From 5cf06a94624e9eb77d35557ece89ad36df1d0e3f Mon Sep 17 00:00:00 2001 From: j8takagi Date: Mon, 9 Dec 2013 01:06:12 +0900 Subject: [PATCH] =?utf8?q?=E3=83=86=E3=82=B9=E3=83=88=E3=81=AB=E3=80=81?= =?utf8?q?=E7=9B=B8=E4=BA=92=E5=8F=82=E7=85=A7=E3=81=AB=E9=96=A2=E3=81=99?= =?utf8?q?=E3=82=8B=E3=82=82=E3=81=AE=E3=82=92=E8=BF=BD=E5=8A=A0=E3=80=82?= =?utf8?q?=E3=81=BE=E3=81=9F=E3=80=81=E3=82=A8=E3=83=A9=E3=83=BC=E3=81=8C?= =?utf8?q?=E7=99=BA=E7=94=9F=E3=81=99=E3=82=8B=E3=83=86=E3=82=B9=E3=83=88?= =?utf8?q?=E3=81=AE=E5=90=8D=E5=89=8D=E3=82=92error=E3=81=A7=E3=81=AF?= =?utf8?q?=E3=81=98=E3=81=BE=E3=82=8B=E3=82=88=E3=81=86=E5=A4=89=E6=9B=B4?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- latex_mk/test/crossref/Makefile | 9 +++ latex_mk/test/crossref/crossref.tex | 73 ++++++++++++++++++ .../{graphics_error => crossref}/latex.mk | 0 latex_mk/test/error_crossref/Makefile | 9 +++ latex_mk/test/error_crossref/crossref.tex | 73 ++++++++++++++++++ .../latex.mk | 0 .../Makefile | 0 .../latex.mk | 0 .../simple.tex | 0 .../Makefile | 0 .../banana.jpg | Bin .../graphics.tex | 0 latex_mk/test/error_no_graphic_file/latex.mk | 1 + .../Makefile | 0 latex_mk/test/error_undef_ctl_seq/latex.mk | 1 + .../simple.tex | 0 16 files changed, 166 insertions(+) create mode 100644 latex_mk/test/crossref/Makefile create mode 100644 latex_mk/test/crossref/crossref.tex rename latex_mk/test/{graphics_error => crossref}/latex.mk (100%) create mode 100644 latex_mk/test/error_crossref/Makefile create mode 100644 latex_mk/test/error_crossref/crossref.tex rename latex_mk/test/{simle_error_no_end_doc => error_crossref}/latex.mk (100%) rename latex_mk/test/{simle_error_no_end_doc => error_no_end_seq}/Makefile (100%) rename latex_mk/test/{simle_error_undef_ctl_seq => error_no_end_seq}/latex.mk (100%) rename latex_mk/test/{simle_error_no_end_doc => error_no_end_seq}/simple.tex (100%) rename latex_mk/test/{graphics_error => error_no_graphic_file}/Makefile (100%) rename latex_mk/test/{graphics_error => error_no_graphic_file}/banana.jpg (100%) rename latex_mk/test/{graphics_error => error_no_graphic_file}/graphics.tex (100%) create mode 120000 latex_mk/test/error_no_graphic_file/latex.mk rename latex_mk/test/{simle_error_undef_ctl_seq => error_undef_ctl_seq}/Makefile (100%) create mode 120000 latex_mk/test/error_undef_ctl_seq/latex.mk rename latex_mk/test/{simle_error_undef_ctl_seq => error_undef_ctl_seq}/simple.tex (100%) diff --git a/latex_mk/test/crossref/Makefile b/latex_mk/test/crossref/Makefile new file mode 100644 index 0000000..6a147f5 --- /dev/null +++ b/latex_mk/test/crossref/Makefile @@ -0,0 +1,9 @@ +TEXTARGETS := crossref.pdf + +all: $(TEXTARGETS) + +include latex.mk + +clean: tex-clean + +distclean: clean tex-distclean diff --git a/latex_mk/test/crossref/crossref.tex b/latex_mk/test/crossref/crossref.tex new file mode 100644 index 0000000..725e291 --- /dev/null +++ b/latex_mk/test/crossref/crossref.tex @@ -0,0 +1,73 @@ +\documentclass[fleqn]{jsarticle} +\usepackage{amsmath} +\begin{document} + +\title{微分積分法の基本定理 fundamental theorem of calculus} +\author{} +\date{} + +\section*{微分積分法の基本定理} + +関数 $f(x)$ の面積を $S(x)$、$x_0$、$x$ を$f(x)$ 上の点とし、$F(x) = \int f(x) dx$ とすると、 + +\begin{align*} + S(x) = \lim_{n \to \infty} \sum^{n}_{k=1} \frac{x - x_0}{n} f(x_0 + \frac{x-x_0}{n} k) = F(x) - F(x_0) +\end{align*} + +$S(x)$ は、次のように表すこともできる。 + +\begin{align*} + \int^{x}_{x_0} f(t) dt &= F(x) - F(x_0) \\ + &= [F(t)]^x_{x_0} +\end{align*} + +\subsection*{証明} +微分の定義により、 +\begin{align} + S'(x) = \lim_{h \to 0} \frac{S(x+h) - S(x)}{h} + \label{df} +\end{align} + +$f(x)$ の $[x, x+h]$ での 最小値・最大値をそれぞれ $\min(f(x+h))$、$\max(f(x+h))$ とすると、 +\begin{align*} + h \cdot \min(f(x+h)) &\leq S(x+h) - S(x) \leq h \cdot \max(f(x+h)) \\ + \min(f(x+h)) &\leq \frac{S(x+h) - S(x)}{h} \leq \max(f(x+h)) +\end{align*} + +$h \to 0$ とすると、$\min(f(x+h)) \to f(x)$、$\max(f(x+h)) \to f(x)$。 + +そのため、はさみうちの原理により、 +\begin{align} + \lim_{h \to 0} \frac{S(x+h) - S(x)}{h} = f(x) + \label{hasami} +\end{align} + + +\ref{df}、\ref{hasami} より、 +\begin{align*} + S'(x) = f(x) +\end{align*} + +$C$ を積分定数とすると、 +\begin{align} + S(x) = F(x) + C + \label{sx} +\end{align} + +また、$x = x_0$ のとき、 +\begin{align*} + S(x_0) = \lim_{n \to \infty} \sum^{n}_{k=1} \frac{x_0 - x_0}{n} f(x_0 + \frac{x_0-x_0}{n} k) = 0 +\end{align*} + +\ref{sx} に$x = x_0$、$S(x_0) = 0$ を代入し、 +\begin{align*} + 0 &= F(x_0) + C \\ + C &= -F(x_0) +\end{align*} + +そのため、 +\begin{align*} + S(x) = F(x) - F(x_0) +\end{align*} + +\end{document} diff --git a/latex_mk/test/graphics_error/latex.mk b/latex_mk/test/crossref/latex.mk similarity index 100% rename from latex_mk/test/graphics_error/latex.mk rename to latex_mk/test/crossref/latex.mk diff --git a/latex_mk/test/error_crossref/Makefile b/latex_mk/test/error_crossref/Makefile new file mode 100644 index 0000000..6a147f5 --- /dev/null +++ b/latex_mk/test/error_crossref/Makefile @@ -0,0 +1,9 @@ +TEXTARGETS := crossref.pdf + +all: $(TEXTARGETS) + +include latex.mk + +clean: tex-clean + +distclean: clean tex-distclean diff --git a/latex_mk/test/error_crossref/crossref.tex b/latex_mk/test/error_crossref/crossref.tex new file mode 100644 index 0000000..af5bbbf --- /dev/null +++ b/latex_mk/test/error_crossref/crossref.tex @@ -0,0 +1,73 @@ +\documentclass[fleqn]{jsarticle} +\usepackage{amsmath} +\begin{document} + +\title{微分積分法の基本定理 fundamental theorem of calculus} +\author{} +\date{} + +\section*{微分積分法の基本定理} + +関数 $f(x)$ の面積を $S(x)$、$x_0$、$x$ を$f(x)$ 上の点とし、$F(x) = \int f(x) dx$ とすると、 + +\begin{align*} + S(x) = \lim_{n \to \infty} \sum^{n}_{k=1} \frac{x - x_0}{n} f(x_0 + \frac{x-x_0}{n} k) = F(x) - F(x_0) +\end{align*} + +$S(x)$ は、次のように表すこともできる。 + +\begin{align*} + \int^{x}_{x_0} f(t) dt &= F(x) - F(x_0) \\ + &= [F(t)]^x_{x_0} +\end{align*} + +\subsection*{証明} +微分の定義により、 +\begin{align} + S'(x) = \lim_{h \to 0} \frac{S(x+h) - S(x)}{h} + \label{df} +\end{align} + +$f(x)$ の $[x, x+h]$ での 最小値・最大値をそれぞれ $\min(f(x+h))$、$\max(f(x+h))$ とすると、 +\begin{align*} + h \cdot \min(f(x+h)) &\leq S(x+h) - S(x) \leq h \cdot \max(f(x+h)) \\ + \min(f(x+h)) &\leq \frac{S(x+h) - S(x)}{h} \leq \max(f(x+h)) +\end{align*} + +$h \to 0$ とすると、$\min(f(x+h)) \to f(x)$、$\max(f(x+h)) \to f(x)$。 + +そのため、はさみうちの原理により、 +\begin{align} + \lim_{h \to 0} \frac{S(x+h) - S(x)}{h} = f(x) + \label{hasami} +\end{align} + + +\ref{noexist}、\ref{hasami} より、 +\begin{align*} + S'(x) = f(x) +\end{align*} + +$C$ を積分定数とすると、 +\begin{align} + S(x) = F(x) + C + \label{sx} +\end{align} + +また、$x = x_0$ のとき、 +\begin{align*} + S(x_0) = \lim_{n \to \infty} \sum^{n}_{k=1} \frac{x_0 - x_0}{n} f(x_0 + \frac{x_0-x_0}{n} k) = 0 +\end{align*} + +\ref{sx} に$x = x_0$、$S(x_0) = 0$ を代入し、 +\begin{align*} + 0 &= F(x_0) + C \\ + C &= -F(x_0) +\end{align*} + +そのため、 +\begin{align*} + S(x) = F(x) - F(x_0) +\end{align*} + +\end{document} diff --git a/latex_mk/test/simle_error_no_end_doc/latex.mk b/latex_mk/test/error_crossref/latex.mk similarity index 100% rename from latex_mk/test/simle_error_no_end_doc/latex.mk rename to latex_mk/test/error_crossref/latex.mk diff --git a/latex_mk/test/simle_error_no_end_doc/Makefile b/latex_mk/test/error_no_end_seq/Makefile similarity index 100% rename from latex_mk/test/simle_error_no_end_doc/Makefile rename to latex_mk/test/error_no_end_seq/Makefile diff --git a/latex_mk/test/simle_error_undef_ctl_seq/latex.mk b/latex_mk/test/error_no_end_seq/latex.mk similarity index 100% rename from latex_mk/test/simle_error_undef_ctl_seq/latex.mk rename to latex_mk/test/error_no_end_seq/latex.mk diff --git a/latex_mk/test/simle_error_no_end_doc/simple.tex b/latex_mk/test/error_no_end_seq/simple.tex similarity index 100% rename from latex_mk/test/simle_error_no_end_doc/simple.tex rename to latex_mk/test/error_no_end_seq/simple.tex diff --git a/latex_mk/test/graphics_error/Makefile b/latex_mk/test/error_no_graphic_file/Makefile similarity index 100% rename from latex_mk/test/graphics_error/Makefile rename to latex_mk/test/error_no_graphic_file/Makefile diff --git a/latex_mk/test/graphics_error/banana.jpg b/latex_mk/test/error_no_graphic_file/banana.jpg similarity index 100% rename from latex_mk/test/graphics_error/banana.jpg rename to latex_mk/test/error_no_graphic_file/banana.jpg diff --git a/latex_mk/test/graphics_error/graphics.tex b/latex_mk/test/error_no_graphic_file/graphics.tex similarity index 100% rename from latex_mk/test/graphics_error/graphics.tex rename to latex_mk/test/error_no_graphic_file/graphics.tex diff --git a/latex_mk/test/error_no_graphic_file/latex.mk b/latex_mk/test/error_no_graphic_file/latex.mk new file mode 120000 index 0000000..5a11d14 --- /dev/null +++ b/latex_mk/test/error_no_graphic_file/latex.mk @@ -0,0 +1 @@ +../../latex.mk \ No newline at end of file diff --git a/latex_mk/test/simle_error_undef_ctl_seq/Makefile b/latex_mk/test/error_undef_ctl_seq/Makefile similarity index 100% rename from latex_mk/test/simle_error_undef_ctl_seq/Makefile rename to latex_mk/test/error_undef_ctl_seq/Makefile diff --git a/latex_mk/test/error_undef_ctl_seq/latex.mk b/latex_mk/test/error_undef_ctl_seq/latex.mk new file mode 120000 index 0000000..5a11d14 --- /dev/null +++ b/latex_mk/test/error_undef_ctl_seq/latex.mk @@ -0,0 +1 @@ +../../latex.mk \ No newline at end of file diff --git a/latex_mk/test/simle_error_undef_ctl_seq/simple.tex b/latex_mk/test/error_undef_ctl_seq/simple.tex similarity index 100% rename from latex_mk/test/simle_error_undef_ctl_seq/simple.tex rename to latex_mk/test/error_undef_ctl_seq/simple.tex -- 2.18.0