From cc2d923d05589b460bbbb58614000c4358462c4d Mon Sep 17 00:00:00 2001 From: eggy Date: Wed, 18 Nov 2020 16:22:52 -0500 Subject: [PATCH] math: cusps --- docs/mhf4u7.md | 8 +++++++- 1 file changed, 7 insertions(+), 1 deletion(-) diff --git a/docs/mhf4u7.md b/docs/mhf4u7.md index d4c6206..200b0c9 100644 --- a/docs/mhf4u7.md +++ b/docs/mhf4u7.md @@ -579,6 +579,8 @@ To find the extrema of a **continuous** function $f(x)$, where $x=a$ is a critic - If the sign is the same on both sides, there is no extrema at $x=a$. - The greatest/least relative/local maximum/minimum is the absolute maximum/minimum. +Alternatively, the second derivative test may be used instead. At the critical point where $x=a$, a positive $f´´(a)$ indicates a **local minimum** while a negative $f´´(a)$ indicates a **local maximum**. If $f´´(x)=0$, the test is inconclusive and the first derivative test must be used. + !!! example The absolute minimum of $f(x)=x^2$ is at $(0,0)$. There is no absolute maximum nor are there any other relative/local maximum/minimum points. @@ -589,13 +591,17 @@ To find the extrema of a **continuous** function $f(x)$, where $x=a$ is a critic ### Concavity !!! definition - A **point of inflection** on a curve is such that $f´´(x)=0 \text{ or DNE}$ and the signs around the point change (e.g., positive to negative). + A **point of inflection** on a curve is such that $f´´(x)=0 \text{ or DNE}$ and the signs of $f´´(x)$ around the point change (e.g., positive to negative). - An interval is **concave up** if it increases from left to right and tangent lines are drawn below the curve, so $f´´(x)>0$. It is shaped like a smile. - An interval is **concave down** if it increases from left to right and tangent lines are drawn **above** the curve, and $f´´(x)<0$. It is shaped like a frown. Changes in concavity only occur at points of inflection. +### Cusps + +A cusp is a point on a continuous function that is not differentiable as the slopes on both sides approach -∞ and ∞. Concavity does not change at a cusp, but they may be considered for local extrema. + ## Resources - [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)