From: Andrew Lorimer Date: Tue, 9 Apr 2019 06:08:16 +0000 (+1000) Subject: Merge branch 'master' of ssh://charles/tank/andrew/school/notes X-Git-Tag: yr12~168 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/9b4ce53d414fedba7ccafc9534dee37f57d74372?hp=d154eec7c3b72f9de6fa03a924917b6b8c172270 Merge branch 'master' of ssh://charles/tank/andrew/school/notes --- diff --git a/spec/calculus.md b/spec/calculus.md index 167a7e7..ddda405 100644 --- a/spec/calculus.md +++ b/spec/calculus.md @@ -1,3 +1,12 @@ +--- +geometry: margin=2cm +columns: 2 +graphics: yes +tables: yes +author: Andrew Lorimer +classoption: twocolumn +--- + # Differential calculus ## Limits @@ -238,8 +247,10 @@ $\int k f(x) dx = k \int f(x) dx$ | $f(x)$ | $\int f(x) \cdot dx$ | | ------------------------------- | ---------------------------- | | $k$ (constant) | $kx + c$ | -| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ | +| $x^n$ | ${x^{n+1} \over {n+1}} + c$ | | $a x^{-n}$ | $a \cdot \log_e x + c$ | +| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ | +| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ | | $e^{kx}$ | ${1 \over k} e^{kx} + c$ | | $e^k$ | $e^kx + c$ | | $\sin kx$ | $-{1 \over k} \cos (kx) + c$ | @@ -247,8 +258,6 @@ $\int k f(x) dx = k \int f(x) dx$ | ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ | | $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)| | $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ | -| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ | -| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ | ### Definite integrals