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semicu
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There are 6 letters in SEMICU ( C3E1I1M3S1U1 )
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Definitions of semicu in various dictionaries:
SEMICU - In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve defined by an equation of the form (A) ...
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In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve defined * by an equation of the form* (A) * * * * * * y * * 2 * * * − * * a * * 2 * * * * x * * 3 * * * * = * * 0 * * , * * a * > * 0 * * . * * * {\displaystyle \quad y^{2}-a^{2}x^{3}\;=\,0\;,\;a>0\;.} * Solving for * * * * y * * * {\displaystyle y} * leads to the explicit form * * (E1) * * * * * y * = * ± * a * * x * * * 3 * 2 * * * * * , * * x * ≥ * 0 * * , * * * {\displaystyle \quad y=\pm ax^{\frac {3}{2}}\;,\;x\geq 0\;,} * which is the cause for the term semicubical parabola. * (A parabola in the common sense can be described by an equation * * * * y * = * a * * x * * 2 * * * * * {\displaystyle y=ax^{2}} * .) * Solving (A) for * * * * x * * * {\displaystyle x} * yields the second explicit form * * (E2) * * * * * x * = * * * ( * * * * * y * a * * * * * * ) * * * * * * * 2 * 3 * * * * * * * , * * y * ∈ * * R * * * . * * * {\displaystyle \quad x={\Big (}{\frac {y}{a}}{\Big )}^{\color {red}{\frac {2}{3}}}\;,\quad y\in \mathbb {R} \;.} * Equation (A) shows, that * * (P) * * * * * x * = * * t * * 2 * * * * , * * y * = * a * * t * * 3 * * * * , * * t * ∈ * * R * * * , * * * {\displaystyle \quad x=t^{2}\;,\quad y=at^{3}\;,\quad t\in \mathbb {R} \;,} * is a parametric representation of the curve. The arc length of the curve was calculated by the English mathematician William Neile and published in 1657 (see section History). . |