From e636dd6c6e4ee1b1d299d9dafde6b678ea338f62 Mon Sep 17 00:00:00 2001 From: Charlie Root Date: Fri, 27 Feb 2026 21:35:42 -0300 Subject: [PATCH] Lots of changes. For example, investigations on the harmonic series, but also bits on how to draw distance graphs from velocity and how to model a periodic phenomenon. --- calculus.pdf | Bin 842402 -> 843785 bytes calculus.tex | 198 +++++++++++++++++++++++++++++++++------------------ 2 files changed, 129 insertions(+), 69 deletions(-) diff --git a/calculus.pdf b/calculus.pdf index 97c029b113308b9edf7094c296fc35b56247ca90..ce6f7b7ad0bf4b95a397b1918f2e33974d0e5ba2 100644 GIT binary patch delta 136195 zcmZs?Q*bU!v@IIj){0kb+qP}5*iOFKwr$(CZQHhU|Gm$tTle9Zj2|v!Izd zypOp~1`?PdQ4krm(ExG}I(2~+1|^l=0*tm1fXogJ%>2JM6FA_oiTz(_jfa1ep;GJP z;ZPeR>0)61?;I^B9Bd*sCu*aNh&CKBORC=*Fj`}mLLEE^D@UsT8Zc_3zaB8+|Bj{p zivXi(?6(<$P4u%tZ%heDf=>+gphn<;p_ehWHFvfkWMbrGPZUh201RtwCTy~y_^hh$ z*&JfR2oyS$I~K1C14Fy z+9}UYhgVs>Fkx9M5s?@5I)PJ;npSml~k2RJ5j~~kt7Lzoz$!bE(K~OIZS9% zn;=KQnA36=)*t;61c*=^!71lZF_AGcrI7kdby(FUH31g5pDh}E-N&Uw<^kl0p9~fx 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Onde ela intercepta o eixo $y$? Coloque $x=0$, obtendo $y = \pm 1$. \begin{exercise*} -Descreva o gráfico da equação $9x^2 + 16y^2 = 144$. Note que $144 = -12^2 = (4\times 3)^2 = 16 \times 9$. Logo, a equação é equivalente a -$(1/16)x^2 + (1/9)y^2 = 1$. O gráfico é uma elipse que intercepta o -eixo $x$ em $\pm 4$ e intercepta o eixo $y$ em $\pm 3$. +Descreva o gráfico da equação $9x^2 + 16y^2 = 144$. \sol. Note que +$144 = 12^2 = (4\times 3)^2 = 16 \times 9$. Logo, a equação é +equivalente a $(1/16)x^2 + (1/9)y^2 = 1$. O gráfico é uma elipse que +intercepta o eixo $x$ em $\pm 4$ e intercepta o eixo $y$ em $\pm 3$. \end{exercise*} \subsection{Trigonometria} @@ -1457,7 +1457,7 @@ sobre a hipotenusa, que é $\sqrt{x^2 + 1}$. Conclua então que Há uma série de identidades trigonométricas. -A identidade do círculo unitário é óbvia: aplicando a definição de +A identidade do círculo unitário é óbvia---aplicando a definição de $\sin$ e $\cos$ e usando o Teorema de Pitágoras, obtemos $\sin^2 x + \cos^2 x = 1$. Se divirmos ambos os lados da equação por $\cos^2 x$, obtemos $$\tg^2 x + 1 = \sec^2 x.$$ Se, em vez disso, dividirmos por @@ -1520,8 +1520,9 @@ como. \end{figure} % \begin{exercise*} -Observe a desenho exibido pela Figura~\ref{fg-trig-add-formula}. Use-o -pra provar a fórmula da subtração +Observe o desenho exibido pela Figura~\ref{fg-trig-add-formula} (na +página \pageref{fg-trig-add-formula}). Use-o pra provar a fórmula da +subtração % $$\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta.$$ % @@ -1529,27 +1530,23 @@ pra provar a fórmula da subtração distância entre os pontos $A$ e $B$ e a segunda pela lei dos cossenos. Primeiro note que o ângulo que subtende o lado $c$ é o ângulo $\alpha - \beta$. Agora, executando a estratégia, obtemos o -quadrado da distância de $A$ pra $B$ sendo $c^2 = (\cos\alpha - +quadrado da distância de $A$ pra $B$ como $c^2 = (\cos\alpha - \cos\beta)^2 + (\sin\alpha - \sin\beta)^2$. Pela lei dos cossenos, -obtemos $c^2 = a^2 + b^2 - 2ab \cos(\alpha - \beta)$. Assim obtemos +obtemos $c^2 = a^2 + b^2 - 2ab \cos(\alpha - \beta)$. Mas $a=b=1$, +então obtemos % - $$(\cos\alpha - \cos\beta)^2 + (\sin\alpha - \sin\beta)^2 = a^2 + b^2 - 2ab \cos(\alpha - \beta).$$ + $$(\cos\alpha - \cos\beta)^2 + (\sin\alpha - \sin\beta)^2 = 2 - 2\cos(\alpha - \beta).$$ % Daí expanda os quadrados do lado esquerdo e observe que pode-se -aplicar $\cos^2 x + \sin^2 x = 1$ pra obter a constante $2$. Note -também que, pelo desenho, $a = b = 1$. Daí a gente consegue eliminar -todas as constantes da expressão e obter +aplicar $\cos^2 x + \sin^2 x = 1$ pra obter a constante $2$ no lado +esquerdo. Daí a gente consegue eliminar todas as constantes da +expressão e obter % $$\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta,$$ % como desejado. \end{exercise*} -\begin{remark*} -Teríamos trabalhando menos se tivéssemos notado mais cedo que $a = b = -1$. -\end{remark*} - \begin{exercise*} Use o exercício anterior pra provar a fórmula da adição relativa ao cosseno. \sol. Declare $\beta = -\beta$ pra obter @@ -1587,9 +1584,14 @@ fórmula da adição, terminando completamente o exercício, declare $y= uma vez que o cosseno é uma função par e o seno, ímpar. \end{exercise*} +Estamos em um momento importante porque agora conseguimos deduzir uma +as principais identidades trigonométricas, fazendo parecer assim que é +mais fácil aprender a deduzir as identidades que memorizá-las. Não +sei nem se vale a pena registrar as outras. + \begin{exercise*} Se $\sin x = \sfrac{1}{3}$ e $\sec y = \sfrac{5}{4}$, calcule $\sin(x -+ y)$, sendo ambos $x, y$ ângulos agudos positivos. Aplicando a ++ y)$, sendo ambos $x, y$ ângulos agudos positivos. \sol. Aplicando a fórmula da adição, obtemos $\sin(x + y) = \sin x \cos y + \cos x \sin y$ e daí % @@ -1613,17 +1615,54 @@ como desejado. \begin{exercise*} Encontre todos os valores de $x$ tal que $\sin(2x) = \sin x$ e $x \in -[0,2\pi]$. Usando a fórmula dos ângulos duplos, sabemos que $2\sin x -\cos = \sin x$, logo $2\sin x \cos - \sin x = 0$ e então $\sin x -(2\cos x - 1) = 0$. Daí estamos interessados em $\sin x = 0$, em que -são soluções $0, \pi, 2\pi$. Estamos também interessados em $2\cos x -- 1 = 0$ em que $\cos x = \sfrac{1}{2}$, em que uma solução é +[0,2\pi]$. \sol. Usando a fórmula dos ângulos duplos, sabemos que +$2\sin x \cos = \sin x$, logo $2\sin x \cos - \sin x = 0$ e então +$\sin x (2\cos x - 1) = 0$. Daí estamos interessados em $\sin x = 0$, +em que são soluções $0, \pi, 2\pi$. Estamos também interessados em +$2\cos x - 1 = 0$ em que $\cos x = \sfrac{1}{2}$, em que uma solução é $\sfrac{\pi}{3}$ --- e a última é $\sfrac{5\pi}{3}$ porque o ângulo de referência relativo a $\sfrac{5\pi}{3}$ é $\sfrac{\pi}{3}$. Coletando as soluções, obtemos $\{0, \sfrac{\pi}{3}, \pi, \sfrac{5\pi}{3}, 2\pi\}.$ \end{exercise*} +\subsection{Distância a partir de velocidade} + +Suponha que alguém lhe dê o seguinte problema. Um carro viaja com uma +velocidade crescente determinada pela função $v = t$. Subitamente, +esse carro começa a perder velocidade até a velocidade chegar a zero e +essa perda é determinada pela função $v = -t$. O que estamos fazendo +é descrevendo um gráfico por via de palavras. O problema é que você +desenhe o gráfico da distância percorrida. \sol. Isso não deveria ser +difícil, certo? O carro começa ganhando distância mas de pouco em +pouco ele ganha mais e mais distância porque a velocidade aumenta. De +repente a velocidade começa a cair, mas o carro continua a ganhar +distância, embora a velocidade esteja caindo. Isso é absolutamente +compreensível. Por exemplo, um carro atinge 100 km/h e de repente cai +pra 90 km/h, implicando que ele ainda ganhará mais 90 km dentro de uma +hora. Portanto, esse gráfico da distância estará sempre subindo. + +Quem já estudou Cálculo sabe que o gráfico da distância na primeira +metade do problema cresce de acordo com $1/2 t^2$. Podemos então +desenhar uma perfeita curva correta já que temos essa +ferramenta. Agora, a segunda metade da velocidade produz a curva $-1/2 +t^2$, que é uma parábola pra baixo. Isso indica que podemos conectar +esses dois gráficos de forma perfeita, mas estou em dúvida sobre como +fazer a conexão. Por exemplo, a gente não vai conectar ambos os +gráficos através de suas formas na origem porque na origem os valores +do tempo são zero. Quando o carro pára de aumentar a velocidade e +passa a diminuí-la, ele ainda ganha distância com grande velocidade +(em contraste a com pouca velocidade ou quase parando). E é isso que +precisamos entender de uma vez por todas. + +Quando o tempo passa, estamos sempre falando de um intervalo. Então a +gente computa dois pontos da função distância e vê de onde pra onde +ela foi. Por exemplo, se a velocidade cai de 100 km/h pra 90 km/h +durante o intervalo $t_1--t_2$$, então a distância percorrida tem que +ir de $d(t_1)$ pra $d(t_2)$, ou seja, ela vai de $-1/2 t_1^2$ pra +$-1/2 t_2^2$. Assim a conexão que devemos fazer tem que respeitar os +intervalos certos. + \section{Definitions, limits and continuity} \masciidef{} The absolute value of a number $x$, written $|x|$ is the @@ -1697,13 +1736,14 @@ entre $x$ e $a$ se tornar pequena, então torna-se pequena também a distância entre $f$ e $L$. Se isso ocorre pra qualquer pequena distância arbitrada, então o limite de $f$ é $L$. +Provar limites pela definição é usualmente muito difícil. + \begin{exercise*} -Provar limites pela definição é muito difícil na maioria das vezes. -Por exemplo, prove que $\lim x^2 - 1 = 3$ com $x \to -2$. Precisamos -de um $d$ que possa nos levar à desigualdade $$|x^2 - 1 - 3| = |x^2 - -4| = |(x + 2)(x - 2)| = |x + 2||x - 2| < e$$ pra todo $e$ positivo. -Pra ultrapassar o obstáculo desse produto, observe que estamos -interessados em valores de $x$ {\em muito} próximos de $-2$. +prove que $\lim x^2 - 1 = 3$ com $x \to -2$. +\sol. Precisamos de um $d$ que possa nos levar à desigualdade $$|x^2 - +1 - 3| = |x^2 - 4| = |(x + 2)(x - 2)| = |x + 2||x - 2| < e$$ pra todo +$e$ positivo. Pra ultrapassar o obstáculo desse produto, observe que +estamos interessados em valores de $x$ {\em muito} próximos de $-2$. Portanto, podemos tranquilamente nos restrigir ao universo de busca em que $d$ não é maior que 1, ou seja, podemos assumir que $|x + 2| < 1$, obtendo $-1 < x + 2 < 1$, o que nos dá $-5 < x - 2 < -3 < 5$. Daí @@ -2132,11 +2172,11 @@ this with a technical argument.) Mostre que $|x|$ é contínua em toda a linha real. Depois prove que se $f$ é contínua em um intervalo, então $|f|$ também é. Por fim responda se a continuidade de $|f|$ num intervalo implica a -continuidade de $f$. Se não for, encontre um contra-exemplo. +continuidade de $f$. Se não for, encontre um contra-exemplo. -Que $|x|$ é contínua em toda a linha é certamente verdadeiro: a quina -em $x=0$ não é qualquer problema porque ambos os lados de $x=0$ estão -o quão próximo desejarmos de zero. +\sol. Que $|x|$ é contínua em toda a linha é certamente verdadeiro: a +quina em $x=0$ não é qualquer problema porque ambos os lados de $x=0$ +estão o quão próximo desejarmos de zero. O converso do teorema é falso. Por exemplo, seja $f$ uma função com valor $-1$ para $x < 0$ e valor $1$ para $x \geq 0$, que é discontínua @@ -6220,7 +6260,11 @@ Agora note que a nossa série de interesse pode ser comparada a essa da direita, garantindo que ela converge também. É óbvio que $i^2 - 1 < i^2$. Agora precisamos perceber que $(i - 1)^2 < i^2 - 1$. Note que $i=1$ nem nos interessa. Quando $i=2$, obtemos $1 < 3$. Quando $i = -3$, obtemos $4 < 8$. É óbvio que a desigualdade é verdadeira. +3$, obtemos $4 < 8$. Fica óbvio, assim, que a desigualdade é +verdadeira, mas (pra um argumento menos {\em hand-waving}) note que o +lado esquerdo subtrai antes de computar o quadrado e o direito só +subtrai depois---a única preocupação que poderia haver seria com os +menores números $i$, o que evidentemente não é um problema aqui. Agora, é possível manipular a série de interesse e obter o limite dela. Note que $i^2 - 1 = (i + 1)(i - 1)$, que é o denominador da @@ -6247,6 +6291,34 @@ Então uma estratégia interessante é procurar substituir uma série por uma expressão equivalente que seja telescópica. \end{exercise} +\begin{remark*} +Anos depois retorno aqui pra documentar outras anotações e aproveito +pra mostrar uma outra situação telescópica. Estou interessado em saber +se $1 + 1/2 + 1/3 + \cdots + 1/10^4 > 10$. O fato é que não é, mas um +argumento não é nada fácil de se obter. Durante investigações, +descobri que +% + $$1 + \fr{1}{2} + \cdots + \fr{1}{10^4} < 1 + 9.21 < 1 + \int_1^{10^4} \fr{1}{x} \dx,$$ +% +o que não é suficiente pra concluir. Aprofundando-me, descobri que +% + $$\fr{1}{n} < \int_{n - 1}^n \fr{1}{x} \dx = \ln n - \ln (n - 1).$$ +% +À primeira vista, essa desigualdade parece interessante como +ferramenta pro problema original porque podemos substituir cada termo +da soma de interesse por algo que é maior que o termo substituído. +Isso nos dá uma soma maior do que a de interesse. Mas---poxa!---, o +resultado é +% + $$< 1 + \fr{1}{2} + \fr{1}{3} + \fr{1}{4} + \cdots + \fr{1}{10^4} < 1 + 9.21 < 1 + \ln{10^4} = 1 + \cancel{\ln 2} - \ln 1 + \ln 3 - \cancel{\ln 2} + \ln 4 - \cancel{\ln 3} + \cdots + \ln 10^4 - \cancel{\ln (10^4 - 1)}$$ +% +e, assim, novamente, obtemos o limite superior de 10.21, que é muito +pro que precisamos---precisamos de uma estimativa mais justa, mais +apertada. +\end{remark*} + + + \subsection{Método de Newton e raízes quadradas} Use o método de Newton. Por exemplo, se você quer uma sequência que @@ -6274,42 +6346,30 @@ Newton nos dá a sequência e daí obtemos $a_0 = 3$, $a_1 \approx 2.7041$, $a_2 \approx 2.7182$, obtendo uma precisão de pelo menos quatro casas decimais. -\section{Lecionando Cálculo} +\subsection{Uma estimativa por via de integrais} -Aqui vão anotações de um curso de Cálculo 1 projetado por outrem. -Começamos o curso investigando as propriedades algébricas da equação -da reta. A reta é um elemento fundamental do Cálculo. Uma curva -$f(x)$, por exemplo, pode e deve ser vista como uma união de pequenos -trechos de retas---as tangentes à curva $f(x)$---porque isso torna a -curva inteira analisável, que é um dos objetivos centrais do Cálculo -de Leibniz e Newton. +Séries podem ser estimadas por integrais, já que integrais são +séries. Considere o problema de estimar se a soma +% + $$1 + \fr{1}{2} + \fr{1}{3} + \cdots + \fr{1}{10^4}$$ +% +é maior que 13. Desenhe o gráfico de $1/x$ e considere a soma +inferior de Riemann de 1 até $10^4$ usando-se $\D x = 1$. Obtemos +% + $$\fr{1}{2} + \fr{1}{3} + \cdots + \fr{1}{10^4} < \int_1^{10^4} \fr{1}{x} \dx < 9.21$$ +% +e aí basta somar 1 a ambos os lados pra obter a conclusão de que +% + $$1 + \fr{1}{2} + \cdots + \fr{1}{10^4} < 10.21 < 13.$$ +% +Entretanto, um problema mais difícil é provar que essa soma é menor +que 10, que é bom o que nos preocupamos agora. -\subsection{A reta} - -O curso começa aparentemente fazendo uma distinção entre relações -proporcionais e relações afins. Isso me parece uma perda de tempo -porque é o tipo de distinção que só mesmo aos matemáticos parece -interessar. Não me entenda mal---acho legal que os estudantes -obtenham alguma matemática, mas {\em alguma} significa que escolhe-se -sabiamente que elementos dela se apresenta primeiro. - -A diferença entre uma função proporcional e uma função afim é que a -proporcional sempre cruza a origem e a afim nem sempre. É uma -distinção provavelmente mais importante em Álgebra Linear ou em -Geometria Analítica, mas não creio que seja importante em Cálculo. Na -verdade, não creio que o curso esteja realmente interessado -nisso---então vamos em frente. - -Quais coeficientes determinam o crescimento de uma variável em relação -a outra nas retas? É o coeficiente líder porque a constante $b$ da -reta $y = mx + b$ é nada mais que uma translação da reta no plano. -Pra mostrar isso à audiência, primeiro precisamos saber calcular o -crescimento, o que nos exige uma definição de crescimento. O -crescimento pode ser medido com relação a uma variável---existe o -crescimento em $x$ e o crescimento em $y$---o que chamamos de -``delta''. - -Quanto cresce $y$ com respeito a $x$? +Comecemos por ter certeza da soma em si. A soma é bem conhecida, que +é a série harmônica até o $10^4$-ésimo termo. É tipicamente chamada de +$H_{10^4} = 1 + 1/2 + 1/3 + \cdots + 1/10^4 \approx 9.78$. E como +fazer pra concluirmos que ela é menor que 10? Considere a integral da +função inversa de $n - 1$ a $n$ e veja o que você consegue descobrir. \newpage \tableofcontents \newpage \printbibliography