ray Leaps Into Boat, Stings Florida Man in Heart.

A dream is a wish your heart makes,
When you're fast asleep,
In dreams you lose your heartaches,
Whatever you wish for, you keep.

Have faith in your dreams and someday,
Your rainbow will come smiling through,
No matter how your heart is grieving,
If you keep on believing,
The dream that you wish will come true.

A dream is a wish your heart makes,
When you’re feeling small,
Alone, In the night you whisper,
Thinking no one can hear you at all.

You wake with the morning sunlight,
To find fortune that is smiling on you,
Don't let your heart be filled with sorrow,
For all you know tomorrow,
The dream that you wish will come true.

A dream is a wish your heart makes,
[sax]
A dream is a wish your heart makes,
[sax]

You Wake with the morning sunlight,
To find fortune that is smiling on you,
Don't let your heart be filled with sorrow,
For all you know tomorrow,
The dream that you wish will come true.

No matter how your heart is grieving
If you keep on believing
The dream that you wish will come true

Asymptotes

Definition of a horizontal asymptote: The line y = y₀ is a "horizontal asymptote" of f(x) if and only if f(x) approaches y₀ as x approaches + or - .

Definition of a vertical asymptote: The line x = x₀ is a "vertical asymptote" of f(x) if and only if f(x) approaches + or - as x approaches x₀ from the left or from the right.

Definition of a oblique asymptote: the line y = ax + b is a "slant asymptote" of f(x) if and only if lim _(x-->+/-) f(x) = ax + b.

Definition of a concave up curve: f(x) is "concave up" at x₀ if and only if f '(x) is increasing at x₀

Definition of a concave down curve: f(x) is "concave down" at x₀ if and only if f '(x) is decreasing at x₀

The second derivative test: If f ''(x) exists at x₀ and is positive, then f ''(x) is concave up at x₀. If f ''(x₀) exists and is negative, then f(x) is concave down at x₀. If f ''(x) does not exist or is zero, then the test fails.

Definition of a critical point: a critical point on f(x) occurs at x₀ if and only if either f '(x₀) is zero or the derivative doesn't exist.

Local (Relative) Extrema

Definition of a local maxima: A function f(x) has a local maximum at x₀ if and only if there exists some interval I containing x₀ such that f(x₀) >= f(x) for all x in I.

Definition of a local minima: A function f(x) has a local minimum at x₀ if and only if there exists some interval I containing x₀ such that f(x₀) <= f(x) for all x in I.

Occurrence of local extrema: All local extrema occur at critical points, but not all critical points occur at local extrema.

The first derivative test for local extrema: If f(x) is increasing (f '(x) > 0) for all x in some interval (a, x₀] and f(x) is decreasing (f '(x) <>0, b), then f(x) has a local maximum at x₀. If f(x) is decreasing (f '(x) <>0] and f(x) is increasing (f '(x) > 0) for all x in some interval [x₀, b), then f(x) has a local minimum at x₀.

The second derivative test for local extrema: If f '(x₀) = 0 and f ''(x₀) > 0, then f(x) has a local minimum at x₀. If f '(x₀) = 0 and f ''(x₀) <>0.

Absolute Extrema

Definition of absolute maxima: y₀ is the "absolute maximum" of f(x) on I if and only if y₀ >= f(x) for all x on I.

Definition of absolute minima: y₀ is the "absolute minimum" of f(x) on I if and only if y₀ <= f(x) for all x on I.

The extreme value theorem: If f(x) is continuous in a closed interval I, then f(x) has at least one absolute maximum and one absolute minimum in I.

Occurrence of absolute maxima: If f(x) is continuous in a closed interval I, then the absolute maximum of f(x) in I is the maximum value of f(x) on all local maxima and endpoints on I.

Occurrence of absolute minima: If f(x) is continuous in a closed interval I, then the absolute minimum of f(x) in I is the minimum value of f(x) on all local minima and endpoints on I.

Alternate method of finding extrema: If f(x) is continuous in a closed interval I, then the absolute extrema of f(x) in I occur at the critical points and/or at the endpoints of I.
(This is a less specific form of the above.)

Definition of an increasing function: A function f(x) is "increasing" at a point x₀ if and only if there exists some interval I containing x₀ such that f(x₀) > f(x) for all x in I to the left of x₀ and f(x₀) <>0.

Definition of a decreasing function: A function f(x) is "decreasing" at a point x₀ if and only if there exists some interval I containing x₀ such that f(x₀) <>0 and f(x₀) > f(x) for all x in I to the right of x₀.

The first derivative test: If f '(x₀) exists and is positive, then f '(x) is increasing at x₀. If f '(x) exists and is negative, then f(x) is decreasing at x₀. If f '(x₀) does not exist or is zero, then the test tells fails.

Power of x.

c = 0

x = 1

xn = n x(n-1) Proof

Exponential / Logarithmic

ex = ex Proof

bx = bx ln(b) Proof

ln(x) = 1/x Proof

Trigonometric

sin x = cos x Proof	csc x = -csc x cot x Proof
cos x = - sin x Proof	sec x = sec x tan x Proof
tan x = sec2 x Proof	cot x = - csc2 x Proof

Inverse Trigonometric

arcsin x = 1 (1 - x2)	arccsc x = -1 |x| (x2 - 1)
arccos x = -1 (1 - x2)	arcsec x = 1 |x| (x2 - 1)
arctan x = 1 1 + x2	arccot x = -1 1 + x2

sinh x = cosh x Proof	csch x = - coth x csch x Proof
cosh x = sinh x Proof	sech x = - tanh x sech x Proof
tanh x = 1 - tanh2 x Proof	coth x = 1 - coth2 x Proof