5.Reference

5.1.2The Pythagorean theorem

H2 = A2 + O2

5.2Non-right triangle trigonometry

b

C A

B

5.2.1The Law of Sines (for any triangle)

5.2.2The Law of Cosines (for any triangle)

A2 = B2 + C2 - (2BC)(cos a)

B2 = A2 + C2 - (2AC)(cos b)

C2 = A2 + B2 - (2AB)(cos c)

5.3Trigonometric equivalencies

5.4Hyperbolic functions

ex - e-x

sinh x =

2

ex + e-x

cosh x =

2

tanh x =

sinh x

cosh x

Note: all angles (x) must be expressed in units of radians for these hyperbolic functions. There are 2¼ radians in a circle (360o).

6.6Rules for derivatives

6.6.1Constant rule

6.6.2Rule of sums

6.6.3Rule of di®erences

6.6.4Product rule

6.6.5Quotient rule

6.6.6Power rule

6.6.7Functions of other functions

d f[g(x)] dx

Break the function into two functions:

6.7The antiderivative (Inde¯nite integral)

Then:

g(x) is the derivative of f(x) f(x) is the antiderivative of g(x)

òg(x) dx = f(x) + c

Notice something important here: taking the derivative of f(x) may precisely give you g(x), but taking the antiderivative of g(x) does not necessarily give you f(x) in its original form. Example:

f(x) = 3x2 + 5

ò6x dx = 3x2 + c

Note that the constant c is unknown! The original function f(x) could have been 3x2 + 5, 3x2 + 10, 3x2 + anything, and the derivative of f(x) would have still been 6x. Determining the antiderivative of a function, then, is a bit less certain than determining the derivative of a function.

6.8Common antiderivatives

ò 1x dx = (ln |x|) + c

Where,

c = a constant

6.9Antiderivatives of power functions of e

òex dx = ex + c

Note: this is a very unique and useful property of e. As in the case of derivatives, the antiderivative of such a function is that same function. In the case of the antiderivative, a constant term "c" is added to the end as well.

6.10Rules for antiderivatives

6.10.1Constant rule

òcf(x) dx = c òf(x) dx

6.10.2Rule of sums

ò[f(x) + g(x)] dx = [òf(x) dx ] + [òg(x) dx ]

6.10.3Rule of di®erences

ò[f(x) - g(x)] dx = [òf(x) dx ] - [òg(x) dx ]

6.11De¯nite integrals and the fundamental theorem of calculus

Then:

x

òf(x) dx = g(x)

0

6.12Di®erential equations

As opposed to normal equations where the solution is a number, a di®erential equation is one where the solution is actually a function, and which at least one derivative of that unknown function is part of the equation.

As with ¯nding antiderivatives of a function, we are often left with a solution that encompasses more than one possibility (consider the many possible values of the constant "c" typically found in antiderivatives). The set of functions which answer any di®erential equation is called the "general solution" for that di®erential equation. Any one function out of that set is referred to as a "particular solution" for that di®erential equation. The variable of reference for di®erentiation and integration within the di®erential equation is known as the "independent variable."