From dd4e84de3df693a2a1eb188d401643f7bfa491fe Mon Sep 17 00:00:00 2001 From: Charlie Root Date: Mon, 30 Mar 2026 21:00:49 -0300 Subject: [PATCH] A years-long explanation on differentials. --- calculus.pdf | Bin 922112 -> 922394 bytes calculus.tex | 32 +++++++++++++++++++++++--------- 2 files changed, 23 insertions(+), 9 deletions(-) diff --git a/calculus.pdf b/calculus.pdf index 7c16cdbe59ac9adefbb0c7eddb3fe77ce69ffcca..4782b9e3534e0cd86bd13e727af3a648f6f970a6 100644 GIT binary patch delta 81320 zcmV(+K;6H9q&S+UIDmu!gaWh!5?=x_HJ6dB1S)?V%aY@^@xEW7TQrp&7;k`fd`SFA zt;+<5OntPb|x8;OwSbKsL=3cIs0+Wkhr=ggo^o>^-q|ws;#zHjLdg+ zx$b|iet!AQCtou*D^MY+G=1RNHPMJ^?SB5^*0qr&YC|xL!=R~c^47HwI%-iM!bsoX z^IR9iPSv-#&2%9Etp58?Sl~=23~4pAd6Vr!7Ch>NWB@ju3~ioYU2~%6uUCMk+<+2s z%to_LdV@PYG^ji14wHH5F3Ye0RGS63SyzAV%k6%<*wxL}55HSM6ZnA5-QjPQUA)Zn zDB&oKzI>CJAGFZwaDrQRt+&hS{^}ZLSHYc|ru*{Sm5}qYwR0Y3S?)NyxAkV(GZ}%JX-}m04t7`$=@p9#RI9RhR-X%<#%K@sld8jZGG2;EiJqXJR2IAXl z=_XnCkX7xb8-U62Q2|327a;UzK!wAbT|v$)SiRbphAMLqV_+>p()P=oBrppe zc<}SQG!aCNm`aM7M1=wo(Tf6(2w{J&4bF^1k%C6oQL7U|gTI5&INnJX>DN>cBya&J zlF-QqFy-`%YKBqh+%x`$2I z9<|5DCqQmepW_zAb+W|J+p>Q$g0u$lioD7a3Q46>0rtcG-XXB*toSE`jvLJbR*Dn@ z)qj_p2CQYOXH9p8v?QMULF$2F;gjXR&}6Z1f4j0vd^mc+0SW98Y$yIu+eslFmg9B3 zTR9jOjU~^fT^edKFaaj580s8!N9LqK2{C?(Z<0yn>j?SJ-tp0Kzq#)RcX0NmX>)Yr~VbYqq7)4Ja6)?VQ=1p-x!6#nhL1rGJnp&6uu zSvb#?jx2Tm=$9NgJ6`f~S9Li{KuV-2dnoc1D)6wve}z#gg}!i3OM@drWpEHc2NJ2W zQ~^2~EAmix!%R)R)6{>a5ZMt^M1i~;=pm;%0kmNKeOu+I4z#O|?AJGWJ5F*H>^N9$ zeF8X5^khB;n;X0bV{>nxkMzXC5ZkKU-gymG=Zw(;M0vnvPAO<4Bu(}02)t{`!9XX# z5ERS^*<(Em9=OMtp7MS*!nnBDel->n9RM9v$q0d7ZaMs`rny9Mttm{H$nXQ7phB#9t&kR-4)wIB0! zueIR5+u!Wkaj{4c?ZnM8sL?W{4c6dsfD`^- z7e=wK@_tQ$5s=(dv0GCIj?Z7OY6q$(fu?mgWpDd!7iAz6?@Jo1(XD}pJh2^lWIOgg zSH1I&xLZ~#P2hv3%i66Q@8don$Q6FOUoSI`@w#m9dUF=+?QPDip+Iu6Q94=UkL(%B z>6k1*YZ!lfH<|wn6KUnbhJ@kbQKtsfnU~7}ZJf*bU0dF-+y{p;b9aa``#Ql6SIjMy z`C`4VH^8~&=HA?{`xfi0K}bPoX)(~sw;9aINRquzf&|=?5WBn%zmuxKNRoQj8#7kVyF{ zf?epEgQXY6s?1}PlPb!1+z%yw6M;^*6kxcn-rTR><~&@`k%GK$0}w2n(222zG&$w1 z2>Ni-R|ds&#Pl_0zaURh${YWyYuw5?zM&+J$i)PddY=jfN9NeZhopm3(5;-THXR0T zrhR_|!$jQb?VWqIu74fW!Dv_)zavG;dGAY1y?e;v0e+O8;775iAfOK|1~#uYMycKS zm{1v+3#{Mis()f!!5?dU?7I4;%(?&sVR{hrJ=|jwXB}>Nc8>`}6A@?kn4(ti(?3=5 z##w$l10$_RQIr>5=l2M@`fls?0X*Vg^|62T9W0M+GQ8<}Gs!8+c3Cc~x@|(M3=4w2 zFIeH7WAb1+{D_gYkIaD=PS!@=KF>$-h|rTZQu(BfRQ~=qQdP{k`K>=t5-{&H3iFjo zL150wsNDT+X$?QZ=RYwMETwV<&!EA6soZOCGh3OoG#o%|P@;2R!N*(Ii?U7HwyuBc zS9|D^0)v5bKmW2?fL~GNFbAfMX2k?^ikw1Q0Sh3{WZxm8aE&<~Rw~^0V1v;a$qB_{De|sfgj&pDbd7R zJK|$trJkIRVXTuv&c6ZTCQ+J2lvtLKKJlcF=WVqsvkrh%AZ+P}aR8=Kf24oO0`(u{ z4}!_?SSHh-PRrw(Oz+crjh5ofo>+eSH8%BsmtG?{44_60!*Wz`exa}Spa4;s?Sn?j z$`J{0NN#mZa=<~@<^$Q)$EL1EnF07nW`KS$T^{5UCo~7HH!j#EL^o|QCR6vwFo*`4 zg5Lj6>bh*|C#Y*|(Ig9&P}hHyWPqu&@#2I6?J>K*awt$pN)-ac4q%9@z8_*I<+Hi^BD;KPF8j5EGMt3gm!M!ASoD7GS18FT6PXBB=So`fTO? z%NS;pKLEUV&7gbzWToOG7)@tH+0y`8UKV1xK!}sn+bLkEYZF*d8IpfumZT0oU@!zI zqKclSC~&ZWptKghC*0zpnRIF#1HVH{*+yhBD)@igV|IL7R4Wgg-5BO zCf#4YqvP_FEV=WWOrPcA@5F5K0vpEx_(ij6`t3ZkbGK}epFvWQis45JiBOCxeKG34 zn$~PRDdFREqybG+{+Uf3a#3p+ z&2ILco489fbD?F^Y~6G&cTdcH=3>=)aI<+Tuw0=|@yA@y9_fFu``&kHS7JZFMZJ5?mDFV1m&B04V2v!%@(YAG7UTZ3)c1^ci;mHxRS~-M-k=+dPz(?sJuiG|r8A3`m#R zK8O|*AeCX2{kDI#uYBXVQr<3~uEK&rw#ben89iIC$P^s`G-7!X_KWIl-XTFUY#uy- z%|fBUF+S7Bq*Nl&Ou0Fjiqr~KK_|*olGG^NR_&(T{iQSUR&qvyY7!q2a8iK6k_5x% zhyZ0=oDvZlNikr1Rr(p0=0fLdwt?hT}(Gk|600>rF-q@-aMx|(6qA(1#Y^POG$Q5F1;OYO!o46-?@V; 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b/calculus.tex @@ -3008,7 +3008,7 @@ Veja também a Seção~\ref{sec:sine-cosine} pra compreender o ``co'' em \subsection{Differentials} If $y = f(x)$, where $f$ is a differentiable function, then the -differential $dx$ is an independent variable; that is, $dx$ can be given +differential $dx$ is an independent variable, that is, $dx$ can be given the value of any real number. The differential $dy$ is then defined in terms of $dx$ by the equation % @@ -3026,22 +3026,36 @@ x$. Then, the change in $y$ is % $$\Delta y = f(x + \Delta x) - f(x).$$ % -Certainly, because $f(x + \Delta x)$ is the height of the point $Q$; -then, if we subtract the height of the point $P$, we get the change in +Certainly, because $f(x + \Delta x)$ is the height of the point $Q$. +Then, if we subtract the height of the point $P$, we get the change in $y$ as we moved from $P$ to $Q$. Now, $dy$ expresses the change in the linearization of $f(x)$, which for a concave down curve with positive slope, would be larger than $\Delta y$. -In a concave down graph with positive slope, for example, let $L(x)$ be -a tangent to the curve at $P$. So $\Delta y < dy$. The value of $\Delta -y$ is simply $f(x + \Delta x) - f(x)$, so the value of $dy$ should be -something like $L(x + \Delta x) - L(x)$. However, the value of $dy$ is -$f'(x) dx$. So it must follow that +In a concave down graph with positive slope, for example, let $L(x)$ +be a tangent to the curve at $P$. So $\Delta y < dy$. The value of +$\Delta y$ is simply $f(x + \Delta x) - f(x)$, so the value of $dy$ +should be something like $L(x + \Delta x) - L(x)$, embora a definição +seja $dy = f'(x) \dx$. Isso significa que tem que ser verdadeiro que %v - $$L(x + \Delta x) - L(x) = f'(x) \dx.$$ + $$L(x + \Delta x) - L(x) = f'(x) \dx.$$ %e +E de fato é---hoje podemos explicar por quê. Declare $\Delta x = +dx$. Comparemos $\Delta f$ com $dy$. Quando nos movemos numa +quantidade $\Delta x$, a variação na curva $f(x)$ é $\Delta f = f(x + +\Delta x) - f(x)$. Já a variação na reta $L(x)$ é +% +\begin{align*} + L(x + \Delta x) - L(x) &= L(x + dx) - L(x)\\ + &= f(a) + f'(a)(x + dx - a) - (f(a) + f'(a)(x - a))\\ + &= \cancel{f(a)} + \cancel{f'(a)x} + f'(a)\dx - \cancel{f'(a)a} - \cancel{f(a)} - \cancel{f'(a)x} + \cancel{f'(a)a}\\ + &= f'(a) \dx, +\end{align*} +% +como desejado. + %% \subsection{Newton's method}