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Some Quick and Dirty Mathematical References

"Do not worry about your difficulties in mathematics, I assure you that mine are greater" - Albert Einstein ( 1879 - 1955 )

Integration By Parts


integration_by_parts.png


Natural Logarithm, ln x


definition_natural_logarith.png

definition_lnx.png

definition_ln_generalized.png

§ Some implications for natural logs:

implications.png

§ Properties of ln:

ln_rules.png


The Inverse of ln x and the number e


natural_logarithm_e.png

§ Definition of number e:

definition_number_e.png

§ Number e defined numerically:

define_number_e_numerically.png

§ Properties of ln x and e:

inverses.png

§ Laws of exponents for ex:

laws_e_exponents.png


The calculus for ln x and ex


calculus.png


The Hyperbolic Function


Every function that is defined on an interval centered at the origin is comprised of even and odd components as follows:

hyperbolic functions

The importance of this class of functions lies in the fact that they best describe the hanging catenary shapes of electrical powerline cables, wave motion in certain elastic solids, and are involved in the differential equations for heat transfer in metals, as well as demonstrating time and distance dilations in special relativity mathematics.

§ Derived hyperbolic identities:

hyperbolic identities

§ The derivatives for these hyperbolic functions:


Maclaurin Series for Some Trig Functions


§ Small trignometric approximations:

radian measureradian measureradian measureradian measure




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