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Algebra and Calculus
Dr. Jay Maron


Algebra
A (B C)   =  (A B) C         Associativity
A B       =  B A             Commutivity
(A + B) C =  A C + B C       Distributivity
CAB       =  CA CB
logC(A B) =  logC(A) + logC(B)

Factorial
1!  =  1
2!  =  2  =  2⋅1
3!  =  6  =  3⋅2⋅1
3!  = 24  =  4⋅3⋅2⋅1
x!  =  x (x-1) (x-2) ... 1

Binomial expansion
(x+y)2  =  x2 + 2 xy + y2
(x+y)3  =  x3 + 3 x2y + 3 xy2 + y3
(x+y)4  =  x4 + 4 x3y + 6 x2y2 + 4 xy3 + y4

Exponential function

The exponential function is

exp(x)  =  ex
e       =  2.718281828459...
The exponential function has the property that
x ex  =  ex
e  =  ∑n=0 1 / n!  =  1 + 1/2 + 1/6 + 1/24 + ...

Logarithm

The logarithm is the inverse of the exponential function.

10-2  =  .01                  log10 .01 =  -2
10-1  =  .1                   log10 .1  =  -1
100   =  1                    log10  1  =   0
101   =  10                   log10 10  =   1
102   =  100                  log10 100 =   2

10log10x  =  log10 10x  =  1
eln x  =  ln ex  =  1

Relative primality

Two integers are "relatively prime" if they have no common factors that are larger than 1. For example, "4" and "6" are not relatively prime because they share the factor "2", and the numbers "3" and 5" are relatively prime because they have no common factors.


Complex numbers
i   =  √-1

X   =  Xr + Xi i =  Xm eiXθ  =  cos(θ) + sin(θ) i
Xm  =  (Xr2 + Xi2)½
tanθ=  Xi/Xr

A   =  Ar + Ai i =  Am eiAθ
B   =  Br + Bi i =  Bm eiAθ
A B =  Ar Br - Ai Bi + (Ai Br + Ar Bi) i
= Am Bm ei(Aθ+Bθ)
Radians


θ  =  Angle in radians   (dimensionless)
X  =  Arc distance around the circle in meters (the red line in the figure)
R  =  Radius of the circle in meters

X  =  θ R

Pi is defined as the ratio of the circumference to the diameter.

Full circle  =  360 degrees  = 2 π radians

1 radian  =  57.3 degrees
1 degree  = .0175 radians

Polar coordinates

Radius  =  R
Angle   =  θ
X coordinate  =  X  =  R cos(θ)
Y coordinate  =  Y  =  R sin(θ)
tan(θ)  =  sin(θ) / cos(θ)

Small angle approximation

Let (X,Y) be a point on a circle of radius R.

θ   =  Angle of the point (X,Y) in radians
X   =  R cos(θ)
Y   =  R sin(θ)
Y/X =    tan(θ)
If θ is close to zero then
X ~ R
Y << X
Y << R
sin(θ) ~ θ
tan(θ) ~ θ
The "small angle approximation" is
Y/X ~ θ

Base
  10      2    4    8   16

   0      0    0    0    0
   1      1    1    1    1
   2     10    2    2    2
   3     11    3    3    3
   4    100   10    4    4
   5    101   11    5    5
   6    110   12    6    6
   7    111   13    7    7
   8   1000   20   10    8
   9   1001   21   11    9
  10   1010   22   12    A
  11   1011   23   13    B
  12   1100   30   14    C
  13   1101   31   15    D
  14   1110   32   16    E
  15   1111   33   17    F
  16  10000   40   20   10
  17  10001   41   21   11
  18  10010   42   22   12
  19  10011   43   23   13
  20  10100   44   24   14

Motion in 1D

Constant velocity

If an object starts at X=0 and moves with constant velocity,

Time              =  T             (seconds)
Velocity          =  V             (meters/second)
Distance traveled =  X  =  V T     (meters)

Constant acceleration

In the previous case the acceleration is 0.

If an object starts at rest with X=0 and V=0 and moves with constant acceleration,

Time              =  T             (seconds)
Acceleration      =  A             (meters/second2)
Final velocity    =  V  =  A T     (meters/second)
Average velocity  =  Va = .5 V     (meters/second)
Distance traveled =  X  =  Va T  =  .5 V T  =  .5 A T2        (meters)
All of these equations contain the variable T. We can solve for T to obtain an equation in terms of (X, A, V).
V2 = 2 A X
Sim:   
Position, velocity, and acceleration
Equations for constant acceleration

There are four variables (X, V, A, T) and four equations, and each equation contains three of the variables.
At T=0, X=0 and V=0.


Equations                       Variables in       Variable not
                                the equation      in the equation

V = A T                             V  A  T             X

X = .5 A T2                      X     A  T             V

X = .5 V T                       X  V     T             A

V2 = 2 A X                       X  V  A                T

The figure shows the position of a ball at regular time intervals and the green arrow shows the direction of the acceleration.

Top row        Zero acceleration (constant velocity)
Second row     Positive acceleration
Third row      Negative acceleration (deceleration)
Fourth row     Free-fall in gravity
In the language of calculus,
Time         =  T
Position     =  X  =         =   V dT
Velocity     =  V  =  ∂X/∂T  =   A dT
Acceleration =  A  =  ∂V/∂T
Examples of position, velocity, and acceleration.

Sim:    Position, velocity, and acceleration #2


Force

Mass         =  M
Acceleration =  A
Force        =  F  =  M A         (Newton's law)

Gravitational acceleration

For an object falling in gravity, the acceleration doesn't depend on mass and the acceleration is the same everywhere on the surface of the Earth.

Mass                 =  M
Gravity constant     =  g  =  9.8 m/s2
Gravity acceleration =  A
Gravity force        =  F  =  M g       (Law of gravity)
                           =  M A       (Newton's law)
Cancelling the "M's", the acceleration experienced by the object is
A = g
If gravity is the only force involved, then all objects experience the same gravitational acceleration.

We can distinguish between gravitational mass and inertial mass.

Mgrav    =  Gravitational mass
Minertial =  Inertial mass
F        =  Mgrav    g            Gravitational mass causes gravitational force
F        =  Minertial A            Inertial mass governs the response to force
For all known forms of matter,
Mgrav  =  Minertial

Falling

For this example we set g=10 m/s2 and assume there is no air drag. If an object starts at rest and falls under gravity, the distance fallen is

 Time   Velocity   Average   Distance   Acceleration
 (s)     (m/s)     velocity   fallen      (m/s2)
                    (m/s)      (m)

  0        0         0         0           10
  1       10         5         5           10
  2       20        10        20           10
  3       30        15        45           10
  4       40        20        80           10

Distance fallen  =  ½ * Acceleration * Time2

Calculus

The limit
limx→∞ x-1   =  0
limx→0 x-1   =  ∞
limx→∞ x     =  ∞
limx→∞ ex    =  ∞
limx→∞ e-x   =  0
limx→∞ ln x  =  ∞
limx→∞ sin x =  DNE
limx→∞ sin x =  DNE
limx→∞ cos x =  DNE
DNE: The limit does not exist


Derivative

The derivative is defined as

x f(x)  =  limδ→0 [f(x+δ) - f(x)] / δ        if this limit exists
For example,
x x  =  limδ→0 [(x+δ) - x] / δ  =  1
Notation:
x [∂x f(x)]  =  ∂x2 f(x)

∂x  f(x) = f'(x)
∂x2 f(x) = f"(x)
∂x3 f(x) = f‴(x)
∂x4 f(x) = f""(x)

Position, velocity, and acceleration
Time         =  t
Position     =  x(t)
Velocity     =  v(t)  =  ∂t x(t)
Acceleration =  a(t)  =  ∂t v(t)  =  ∂t2 x(t)

Common derivatives

"a" is a constant and "x" is the veriable being differentiated

x a   =  0
∂x x   =  1
∂x x2  =  2 x
∂x x3  =  3 x2x x4  =  4 x3x xn  =  n xn-1x ex  =  exx ln(x)  =  x-1x sin(x) =  cos(x)
∂x cos(x) = -sin(x)

Taylor series

f(x)  =  f(0) + f'(0) x + (2!)-1 f"(0) x2 + (3!)-1 f‴(0) x3 + (4!)-1 f""(0) x4 + ...

ex  =  1 + x + (2!)-1 x2 + (3!)-1 x3 + (4!)-1 x4 + ...

ln (1+x)  =  x - 2-1 x2 + 3-1 x3 - 4-1 x4 + ...

sin x  =  x - (3!)-1 x3 + (5!)-1 x5 - (7!)-1 x7 + ...

cos x  =  1 - (2!)-1 x2 + (4!)-1 x4 - (6!)-1 x6 + ...

Properties of derivatives
x (A B)  =  (∂xA) B + A (∂xB)

Differential equations

"C" denotes an arbitrary constant.

x  Y =  Y      →     Y  =  C ex2x Y = -Y      →     Y  =  C1 sin(x) + C2 cos(x)

Integral


Exponential function
ex  =  limn→∞ (1 + x/n)n

Problems from mathematics competitions

Tower of power

Let Tn(x) be a sequence such that    T1(x) = x    and    Tn+1(x) = xTn(x).    For example,

T1(x) = x
T2(x) = xx
T3(x) = xxx
For what positive values of x does limn→∞(x) converge?
Tiles

Using tiles of size (4,6) and (5,7), how would you tile a floor of size (1000,1000)? The tiles cannot overlap and there can be no untiled gaps.


Football

If a game has two possible ways to score points, the values being "3" and "5", then the unattainable scores are (1, 2, 4, 7). If the possible values are relatively prime integers "A" and "B" then how many unattainable scores are there?


Holy Trinity

What are the last 2 digits in the base-10 representation of

33333


Cube in a barrel

A cylindrical barrel 1 meter tall and 1 meter in diameter is full to the brim with water. If a cube with side length 2 is placed in the barrel as far as it can go and with its diagonal being vertical. How much water spills out?


Trees

A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let (m/n) in lowest terms be the probability that no two birch trees are next to one another. Find m+n.


Even number

What is the largest even integer that cannot be written as the sum of two odd composite numbers?


Coin

A fair coin is to be tossed 10 times. Let (i/j), in lowest terms, be the probability that heads never occur on consecutive tosses. Find i+j.


Complex plane

Consider the region "A" in the complex plane that consists of all points z such that both z/40 and 40/z have real and imaginary parts between 0 and 1, inclusive. What is the integer that is nearest the area of A?










Putnam


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