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

Time         =  T
Position     =  X  =         =  $\int$ V dT
Velocity     =  V  =  ∂X/∂T  =  $\int$ A dT
Acceleration =  A  =  ∂V/∂T

Sim:    Position, velocity, and acceleration #2

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

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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

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Calculus

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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

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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)

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Position, velocity, and acceleration

Time         =  t
Position     =  x(t)
Velocity     =  v(t)  =  ∂t x(t)
Acceleration =  a(t)  =  ∂t v(t)  =  ∂t2 x(t)

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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 x2
∂x x4  =  4 x3
∂x xn  =  n xn-1
∂x ex  =  ex
∂x ln(x)  =  x-1
∂x sin(x) =  cos(x)
∂x cos(x) = -sin(x)

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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 + ...

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Properties of derivatives

∂x (A B)  =  (∂xA) B + A (∂xB)

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Differential equations

"C" denotes an arbitrary constant.

∂x  Y =  Y      →     Y  =  C ex
∂2x Y = -Y      →     Y  =  C1 sin(x) + C2 cos(x)

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Integral

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Exponential function

ex  =  limn→∞ (1 + x/n)n

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Mathematics competition problems

Getting the correct answer is only worth half the points. For full credit, you must give a rigorous proof.

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Problems with Calculus not required

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Holy Trinity

What are the last 2 digits of

3³³³³

Note that exponentials are evaluated top down. 3³³ = 3⁽³³⁾ = 3²⁷

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Tiles

Using tiles of size 4x6 and 5x7, how would you tile a floor of size 1999x1999? The tiles cannot overlap and there can be no gaps.

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Problems with Calculus required

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Tower of power

x^xxxx

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

T1(x) = x
T2(x) = xx
T3(x) = x(xx)

For what positive values of x does lim_n→∞ T_n(x) converge?

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Limit

f(x) is a continuous function and f(x) >= 0 for all x. If

∫_-∞^∞ f(x) dx

is finite, does it follow that

∫_-∞^∞ [f(x)]² dx

is finite?

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Main page

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