From 2800efac76831ca7c242568e156cd5757f1026db Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Tue, 2 Apr 2019 14:35:36 +1100 Subject: [PATCH] [spec] add table of graphs for points of inflection --- spec/calculus.md | 7 ++++--- spec/graphics/second-derivatives.png | Bin 0 -> 162533 bytes 2 files changed, 4 insertions(+), 3 deletions(-) create mode 100644 spec/graphics/second-derivatives.png diff --git a/spec/calculus.md b/spec/calculus.md index c9a364e..5c3b3fc 100644 --- a/spec/calculus.md +++ b/spec/calculus.md @@ -194,15 +194,16 @@ $$\therefore y \implies {dy \over dx} \implies {d({dy \over dx}) \over dx} \impl Order of polynomial $n$th derivative decrements each time the derivative is taken -### Maxima and minima +### Points of Inflection -- if $f^\prime (a) = 0$ and $f^{\prime\prime}(a) > 0$, then point $(a, f(a))$ is a local min (curve is concave up) +*Point of inflection* - point of maximum gradient (either +ve or -ve). Occurs where $f^{\prime\prime} = 0$ +- if $f^\prime (a) = 0$ and $f^{\prime\prime}(a) > 0$, then point $(a, f(a))$ is a local min (curve is concave up) - if $f^\prime (a) = 0$ and $f^{\prime\prime} (a) < 0$, then point $(a, f(a))$ is local max (curve is concave down) - if $f^{\prime\prime}(a) = 0$, then point $(a, f(a))$ is a point of inflection - - if also $f^\prime(a)=0$, then it is a stationary point of inflection -*Point of inflection* - point of maximum gradient (either +ve or -ve) +![](graphics/second-derivatives.png) ## Antidifferentiation diff --git a/spec/graphics/second-derivatives.png b/spec/graphics/second-derivatives.png new file mode 100644 index 0000000000000000000000000000000000000000..3323f4036e2dc8d1ed2efd291f764116cd2d4ead GIT binary patch literal 162533 zcmZs?1wd8Z)-DVikWMM-?(Xi+O?Q`cmvl=vD4hb*CDI@z64Ko%Dc$j}{ho8r{r>wM z-O6UIHEYZR3riGuO$` zayg*@9mdqe(6!-;&4rfR8)3KG`_je!`qideS9n1KH|N=XyLLN%JJ$Is>^>f(u%~hg 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