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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% This file:
% http://anggtwu.net/LATEX/2025-1-C2-S-defs.tex.html
% http://anggtwu.net/LATEX/2025-1-C2-S-defs.tex
% (find-angg "LATEX/2025-1-C2-S-defs.tex")
% Author: Eduardo Ochs <eduardoochs@gmail.com>
%
% «.P» (to "P")
% «.DefDeriv» (to "DefDeriv")
% «.TFC2» (to "TFC2")
% «.DFI» (to "DFI")
% «.MV:reset» (to "MV:reset")
% «.MV» (to "MV")
% «.EDOVS:reset» (to "EDOVS:reset")
% «.EDOVS» (to "EDOVS")
% «.Aipim» (to "Aipim")
% «.eq-if» (to "eq-if")
% «.p-if» (to "p-if")
% «P» (to ".P")
\def\P #1{\left( {#1} \right)}
\def\Pbig#1{ \big( {#1} \big)}
\def\PBig#1{ \Big( {#1} \Big)}
% «DefDeriv» (to ".DefDeriv")
% (c2m251stp 3 "DefDeriv")
% (c2m251sta "DefDeriv")
% 2kT63: (c2m251tnp 4 "defs-DD")
% (c2m251tna "defs-DD")
% (find-angg "MAXIMA/2025-1-s.mac" "TFC2")
\sa {DDL} {\left. \ddx f(x) \, \right|_{x=a}}
\sa {[DD1]} {\CFname{DD1}{}}
\sa {[DD2]} {\CFname{DD2}{}}
\sa {(DD1)} {\P{\D \ga{DDL} = \lim_{ε→0} \frac{f(x+ε)-f(x)}{ε}}}
\sa {(DD2)} {\P{\D \ga{DDL} = \lim_{a→x} \frac{f(a)-f(x)}{x-a}}}
\sa {[DefAt]} {\CFname{DefAt}{}}
\sa {(DefAt)} {\PBig{\left. f(x) \right|_{x=a} \;=\; f(a)}}
\sa {[DefDeriv1]} {\CFname{DefDeriv1}{}}
\sa {[DefDeriv2]} {\CFname{DefDeriv2}{}}
\sa {(DefDeriv1)} {\P{\D \ga{DDL} = \lim_{ε→0} \frac{f(x+ε)-f(x)}{ε}}}
\sa {(DefDeriv2)} {\P{\D \ga{DDL} = \lim_{a→x} \frac{f(a)-f(x)}{x-a}}}
% «TFC2» (to ".TFC2")
% (c2m251stp 3 "TFC2")
% (c2m251sta "TFC2")
% (c2m251sda "TFC2")
\sa {[DefDif]}{\CFname{DefDif}{}}
\sa {(DefDif)}{\PBig{\D \difx{a}{b}{F(x)} \;=\; F(b)-F(a)}}
\sa {[TFC2]} {\CFname{TFC2}{}}
\sa {(TFC2)} {\P{\D \Intx{a}{b}{F'(x)} \;=\; \difx{a}{b}{F(x)}}}
\sa {[II]} {\CFname{II}{}}
\sa {(II)} {\P{\D \intx{F'(x)} \;=\; F(x)}}
\sa {[RDC]} {\CFname{RDC}{}}
\sa {(RDC)} {\PBig{\ddx c \;=\; 0}}
\sa {[RMC]} {\CFname{RMC}{}}
\sa {(RMC)} {\PBig{\ddx(cf(x)) \;=\; c\ddx f(x)}}
\sa {[RPot]} {\CFname{RPot}{}}
\sa {(RPot)} {\PBig{\ddx x^n \;=\; nx^{n-1}}}
\sa {[RSoma]} {\CFname{RSoma}{}}
\sa {(RSoma)} {\PBig{\ddx(f(x)+g(x)) \;=\; \ddx f(x) + \ddx g(x)}}
\sa {[RProd]} {\CFname{RProd}{}}
\sa {(RProd)} {\PBig{\ddx(f(x)g(x)) \;=\; f(x) \ddx g(x) + g(x) \ddx f(x)}}
\sa {[RC]} {\CFname{RC}{}}
\sa {(RC)} {\PBig{\ddx f(g(x)) \;=\; f'(g(x))g'(x)}}
% «DFI» (to ".DFI")
% (c2m251stp 4 "DFI")
% (c2m251sta "DFI")
\sa {[DFI]} {\CFname{DFI}{}}
\sa {(DFI)} {\P{
\begin{array}{rcl}
f(g(x)) &=& x \\
\ddx f(g(x)) &=& \ddx x \\
&=& 1 \\
\ddx f(g(x)) &=& f'(g(x))g'(x) \\
f'(g(x))g'(x) &=& 1 \\
g'(x) &=& \D \frac{1}{f'(g(x))} \\
\end{array}}}
% «MV:reset» (to ".MV:reset")
% (c2m251sta "MV-defaults")
% (c2m251mvdefsa "mv-defaults")
% (c2m241dip 7 "MVDs-e-MVIs-color")
% (c2m241dia "MVDs-e-MVIs-color")
% \def\MVf #1{f (#1)}
% \def\MVfp#1{f'(#1)}
% \def\MVg #1{g (#1)}
% \def\MVgp#1{g'(#1)}
\sa{MV:reset}{
\sa {.fg}{}
\sa {a}{a}
\sa {b}{b}
\sa {u}{u}
\sa {x}{x}
\sa {f(u)}{f (\ga{u})}
\sa {g(a)}{g (\ga{a})}
\sa {g(b)}{g (\ga{b})}
\sa {g(x)}{g (\ga{x})}
\sa {f'(u)}{f'(\ga{u})}
\sa {g'(x)}{g'(\ga{x})}
\sa {f(g(a))}{f (\ga{g(a)})}
\sa {f(g(b))}{f (\ga{g(b)})}
\sa {f(g(x))}{f (\ga{g(x)})}
\sa {f'(g(x))}{f'(\ga{g(x)})}
\sa{f'(g(x))g'(x)}{\ga{f'(g(x))} \ga{.fg} \ga{g'(x)}}
}
\ga{MV:reset}
% «MV» (to ".MV")
\sa {[MVD4]} {\CFname{MVD4}{}}
\sa {(MVD4)} {\P{\begin{array}{rcl}\ga{MVD4}\end{array}}}
\sa {MVD4} {
\D \Intx {\ga{a}} {\ga{b}} {\ga{f'(g(x))g'(x)}}
&=& \D \Difx {\ga{a}} {\ga{b}} {\ga{f(g(x))}} \\
&=& \D \ga{f(g(b))} - \ga{f(g(a))} \\
&=& \D \Difu {\ga{g(a)}} {\ga{g(b)}} {\ga{f(u)}} \\
&=& \D \Intu {\ga{g(a)}} {\ga{g(b)}} {\ga{f'(u)}} \\
}
\sa {[MVD1]} {\CFname{MVD1}{}}
\sa {(MVD1)} {\P{\begin{array}{rcl}\ga{MVD1}\end{array}}}
\sa {MVD1} {
\D \Intx {\ga{a}} {\ga{b}} {\ga{f'(g(x))g'(x)}}
&=& \D \Intu {\ga{g(a)}} {\ga{g(b)}} {\ga{f'(u)}} \\
}
\sa {[MVI3]} {\CFname{MVI3}{}}
\sa {(MVI3)} {\P{\begin{array}{rcl}\ga{MVI3}\end{array}}}
\sa {MVI3} {
\D \intx {\ga{f'(g(x))g'(x)}}
&=& \D \ga{f(g(x))} \\
&=& \D \ga{f(u)} \\
&=& \D \intu {\ga{f'(u)}} \\
}
\sa {[MVI1]} {\CFname{MVI1}{}}
\sa {(MVI1)} {\P{\begin{array}{rcl}\ga{MVI1}\end{array}}}
\sa {MVI1} {
\D \intx {\ga{f'(g(x))g'(x)}}
&=& \D \intu {\ga{f'(u)}} \\
}
% «EDOVS:reset» (to ".EDOVS:reset")
\sa{EDOVS:reset}{
\sa {G(x)} {G(x)}
\sa {H(y)} {H(y)}
\sa {g(x)} {g(x)}
\sa {h(y)} {h(y)}
\sa {Hinv(u)} {H^{-1}(u)}
\sa {Hinv(H(y))} {H^{-1}(H(y))}
\sa {Hinv(G(x)+C_3)} {H^{-1}(G(x)+C_3)}
}
\sa{EDOVS:reset-S1}{
\sa {g(x)} {-2x}
\sa {h(y)} {2y}
\sa {G(x)} {-x^2}
\sa {H(y)} {y^2}
\sa {Hinv(u)} {\sqrt{u}}
\sa {Hinv(H(y))} {\sqrt{y^2}}
\sa {Hinv(G(x)+C_3)} {\sqrt{-x^2+C_3}}
}
\ga{EDOVS:reset}
% «EDOVS» (to ".EDOVS")
\sa {[M]} {\CFname{M}{}}
\sa {(M)} {\P{\begin{array}{rcl}\ga{M}\end{array}}}
\sa {M} {
\D \dydx &=& \D \frac{\ga{g(x)}}{\ga{h(y)}} \\
\ga{h(y)}\,dy &=& \ga{g(x)}\,dx \\ \\[-10pt]
\inty{\ga{h(y)}} &=& \intx{\ga{g(x)}} \\
\mcc{\veq} & & \mcc{\veq} \\
\mcc{\ga{H(y)}+C_1} & & \mcc{\ga{G(x)}+C_2} \\ \\[-10pt]
\ga{H(y)} &=& \ga{G(x)}+C_2-C_1 \\
&=& \ga{G(x)}+C_3 \\ \\[-10pt]
\ga{Hinv(H(y))} &=& \ga{Hinv(G(x)+C_3)} \\
\mcc{\veq} & & \\
\mcc{y} & & \\
}
\sa {[F3]}{\CFname{F}{_3}}
\sa {[F2]}{\CFname{F}{_2}}
\sa {[S1]}{\CFname{S}{_1}}
\sa{(F3)}{
\left(\begin{array}{rcl}
\D \dydx &=& \D \frac{\ga{g(x)}}{\ga{h(y)}} \\ \\[-10pt]
\ga{Hinv(H(y))} &=& \ga{Hinv(G(x)+C_3)} \\
\mcc{\veq} & & \\
\mcc{y} & & \\
\end{array}
\right)
}
\sa{(F2)}{
\left(\begin{array}{rcl}
\D \dydx &=& \D \frac{\ga{g(x)}}{\ga{h(y)}} \\ \\[-10pt]
y &=& \ga{Hinv(G(x)+C_3)} \\
\end{array}
\right)
}
\sa{(S)}{
\left[\begin{array}{rcl}
g(x) &:=& \ga{g(x)} \\
h(y) &:=& \ga{h(y)} \\
G(x) &:=& \ga{G(x)} \\
H(y) &:=& \ga{H(y)} \\
H^{-1}(u) &:=& \ga{Hinv(u)} \\
\end{array}
\right]
}
% «Aipim» (to ".Aipim")
% (c2m251stp 7 "Aipim")
% (c2m251sta "Aipim")
% (c2m251introp 5 "defs-Aipim")
% (c2m251introa "defs-Aipim")
\def\und#1#2{\underbrace{#1}_{#2}}
\def\setdepthto#1#2{\setbox1\hbox{$#2$}%
\dp1=#1%
\box1}
%
\sa{[Aipim]} {\CFname{Aipim}{}}
\sa{(Aipim)} {\Pbig{\sqrt{a^2+b^2} \;=\; a+b}}
\sa{(Aipim34)} {\Pbig{\sqrt{3^2+4^2} \;=\; 3+4}}
\sa {Aipim} {\sqrt{a^2+b^2}=a+b}
\sa {Aipim u}{ % Aipim with "\und"s
\sa{a 0} {\setdepthto{2pt}{a}}
\sa{b 0} {\setdepthto{2pt}{b}}
\sa{a} {\und{\ga{a 0}}{3}}
\sa{b} {\und{\ga{b 0}}{4}}
\sa{a^2} {\und{\ga{a}^2}{9}}
\sa{a^2} {\und{{\ga{a}}^2}{9}}
\sa{b^2} {\und{{\ga{b}}^2}{16}}
\sa{a^2+b^2 0} {\und{\ga{a^2}+\ga{b^2}}{25}}
\sa{a^2+b^2} {\setdepthto{0pt}{\ga{a^2+b^2 0}}}
\sa{sqrt(a^2+b^2) 0} {\sqrt{\ga{a^2+b^2}}}
\sa{sqrt(a^2+b^2) 1} {\setdepthto{50pt}{\ga{sqrt(a^2+b^2) 0}}}
\sa{sqrt(a^2+b^2)} {\und{\ga{sqrt(a^2+b^2) 1}}{5}}
\sa{a+b} {\und{\ga{a}+\ga{b}}{7}}
\sa{sqrt(a^2+b^2)=a+b} {\und{\ga{sqrt(a^2+b^2)} = \ga{a+b}}{\False}}
\ga{sqrt(a^2+b^2)=a+b}
}
% «eq-if» (to ".eq-if")
% (c2m251stp 7 "eq-if")
% (c2m251sta "eq-if")
% (c2m251sda "eq-if")
\Sa {=}#1{\sa{1}{#1}\ga{= after 1}}
\sa{= after 1}{\ga{eq after 1}}
\sa{= after 1}{\ga{eqnp after 1}}
\sa{= after 1}{\ifgaundefined{=.\ga{1}.}\ga{eq after 1}
\else \ga{eqnp after 1}
\fi}
\Sa{eq after 1}{=}
\Sa{eqnp after 1}{\overset{\scriptscriptstyle(\ga{1})}{=}}
\Sa{eqnp after 1}{\standout{$\overset{\scriptscriptstyle(\ga{1})}{=}$}}
% «p-if» (to ".p-if")
% (c2m251stp 6 "p-if")
% (c2m251sta "p-if")
% (c2m251sda "p-if")
\Sa {p}#1#2{\sa{1}{#1}\sa{2}{#2}\ga{p after 2}}
\Sa{p after 2}{\ifgaundefined{p.\ga{1}.}\ga{pnormal after 2}
\else \ga{pbox after 2}
\fi}
\Sa{pnormal after 2}{\ga{2}}
\Sa{pbox after 2}{\standout{$\ga{2}$}}
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2sd"
% ee-tla: "c2m251sd"
% End: