From 681cbfda10f2e1bed5cbef8dd853ccd3c7f90ba8 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Tue, 11 Sep 2018 19:16:39 +1000 Subject: [PATCH] finish calculus reference sheet --- spec/calculus-ref.md | 16 +++++++++++++--- spec/calculus-ref.pdf | Bin 173477 -> 182481 bytes 2 files changed, 13 insertions(+), 3 deletions(-) diff --git a/spec/calculus-ref.md b/spec/calculus-ref.md index 88ad371..eba250f 100644 --- a/spec/calculus-ref.md +++ b/spec/calculus-ref.md @@ -4,8 +4,11 @@ geometry: margin=2cm graphics: yes tables: yes author: Andrew Lorimer +classoption: twocolumn + --- + # Spec - Calculus ## Gradients @@ -33,11 +36,13 @@ $$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$ ## Derivatives + + | $f(x)$ | $f^\prime(x)$ | -| ------ | ------------- | +| --- | --- | | $kx^n$ | $knx^{n-1}$ | | $g(x) + h(x)$ | $g^\prime (x) + h^\prime (x)$ | -| $c$ | $0$ | +| $c$ | $0$ | | ${u \over v}$ | ${{v{du \over dx} - u{dv \over dx}} \over v^2}$ | | $uv$ | $u{dv \over dx} + v{du \over dx}$ | | $f \circ g$ | ${dy \over du} \cdot {du \over dx}$ | @@ -50,6 +55,8 @@ $$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$ + + ## Product rule for $y=uv$ $${dy \over dx} = u{dv \over dx} + v{du \over dx}$$ @@ -65,7 +72,7 @@ $$\int f(x) dx = F(x) + c$$ - area enclosed by curves | $f(x)$ | $\int f(x) \cdot dx$ | -| --------------- | ------------------ | +| ----|--- | | $k$ (constant) | $kx + c$ | | $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ | | $a x^{-n}$ | $a \cdot \log_e x + c$ | @@ -92,9 +99,12 @@ $$\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)_{}$$ **acceleration $a$** - change in velocity **speed** - magnitude of velocity + + | | no | | - | -- | | $v=u+at$ | $s$ | | $s=ut + {1 \over 2} at^2$ | $v$ | | $v^2 = u^2 + 2as$ | $t$ | | $s= {1 \over 2}(u+v)t$ | $a$ | + diff --git a/spec/calculus-ref.pdf b/spec/calculus-ref.pdf index d2ede0431dbcba2933be77687bee634f16351467..cb652fa5a2c7da7c1daaa9db85a845efeca61e68 100644 GIT binary patch delta 152447 zcmV)0K+eCV%L>tv3n(Q}L`E$!E;R}z(A34)3N<$%Fd%PYY6?6&3NK7$ZfA68ATu~H zG9Z^R8v-YP?Oa=L8@UyJ&##!i6cW;5$-_H}MS*Rbpo>M(cJtKTJSesj3v0_;`LfBs z?;&TTAxCFsC`(q{qEF6+KIFMy_;J2{_v-g=fLB5>sSH%_K33W=<7H(*<3FkD-DUOT z?1x|GwG^}E1^%#lRyRNXw79-_xPG{szkl~#Lq#xuuXUutt6BoHMmGfBZWfoT<$8`p z?%VZ}SDtwTCM;K$NonzSB(tL3^roe&lXZeK#T`Kw;nhE0eH%3dX#jDU0iK1!S2wHsCO+I>EpIo=o7Ue^^xZa=2{aFM2aGZ8CW6gt z&H3!j+{xMP21!UcySQE7-7nU2aI^d7Up_vpFYZ^jYtr*Q+JZQNRjOY2=+Q7*!qv}&AT-l_cjDg>y9H<5QMg{UK@l}$FtVU&v_S^ht_7N<&cmKcmMJvuT`c+y zLB}$$WY1YbMKcf-ck=y2ZX~#WKe_dcTM!5gbh}f%Ay~tp$+E(* z+)!p5s54Rc2D=%pWvtZ4YAnrEtU}TnE0Y=0!VXJ^rU!B(%uglcNmz6HE|aN$xhcXV zQ&Q)ZXntsh&WYLvnOJB~NGdKWkda})<(PwS9iwiSCzc@J7W>_lM* zIq>a&e$#jkCWu?)hhS7DUq-+9)<71!|SP-dqtkYs+ZVw7bn?%(=8ZS zQ7LS`t<$59?xq?}4$NWrQO3rxOeVvg=PAPkb|Aq^wyo*AxmUBr`qOTIwiJEtx6O}F z`%@<+Gly?FP}tEjTUrqNO{dVx=VG>A5{cAc@1}QZubcMta`_XNYQ4N{{`z4KcJ_sG zcon_0{pBx&9vmin-$f*H(51yLLEKQf6owhCi%s`}?9`3i+1#&J%MBI20JYp5mK@v( zJgn-o(@QW8*D9ve*sQ~U;WCZp!qEoC49~RzW+=mSgz)#Q%m*ErmDB*biD?dbXzQiF z-EM9c*TF<2nh0$lXxK8b56b6F7vug&kU*Jqk>|cK&2AJj;Nh5;1lYbZ(%9-S9Jy>> z5N0ey%ETPvP1^BeU=q}p%sZIc(PlfYQGitP!pMS4l7-*Um2hHz`dzf_g>Bo7=>#F& zaB8t1mV3(QXeZ9()$$L^i$xnG_V>l;Bqj{q3J8tWwzL^4Y;8gHCyT4U*8~OAjzqJl z+dl@Cy427ylGK%N1f5Nuhu?wR#c>hGO4jihfyV~M79)}$t5TuqHl;;n8b6l1j7QQs zrY`T=e&)}6mEqlg{(0N$7>SM{xfty`(D6;2g$?D}EiI0LOu2K<@QAR1F}!op!FpFR zBg|yIo2)wNnQ3)k)#6d(r%(r2VBpktf*CH_KGR`C;OP=JsTsKDZ6nbwYe=p?4;1T` zq?y<19yAebUC+NwFW6=Wt39SA4`CeJ8Q(Fpm-A073o846XLA+oUR{4=)$^1fwxCH} zJV_W2Rb~E0V9)KquHTrEDFoiW-wpoOaddK3>YKcFyl^uyp8>8kNy%6Z|#I z;uPa;OY=ZP9n3^%s{kB-(~jmWtSv?|mUdVh8mzK3L*|M+pIWgL!oE;~{bhdmqvmKwPZOZ-$#`ROB(05JYIn`ROqT}J|wx=jLY!SwJrueH#U zl&x`b_%bD3?9ddN1#pSFP)A9(B_%xOM&U-Ji}u5R<4ANT-^m91=?9(*<0qxG#sq9LpetTF}t(RK?lkK$(gV( zyoZ^ndVY5a`0t5Ck!AP8V(E=GkEBW4ON5P z3#x{P8Z32~C6$nwky%H%po#=6Mgd%=qXk6`e}ij^T5j`JFl&2Lg6mtVUcjX6w8fQO z{YBoaB9p#K7A?rZp~CIbB1{aI9@a*b=x7txj3K8g4`fz5Kdjf1L^)CT1g?-3 z*5pQ?0Olf}f(6MY)gqWT7X1c|)40~={=T`5L&VF6ix%audAMFSfBJa4Y3OK?(czfA zl#T!kDJ1Q5~W6cfTEA-atlb{fSMHkPI`B1|BUv#>3xaQn24b;mf3hzKFH zC+SBR!1gRGC}UvUpthtk1|H~pbMTpq4BVSH^MaHG(kBE~_Ua@oL{h#$2eshgnmXjv z0Dd?CNOsBr(dfkxeH9%}G69f((@ck=Dp}P6{)UQpg;Exiz0CfunGAHq^W?C+(+Z-X z#UDm(4Bsg|L7D?8&OSkgPlqakMf{i~Uqn#bQLqcnFyjvyyM` zBEG-v{AtZ5v z$EpEfKs*59-Oa(9exXx;xx@;?<`0h!)k+bw`CW7Lr+sV*VYIy5UY{1Hxw! 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