi^n +Y^n
31
i in matrix (Aij)
Row
32
j in matrix (Aij)
Column
33
Clock Problem: Stat 2 Caltech
x|y
0|T(-30deg)
60|T(-30deg) +330
34
Clock Problem: First
-Theta
35
Clock Problem: 2nd
Theta
36
Clock Problem: 3rd
- theta + 360
37
Clock Problem: 4th
Theta + 360
38
Clock Problem: 5th
-theta + 720
39
Clock Problem: 6th
Theta + 720
40
Clock Problem: Together/Coincide
Theta = 0
41
Clock Problem: Perpendicular
90deg
42
Clock Problem: Align
180 deg
43
Rowing Boat Problem: d=v x t
| V is
Still water
44
Rowing Boat Problem: d=v x t
| w is
Current speed
45
Rowing Boat Problem: d=v x t
| Upstream
v-w
46
Rowing Boat Problem: d=v x t
| Downstream
v + w
47
Plane Problem: d=v x t
| Tail wind
Same direction of wind & Plane
48
Plane Problem: d=v x t
| Downwind
Opposite direction of wind and plane
49
Trains and tunnels Problem:
La + Lb= (Va +- Vb) t
+ for opposite
- for same direction
50
Diophantine Equation
2 equation 3 unknowns
51
TRIGO: Arc length formula
s=r@
52
TRIGO: Angular Velocity Formula
v=wr
53
Csc@
1/sin@
54
Sec@
1/cos@
55
Cot@
1/tan@
56
Pythagorean:
sin2@ +cos2@ = 1
tan2@ + 1 = sec2@
Cot2@ + 1 = csc2 @
57
Double angle: sin2@
2sin@cos@
58
Double angle: cos2@
Cos2@ - sin2@
59
Half Angle: Sin (1/2 @)
Sqr root of (1/2)(1-cos@)
60
Half Angle: Cos (1/2 @)
Sqr root of (1/2)(1+cos@)
61
Half Angle: tan (1/2 @)
Sqr root of (1-cos@)/(1+cos@)
62
Negative angles: sin(-@)
-sin@
63
Negative angles: cos(-@)
Cos@
64
Negative angles: tan(-@)
-tan@
65
Sum & Difference: cos (theta +- Beta)
Cos@ cosB -+ sin@ sinB
66
Sum & Difference: sin (theta +- Beta)
Sin@ cos B +- cos@ Sin B
67
Sine Law: 2 angles are known
(a/sinA) = (b/sinB) = (c/sinC)
68
Cosine Law:
a^2 = b^2 + c^2 -2bc cos A
69
Heron’s formula: for oblique triangles
| Semi perimeter formula
s = (a+b+c)/2
70
Heron’s formula: for oblique triangles
| Area of Triangle formula
A= sqr root of s(s-a)(s-b)(s-c)
71
For Cyclic Quadrilateral: Brahmagupta’s equation
| Semi Perimeter Formula
s = (a+b+c+d)/2
72
For Cyclic Quadrilateral: Brahmagupta’s equation
| Area Formula
A = sqr root of (s-a)(s-b)(s-c)(s-d)
73
Spherical Triangle
A= piR^E/180
Epsilon, E = A + B + C - 180
74
Napier’s Triangle for Right Spherical
| SIN-TAAD
Sin(B)= tanc tan a
75
Napier’s Triangle for Right Spherical
| SIN-COOP
Sin(a) = cosc cosA
76
Cosine Law for Oblique Spherical Triangle
cosa = cosbcosc + sinbsinc cos A
77
Sine law for oblique spherical Triangle
(sina/sinA) = (sinb/sinB) = (sinc/sinC)
78
Terrestrial Sphere: horizontal line
Equator
79
Terrestrial Sphere: parallel to equator
Latitude
80
Terrestrial Sphere: perpendicular to equator
Longitude
81
1 deg is equal to blank nm
60 nautical miles
82
1 deg is equal to blank mins
4 mins
83
1 nautical miles is equal to blank meters
1852 meters
84
1 knots is equal to
nautical miles/hr
85
Which quadrant is sin cos tan positive?
1st quadrant
86
Which quadrant is sin positive?
2nd quadrant
87
Which quadrant is cos positive?
4th quadrant
88
Which quadrant is tan positive?
3rd quadrant
89
Angle less then 90 deg
Acute
90
Angle more than 90 deg
Obtuse
91
Sum of 90 deg
Complement angle
92
Sum of 180 deg
Supplement angle
93
Sum of all angles of triangle are equal to
180 deg
94
Vectors has
Both magnitude and direction
95
Components of vectors
A= axi +ayj+azk
96
Magnitude of vectors
|A| = sqr root of ax2 + ay2 +az2
97
Unit vector
a = component/magnitude
98
Dot product is
Scalar
99
Cross product is
Vector
100
Dot product A • B =
|A| |B| cos @
101
Cross product |A x B| =
|A| |B| sin @
102
Area by three coplanar points
A plane = 1/2 det (V1 x V2)
103
Area by three coplanar points if 3D
1/2 x Abs (Vct A x Vct B)
104
Distance formula
d= sqr root of (X2-X1)2 + (Y2-Y1)2
105
Distance formula Caltech
Cmplx mode
| Abs((X1+Y1i)-(X2+Y2i))
106
Slope, m
m= Y2-Y1/X2-X1 = tan@
107
Inclination Caltech
CMPLX
| arg ((X1+Y1i) - (X2+Y2i))
108
Angle Between 2 Lines Caltech
CMPLX
arg (X1+Y1i/X2+Y2i)
Answer must be acute!
109
Point to a line Caltech
Comp
Ax+By+C/sqr root of A2 + B2
Calc X and Y = Xo,Yo (Point)
110
Distance Between 2 Parallel lines
d= C2-C1/sqr root of A2 + B2
111
Forms of a line:
| Standard form
Ax+By+C=0
112
Forms of a line:
| Slope-intercept form
y=mx+b
113
Forms of a line:
| Point-slope form
y-yo=m(x-xo)
114
Forms of a line:
| Two-point form
y-y1 = (y2-y1/x2-x1)(x-x1)
115
Forms of a line:
| Intercept form
x/a + y/b = 1
116
Conic Sections Formula
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
117
Locus of points that are equidistant from a fixed point/center
Circle
118
Circle General Form
Ax2 + Cy2 + Dx + Ey + F = 0
| A and C should be equal
119
Circle (h,k) formula of h
h= -D/2A
120
Circle (h,k) formula of k
k= -E/2A
121
Circle Standard Form
(x-h)2 + (y-k)2 = r2
122
Formula of Circle’s Radius from general form
r= sqr root of (D2 + E2 -4AF)/4A2
123
Locus of points that are equidistant from a fixed point (focus) and directrix (line)
Parabola
124
Parabola Vertical Formula
4p(y-k)=(x-h)2
125
Parabola Horizontal Formula
4p(x-h)=(y-k)2
126
Latus rectum of Parabola
4f (f is focus to vertex)
127
Locus of points such that the sum of its distance from 2 fixed points (foci) is constant
Ellipse
128
Area of ellipse
A=pi(ab)
129
Vertical Ellipse Formula
(y-k)2/a2 + (x-h)2/b2 = 1
130
Horizontal Ellipse Formula
(x-h)2/a2 + (y-k)2/b2 = 1
131
Ellipse eccentricity
e = c/a < 1
132
Ellipse Latus Rectum
2b2/a
133
Ellipse Flatness
f=a-b/a
134
Ellipse apogee
a + c
135
Ellipse directrix
a/e
136
Locus points such that the difference of the distance between two fixed points (foci) is at constant 2a
Hyperbola
137
Ellipse general form/length
b2 + c2 = a2
138
Hyperbola General form
b2 = c2 -a2
139
Axis in hyperbola that passes two foci
Transverse axis
140
Axis in hyperbola that is perpendicular to transverse
Conjugate axis
141
Axis in hyperbola formed by transverse and conjugate
Auxiliary rectangle
142
Hyperbola Vertical Formula
(y-k)2/a2 - (x-h)2/b2 = 1
143
Hyperbola Horizontal Formula
(x-h)2/a2 - (y-k)2/b2 = 1
144
Hyperbola eccentricity
e= c/a
145
Hyperbola Latus Rectum
2a^2/b
146
Caltech for Parabola
Stat 3 (3 points, 1 point is mirrored)
147
Caltech of Ellipse/Hyperbola
Stat 2 (2 points but square the other)
148
Permutations: Circular w/ flip
p= (n-1)!/2
149
Permutations: Circular w/ no flip
P=(n-1)!
150
Permutation repetition
P=n!/(k1! x k2! x k3!…)
151
Permutation Alteration (equal number per group)
P= g! (a!)^g (g is number of groups)
152
Permutation Alteration (not equal number per group)
P=a!b!
153
Permutation proximity
P= (n+1) x (n!p!)
p- together
n- not together
154
Probability Formula
of Favorable outcomes/ total outcomes
155
Law of total probability
P(A U B) = P(A) + P(B) - P(AŪB)
156
If P(AŪB) = 0 what does it mean?
It means events are mutually exclusive
157
2 types Discreet Probability
```
Binomial Probability (n<20)
and
Poisson Probability (n>20)
```
158
Binomial Probability Formula
P(x) = nCx (P)^x (1-P)^(n-x)
159
Poisson Probability Formula
P(x) = (e^-np (np)^x)/x!
```
n= # of trials
p= success rate
x= interval
```
160
Geometric Mean
Xg= nth root of X1X2X3…Xn
161
Mean of two middle values
Median
162
Value that appears frequently
Mode
163
Perimeter
P=sn
164
Interior angle
Theta= 180(n-2)/n
165
Sum of interior angles
180(n-2)
166
Diagonals
D=n(n-3)/2
167
Area
(ns^2)/4tan(180/n)
168
Apothem, a
s/2tan(180/n)
169
Volume of Cylinder
V=pi r^2 h
170
Lateral Surface area of cylinder
2 pi r h
171
Volume of cone
1/3 pi r^2 h
172
Slant height of cone
Sqr root of (h^2 + r^2)
173
Lateral Surface area of cone
Pi r Sl
174
Total surface area of pyramid
TSA=Area of B + LSA
175
Lateral Surface area of Pyramid
1/2 (P)(SI)
176
Volume of Pyramid
1/3 (Area of Base) (h)
177
Volume of Sphere
4/3 pi r^3
178
Surface area of Sphere
4pir^2
179
For spherical segment of 1base
(Pih^2/3)(3r-h)
180
If two solids have the same height and cross sectional area at every level then they have the same volume
Cavalieri’s principle
181
Area of Polygon Inscribed in a Circle
(1/2)(nr^2 sin360/n)
182
Area of Polygon circumscribed on a Cirle
nr^2 tan180/n
183
Perimeter of a parallelogram
2a + 2b
184
Area of a parallelogram
Bh=absin@=1/2 (d1d2)(sin @)
185
2 diagonals of a parallelogram
```
p^2 = a^2 + b^2 -2abcos@
q^2 = a^2 + b^2 -2abcos (180-@)
```
186
Rhombus Perimeter
4b
187
Rhombus area
1/2 (d1d2)
188
Trapezoid Area
h(b1 + b2)/2
189
Area of a circle
Pi r ^2
190
Circumference of a circle
2 pi r
191
Sector area of a circle
1/2 (theta) (r^2)
| Should be in rad
192
Segment (zone) area of a circle
1/2 r^2 (@-sin@)
193
Area of a triangle
1/2 bh
194
Incenter (circle on a triangle)
r=area of traingle/semi perimeter
195
Circumcenter (triangle on a circle)
r= abc/4(Area of a triangle)
196
Type of polyhedra that has two identical faces that are parallel to each other
Prisms
197
Rectangular parallelipiped, LSA
2lh + 2wh
198
Rectangular parallelipiped, TSA
2lh + 2wh +2lw
199
Rectangular parallelipiped, Volume
Lwh
200
Area of a quadrilateral
A= sqr root of (s-a)(s-b)(s-c)(s-d)-abcdcos^2 @
201
Lune Surface Area
(Pi r^2 theta)/90
202
Wedge Volume
Pi r ^3 theta / 270
203
Spherical sector
V=(pi/6)(h)(3a^2 +3b^2 +h)
204
Spheroids Prolate (rotate on major axis)
4/3 pi a b^2
205
Spheroids Oblate (rotate on minor axis)
4/3 pi a^2 b
206
Frustum Volume
1/3 h (B1 + B2 + sqr root of B1B2)
207
Truncated Cylinder Formula
Pi r^2 [(h1 + h2)/2]
208
Truncated Prism Formula
Area of base x [(h1+h2 +…+hn)/n]
209
Platonic Solids: Tetrahedron
(Sqr root of 2 /12) (a^3)
210
Platonic Solids: Hexahedron
a^3
211
Platonic Solids: octahedron
(Sqr root of 2 / 3) (a^3)
212
Platonic Solids: Dodecahedron
7.66a^3
213
Platonic Solids: Hexahedron (20)
2.18a^3
214
Paraboloid Volume
Pi/8 d^2 h
215
Paraboloid Area
2pi/3 d h
216
Torus Volume
2pi^2 R r^2
217
Torus Area
4pi^2 R r
218
Pappus’ No. 1
SA= (2 pi R) (C)
219
Pappus’ No. 2
V= 2piR (area)
220
Hydrostatic Pressure and Force Formula
P= rho g h / gc
221
Centroid Technique
F= gamma hc A
222
Derivative of a^u
a^u (ln a)
223
Derivative of log(a) u
1/ulna
224
Largest rectangle inscribed in a triangle
x=b/2 y=h/2
225
Sector of area but minimum perimeter
A sec= (1/2) theta r^2 (in rad)
s=r theta =2r
Theta =2 rad
r= sqr root of A
226
Rectangle to be fenced at 3 sides with given area at min. Perimeter
x=2y
P=4y
Area=2y^2
227
Largest square inscribed in a circle
a=r sqr root of 2
228
Largest rectangle inscribed in a semi circle
a=r sqr root of 2
229
Largest rectangle inscribed in a triangle w/ one side lying on the base of the triangle
x=b/2
y=h/2
Area of rectangle = 1/2 area of triangle
230
Triangle with 2 equal sides
Isosceles traingle
231
Max light on norman window
x=2y
h=x
P=x(2+pi/2)
232
Longest segment that can fit on a perpendicular intersection
L= (a^2/3 + b^2/3)^3/2
233
Sweet spot
x= sqr root of y1y2
234
Location of a stake on the ground to minimize wire length
x=dh1/h1+h2
| L1 + L2 = (sqr root of h1^2 + x^2) + (sqr root of h2^2 + (d-x)^2)
235
Maximum Cylinder Inscribed in a Cone
h = (1/3)H
| r=2/3 R
236
Cylinder on a Sphere
V of sphere = (4/ cube root of 27) (pi r^3)
| V of cylinder = (4/ sqr root of 27) (pi r^3)
237
Integral of sin x
-cos x
238
Integral of cos x
Sin x
239
Integral of tan x
ln(secx)
240
Integral of cot x
ln(sinx)+c
241
Integral of sec x
ln|secx + tanx| + c
242
Integral of csc x
ln|cscx - cotx| + c
243
Integral of sec^2 x
tan x
244
Integral of csc^2 x
-cot x
245
Integral of tan x sec x
Sec x
246
Integral of cot x csc x
-csc x
247
What kind of polar graph is
| r=a+-b sin theta or r= a +-b cos theta?
Limacons
248
What kind of polar graph is
| r=a sin (n@) or r= a cos (n@) ?
Roses
249
What kind of polar graph is
| r^2 =a^2 sin theta or r^2 = a^2 cos theta?
Lemniscate
250
Roses with even petals
Petals =2n
251
Roses with odd petals
petals = n
252
Guess what type of ODe (Ordinary Differential Equation):
dy/dx = f(x)g(y)
Separable ODe
253
Guess what type of ODe (Ordinary Differential Equation):
| dy/dx + P(x)y=Q(x)
Linear ODe
254
Guess what type of ODe (Ordinary Differential Equation):
| dy/dx = f(y/x)
Homogenous ODe
255
Guess what type of ODe (Ordinary Differential Equation):
| dy/dx + P(x)y =Q(x)y^n
Bernoulli ODe
256
Guess what type of ODe (Ordinary Differential Equation):
| M(dx) +N(dy) = 0
Exact Equation
257
```
Application DE (formula and caltech):
For Nuclear Decay
```
```
Formula:
m=mo e*-kt
ln|m/mo|=-kt
k=ln2/t1/2
Caltech stat 5
x- 0 and half life (time)
y- 100 and 50 (mass remaining)
```
258
```
Application DE (formula and caltech):
For Mixing with same or constant flowrate q
```
(C-Ci/Co-Ci)=e^-(q/v)(t)
259
```
Application DE (formula and caltech):
For Mixing with qo not equal to qi
```
V=(qi-qo)t+Vo
| (C-Ci/Co-Ci)=(V/Vo)^-qi/qi-qo
260
```
Application DE (formula and caltech):
For Population Growth
```
P=Poe^kt
k=ln2/t2
dp/dt= mortality(Kp) - morbidity (death) + migration - immigration
Caltech stat 5
x= 0 and t
Y= 100 and population @ t
261
```
Application DE (formula and caltech):
For Newton’s law of cooling/heating
```
(T-Ta/To-Ta) = e^kt
```
Ta= ambient T
Caltech Mode Stat 5 (x time, y temp)
For cooling (T-Ta)
For heating (Ta-T)
```
262
Econ: simple interest formula
F=P(1+in)
263
Econ: Compound interest
F=P(1+r/m)^mt
264
Econ: i is equal to
r/m
265
Econ: r stands for
Nominal interest rate
266
Econ: ieff for compounded
ieff= [(1+r/m)^m] - 1
267
Econ: ieff for continuous
ieff=e^r - 1
268
Econ: continuous interest
F=Pe^rn
269
Econ: Interest formula
I=Pin
270
Depreciation: Caltech mode
| Straight line method
Stat 2
271
Depreciation: Caltech mode
| Declining Balance Method
Stat 6
272
Depreciation: Caltech mode
| Double Declining Balance
Stat 6
273
Depreciation: Caltech mode
| Sum of Years digit
Stat 3
274
Depreciation: Caltech mode
| Sinking Fund Method
Stat 6
275
Break even in econ means
No loss or gain (balik capital)
276
Econ: book value formula for SLM, DBM, DDBM and SOYD
BVn=nÿ
277
Econ: Depreciation @ nth year formula for SLM, DBM, DDBM and SOYD
dn=(n-1)ÿ-nÿ
278
Econ: Total Depreciation formula for SLM, DBM, DDBM and SOYD
Dtn=FC-nÿ
| FC- first cost
279
Econ: book value formula for SFM
BVn=(FC-SV)x(1-nÿ)/(1-Lÿ)
| L- economic life
280
Econ: depreciation @ nth year formula for SFM
dn=(FC-SV)x(0ÿ-1ÿ)/(1-Lÿ)
281
Econ: total depreciation formula for SFM
Dtn=FC-BVn
282
Perpetuity Formula
P’=A/i
283
Ordinary Annuity Present Formula
P= A( (1+i)^-1/i(1+i)^n )
284
Ordinary Annuity Future Formula
F= A ( (1+i)^n-1/i )
285
Deferred Annuity Formula
P= A ( (1+i)^n -1/i (1+i)^n) (1+i)^-(k-1)
