From 6c01c03f2a2eafd1684640e503f0bc555ff10a11 Mon Sep 17 00:00:00 2001 From: Charlie Root Date: Mon, 23 Feb 2026 23:18:30 -0300 Subject: [PATCH] Mere cosmetic changes. --- .gitignore | 1 + Makefile | 2 +- calculus.pdf | Bin 768098 -> 795805 bytes calculus.tex | 43 +++++++++++++++++++++---------------------- 4 files changed, 23 insertions(+), 23 deletions(-) diff --git a/.gitignore b/.gitignore index 75f5da8..c4cc296 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,4 @@ +*~ *.bib *.bbl *.toc diff --git a/Makefile b/Makefile index 0918a2a..1c2f5b1 100755 --- a/Makefile +++ b/Makefile @@ -29,7 +29,7 @@ Makefile calculus.tex refs.bib refs: \ refs.bib - biber calculus + bibtex calculus calculus.ps: \ calculus.dvi Makefile diff --git a/calculus.pdf b/calculus.pdf index 24ff719f45dadebbc22bacdb89362226c33e857d..3c264c2e376a616be76ce697b608a67e21a3f5e6 100644 GIT binary patch delta 318635 zcmV)9K*hh}@GhOnFbE}3L`E$!E;X?TR0jeyHkU!(0x5sRSy^-Bwh?~cUolTHr@X`9 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--git a/calculus.tex b/calculus.tex index 9ef83b8..d8ed8b8 100644 --- a/calculus.tex +++ b/calculus.tex @@ -4944,11 +4944,11 @@ fundamental, obtemos Isso não é lá notícia tão boa porque significa que teremos que memorizar muitas derivadas. Por exemplo, qual a derivada de $\cot x$? Primeiro lembre-se que $\cot x = 1/\tg x$. Daí aplique a regra do -quociente e obtenha $- sec^2 x / \tg^2 x = - 1/\sin^2 x = - \csc^2 x$, -implicando que $\csc^2 x \dx = - \cot x + c$, como +quociente e obtenha $- \sec^2 x / \tg^2 x = - 1/\sin^2 x = - \csc^2 +x$, implicando que $\int \csc^2 x \dx = - \cot x + c$, como desejado. Similarmente, obtém-se assim muitas outras integrais fundamentais como $\int \sec x \tg x \dx = \sec x + c$ uma vez que a -derivada de $\sec x = \sec x \tg x$. Infelizmente, o único jeito de +derivada de $\sec x$ é $\sec x \tg x$. Infelizmente, o único jeito de fazer uso dessas regras é memorizar cada uma por prática. Matemáticos fazem manuais de integração em que eles catalogam as importantes. A gente não vai fazer isso, não, porque seria um trabalho muito mais @@ -4973,24 +4973,26 @@ derivada $f'(g(x)) g'(x)$ e então obter a regra A substituição em si é obtida por identificar na função a ser integrada uma expressão qualquer que possa tomar o lugar de $g'(x)$ e $g(x)$, fazendo assim a substuição $u = g(x)$ e substuindo por -consequinte $g'(x) \dx = \du$, terminando por integrar $\int f'(u) -\du$. +conseguinte $g'(x) \dx = \du$, terminando com o problema resultante +$\int f'(u) \du$, que é tipicamente muito mais fácil que o original. A integração por partes vem da regra do produto. Uma vez que -$(f(x)g(x))' = f'(x) g(x) + f(x)g'(x)$, então % + $$(f(x)g(x))' = f'(x) g(x) + f(x)g'(x),$$ +% +então \begin{align*} \int f(x)g(x) \dx &= \int f'(x) g(x) \dx + \int f(x)g'(x) \dx\\ f(x)g(x) &= \int g(x) f'(x) \dx + \int f(x)g'(x) \dx. \end{align*} % -Agora declare $u f(x) = u$ e $v = g(x)$ de forma que $\du = f'(x)\dx$ -e $\dv = g'(x)\dx$. Assim, pelo teorema fundamental, obtemos +Agora declare $u = f(x)$ e $v = g(x)$ de forma que $\du = f'(x)\dx$ e +$\dv = g'(x)\dx$. Assim, pelo teorema fundamental, obtemos % $$f(x)g(x) = \int v\du + \int u \dv,$$ % -permitindo-nos integral a difícil $\int v\du$ se $\int u \dv$ for -fácil, conversamente permitindo-nos integral a difícil $\int u \dv$ +permitindo-nos integrar a difícil $\int v\du$ se $\int u \dv$ for +fácil, conversamente permitindo-nos integrar a difícil $\int u \dv$ desde que $\int v \du$ seja fácil. Em outras palavras, uma das duas precisa ser fácil de integrar e é por isso que precisamos de alguma sabedoria ao escolher $u$ e $v$ durante a integração por partes. @@ -5007,12 +5009,10 @@ a bonita fórmula % $$\int x^n e^x \dx = e^x \left(\sum_{i=0}^n (-1)^i \frac{n!}{(n - i)!} x^{n - i}\right) + c$$ % -pra todo natural $n$. - -Por exemplo, $\int x^5 e^x \dx = e^x (x^5 - 5 x^4 + 20 x^3 - 60x^2 + -120x - 120) + c$. Essa fórmula vem da integração -por-partes. Escolhe-se sempre o termo $x^i$ como função $u$ e $e^x$ -como função $v$. Aplica-se a integração por-partes $n$ vezes +pra todo natural $n$. Por exemplo, $\int x^5 e^x \dx = e^x (x^5 - 5 +x^4 + 20 x^3 - 60x^2 + 120x - 120) + c$. Essa fórmula vem da +integração por-partes. Escolhe-se sempre o termo $x^i$ como função $u$ +e $e^x$ como função $v$. Aplica-se a integração por-partes $n$ vezes (recursivamente) e em seguida substitui-se de volta em direção ao problema original. Na verdade o que fizemos pra obter essa fórmula foi computar uma {\em reduction formula} pra integral de $x^n e^x$. @@ -5046,14 +5046,14 @@ trigonométricas. Começamos por observar que integrar $\tg x$ não é difícil porque ela é obtida por uma substituição trivial. Primeiro escreva $\tg x = \sin x -/ \cos x$. Em seguida declare $u = \cos x$ de forma que $-\du/(\sin x) -= \dx$, implicando +/ \cos x$. Em seguida declare $u = \cos x$ de forma que $-\du / (\sin +x) = \dx$, implicando % - $$\int \tg x \dx = -\int 1/u \du = -\ln |\cos x| + c = ln |\sec x| + c,$$ + $$\int \tg x \dx = -\int 1/u \du = -\ln |\cos x| + c = \ln |\sec x| + c,$$ % -como desejado. Agora vamos subir o grau. +como desejado. -Feito essa observação, a gente também observa que já sabemos integrar +Feita essa observação, a gente também observa que já sabemos integrar $\sin x$ e $\cos x$, mas que precisamos pensar em potências delas como $\sin^2 x$, $\cos^2 x$ {\em et cetera}. Nenhma dessas duas sai por por-partes---experimente. Obtemos $\int \sin^2 x \dx = -\sin x \cos x @@ -5063,7 +5063,6 @@ integral de $\sin^2 x$, estabelecendo-se assim um ciclo do qual não conseguiremos sair. Agora, a sorte não é a mesma se desejarmos pelas integrais de $\sin^3 x$ ou $\cos^3 x$. - \section{Studying functions} We're going to look into a series of topics now. We are mostly using

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