From 2a16f5bbb707d52510559c4e85fcededdcb7dbd4 Mon Sep 17 00:00:00 2001 From: Charlie Root Date: Sat, 28 Mar 2026 21:09:44 -0300 Subject: [PATCH] Cosmetics on the derivative of ax^n. --- calculus.pdf | Bin 907776 -> 911975 bytes calculus.tex | 72 ++++++++++++++++++++++++++++----------------------- 2 files changed, 39 insertions(+), 33 deletions(-) diff --git a/calculus.pdf b/calculus.pdf index 496c0e2a48c8364f560c427243d914417dd05824..bff0106755ab7bf212388653f4e38b5f5f4d7863 100644 GIT binary patch delta 158355 zcmV(xK5}- zMQMbYeR!N1Ss3YNR;r?OV)kJ*`(a++Eu=K_b@$ZW{rKT;Uw*6I4E(G6Ee*E#xL@y@kLzYx?HWH#b1Pz%o44WS)1F3-aJTig?GAue&2M#|6Ojhq z^++dhMBErY0RusfK<~1`L4TbPz(gjAoDquwF&l19i`f>!Iu!E&_3myVoH$7vFAK%h z^N2f%%N7>2`#eTHbwp$isQ{MPMd=N1Oz!ew&1)d zL4g5(hQx$`%WP)|bcJO6(A?#GdoPW|TQ%Re6^+$T`^}0THr=iQK7TCy$u10itokZL z0#~fyQXjhW!kzU)0e)Bxk}D7b#9_)pE0@aUo9(XNtm?Mg-Yt~2^9nEMc=t@IO*0`9 zXFv8dpBFt>p!C5g^$C%StBoQj@&o{ELOCD{&2ZlJfynyTJB~4u&)bjdjzG5cPei)c 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expandes esses binômios $(a ++ b)^n$ pra gente. \end{remark*} %% \begin{remark*} @@ -2509,53 +2510,58 @@ Relevante é o Teorema Binomial. %% \end{table} %% \end{remark*} -Outro fato é que $ax^n$ tem derivada $anx^{n - 1}$. É um exercício -interessante provar esse fato por dois ângulos diferentes, que são as -definições de derivada que temos. Uma diz que - - f'(x) = \lim \fr{f(x + h) - f(x)}{h} - -sendo $h \to 0$ e a outra diz que - - f'(x) = \lim \fr{f(x) - f(c)}{x - c} - -com $x \to c$. Ambas são equivalentes, como sabemos, e obter a -derivada por ambas nos dá a informação de que às vezes é mais fácil -trabalhar com uma do que com a outra. Por exemplo, a derivada de -$ax^n$ é +\begin{theorem*} + A derivada de $ax^n$ é $an x^{n - 1}$. +\end{theorem*} +\begin{proof} +É um exercício interessante provar esse fato por dois ângulos +diferentes, que são as definições de derivada que temos. Uma diz que +% + $$f'(x) = \lim_{h \to 0} \fr{f(x + h) - f(x)}{h}$$ +% +e a outra diz que +% + $$f'(x) = \lim_{x \to c} \fr{f(x) - f(c)}{x - c}.$$ +% +Ambas são equivalentes, como sabemos, e obter a derivada por ambas nos +dá a informação de que às vezes é mais fácil trabalhar com uma do que +com a outra. Por exemplo, a derivada de $ax^n$ é % \begin{align*} f'(x) &= \lim \fr{a(x + h)^n - ax^n}{h}\\ - &= a \lim \fr{x^n + C(n,1)x^{n-1}h + C(n,2)x^{n-2}h^2 + ... + C(n,n)h^n - ax^n}{h}\\ - &= a \lim \fr{\cancel{x^n} + C(n,1)x^{n-1}h + C(n,2)x^{n-2}h^2 + ... + C(n,n)h^n - \cancel{ax^n}}{h}\\ + &= a \lim \fr{x^n + C(n,1)x^{n-1}h + C(n,2)x^{n-2}h^2 + \cdots + C(n,i)x^{n-i}h^i + \cdots + C(n,n)h^n - x^n}{h}\\ + &= a \lim \fr{\cancel{x^n} + C(n,1)x^{n-1}h + C(n,2)x^{n-2}h^2 + \cdots + C(n,i)x^{n-i}h^i + \cdots + C(n,n)h^n - \cancel{x^n}}{h}\\ + &= a \lim \left(\fr{C(n,1)x^{n-1}h}{h} + \fr{C(n,2)x^{n-2}h^2}{h} + \cdots + \fr{C(n,i)x^{n-i}h^i}{h} + \cdots + \fr{C(n,n)h^n}{h}\right)\\ &= a n x^{n - 1}, \end{align*} % como desejado. Já usando a outra definição, estamos interessados em % \begin{align*} - f'(x) &= \lim \fr{ax^n - ac^n}{x - c}\\ - &= a \lim \fr{x^n - c^n}{x - c}\\ + f'(x) = \lim \fr{ax^n - ac^n}{x - c} = a \lim \fr{x^n - c^n}{x - c} \end{align*} % e vamos ter que computar uma divisão de polinômios. A divisão de $x^n -- c^n$ por $x - c$ resulta no quociente $x^{n - 1} + c x^{n - 2} + c^2 -x^{n - 3} + ... + c^i x^{n - (i + 1)} + ... + c^{n - 1}$. O padrão é -mais facilmente compreendido se você dividir $x^3 - c^3$ por $x - c$ e -depois $x^4 - c^4$ por $x - c$. Feita a divisão, temos uma função -contínua em toda a linha real e daí obtemos +- c^n$ por $x - c$ resulta no quociente $$x^{n - 1} + c x^{n - 2} + +c^2 x^{n - 3} + ... + c^i x^{n - (i + 1)} + ... + c^{n - 1}.$$ O +padrão é mais facilmente compreendido se você dividir $x^3 - c^3$ por +$x - c$ e depois $x^4 - c^4$ por $x - c$. Feita a divisão, temos uma +função contínua em toda a linha real e daí obtemos % \begin{align*} f'(x) &= a \lim \fr{x^n - c^n}{x - c}\\ - &= a (x^{n - 1} + c x^{n - 2} + c^2 x^{n - 3} + \cdots + c^i x^{n - (i + 1)} + ... + c^{n - 1})\\ - &= a (c^{n - 1} + c c^{n - 2} + c^2 c^{n - 3} + \cdots + c^i c^{n - (i + 1)} + ... + c^{n - 1})\\ + &= a (x^{n - 1} + c x^{n - 2} + c^2 x^{n - 3} + \cdots + c^i x^{n - (i + 1)} + \cdots + c^{n - 1})\\ + &= a (c^{n - 1} + c c^{n - 2} + c^2 c^{n - 3} + \cdots + c^i c^{n - (i + 1)} + \cdots + c^{n - 1})\\ &= a (c^{n - 1} + c^{n - 1} + \cdots + c^{n - 1})\\ &= a n c^{n - 1}. \end{align*} % Note que temos $n$ termos $c^{n - 1}$ porque o contador vai de $n - 1$ -a zero. Obtemos assim a mesma derivada, mas por percursos algébricos -diferentes, sendo o segundo menos óbvio que o primeiro. +a zero. Por fim, como o ponto $x = c$ é um ponto arbitrário, podemos +reescrever $$f'(x) = a n x^{n - 1}.$$ Obtemos assim a mesma derivada, +mas por percursos algébricos diferentes, sendo o segundo menos óbvio +que o primeiro (em nossa humilde opinião). +\end{proof} \begin{theorem*}[regra da soma] Se $f$ e $g$ são deriváveis, então @@ -2580,9 +2586,9 @@ como desejado. A regra do produto é mais envolvente. Stewart oferece uma analogia que é bem útil. Suponha que $f$ e $g$ sejam sempre positivas, o que -nos permite pensar em $fg$ como uma área, como uma a área de um -retângulo. É uma analogia pra dar um sentido geométrico à variação -$\Delta (fg)$ e nos ajudar a escrever a prova do fato. +nos permite pensar em $fg$ como a área de um retângulo. É uma +analogia pra dar um sentido geométrico à variação $\Delta (fg)$ e nos +ajudar a escrever a prova do fato. \begin{theorem*}[regra do produto] Se $u = f(x)$ e $v = g(x)$ são diferenciáveis, então @@ -2700,7 +2706,7 @@ tangentes e coloque $x = 0$, obtendo \end{align*} % o que nos revela que as tangentes relativas aos pontos $x = - 1/2$ e -$x = 1/2$ são perpendiculares se encontram no eixo $y$. Elas têm +$x = 1/2$ são perpendiculares e se encontram no eixo $y$. Elas têm inclinação $-1$ e $1$, logo quando $x=0$, obtemos $y = - 1/4$, como desejado. \end{exercise*} @@ -2709,7 +2715,7 @@ desejado. Considere a hipérbole $xy = c$. Marque qualquer ponto na curva. Trace a reta tangente à curva ao ponto. Essa reta forma um triângulo retângulo sendo os eixos $x, y$ os catetos. Mostre que a coordenada -$x$ do ponto é o ponto médio entre a origem a inserseção entre a reta +$x$ do ponto é o ponto médio entre a origem a intersecção entre a reta tangente e eixo $x$. Em seguida, mostre que a área do triângulo é uma constante~\cite[exercício~76, seção~3.1]{calculo}. %