User:Jonargue/Sandbox

Sandbox Page This is a sandbox page.

The purpose of this page is to demonstrate the ability to represent mathematical notations similar to what is contained in the BC Ministry of Education's Math 10 IRP, in addition to other types of special notations.

The source of this information is the BC Ministry of Education Math 10-12 IRP [1].

It is expected that students will recognize antidifferentiation (indefinite integral) as the reverse of the differentiation process.

It is expected that students will:

• explain the meaning of the phrase “$F(x)$ is an antiderivative (or indefinite integral) of $f(x)$”
• use antiderivative notation appropriately (i.e., $\int f(x)dx$ for the antiderivative of $f(x)$)
• compute the antiderivatives of linear combinations of functions whose individual antiderivatives are known including:

• • $\int x^{r}\,dx={\frac {x^{r+1}}{r+1}}+C$ if $r\neq -1$
  • $\int {\frac {dx}{x}}=\ln |{x}|+C$
• compute $\int f(ax+b)dx$ if $\int f(u)du$ is known
• create integration formulas from the known differentiation formulas
• solve initial value problems using the concept that if $F'(x)=G'(x)$ on an interval, then $F(x)$ and $G(x)$ differ by a constant on that interval

Musical Notations "C♯♯" yields D, when D's sharp is in the signature. Assuming enharmonicity, it is possible that use of accidentals will create equivalences between pitches that are written differently. For instance, raising the note B to B♯ is equal to the note C. Assuming the elimination of all such equivalences, however, the complete chromatic scale adds five additional pitch classes to the original seven lettered notes for a total of 12, each separated by a half-step.

Accidentals

Frequency on a logarithmic scale vs Position on Treble Clef. Each note shown has a frequency of the previous note multiplied by ${\sqrt[{12}]{2}}$

Letter names are modified by the accidentals - sharp (♯, similar to the symbol #) and flat (♭, similar to the letter b). These symbols respectively raise or lower a pitch by a semitone or half-step, which in modern tuning will multiply or divide (respectively) the frequency of the original note by ${\sqrt[{12}]{2}}$, approximately 1.059. They are written after the note name: so, for example, F♯ represents F-sharp, B♭ is B-flat.

--Jonargue (talk) 22:03, 2 September 2008 (UTC)