daviddoji/CalculusWithJuliaNotes.jl
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edits, add dark mode

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jverzani
2024-07-31 11:24:53 -04:00
parent f1e7895946
commit c3998bc835
12 changed files with 128 additions and 94 deletions
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quarto/derivatives/first_second_derivatives.qmd
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@@ -162,9 +162,10 @@ This question can be answered by considering the first derivative.
> *The first derivative test*: If $c$ is a critical point for $f(x)$ and *if* $f'(x)$ changes sign at $x=c$, then $f(c)$ will be either a relative maximum or a relative minimum.
>
> * It will be a relative maximum if the derivative changes sign from $+$ to $-$.
> * It will be a relative minimum if the derivative changes sign from $-$ to $+$.
> * If $f'(x)$ does not change sign at $c$, then $f(c)$ is *not* a relative maximum or minimum.
> * $f$ will have a relative maximum at $c$ if the derivative changes sign from $+$ to $-$.
> * $f$ will have a relative minimum at $c$ if the derivative changes sign from $-$ to $+$.
>
> Further, If $f'(x)$ does *not* change sign at $c$, then $f$ will *not* have a relative maximum or minimum at $c$.
@@ -185,7 +186,7 @@ f(x) = exp(-abs(x)) * cos(pi * x)
plotif(f, f', -3, 3)
```
We can see the first derivative test in action: at the peaks and valleys – the relative extrema – the color changes. This is because $f'$ is changing sign as the function changes from increasing to decreasing or vice versa.
We can see the first derivative test in action: at the peaks and valleys – the relative extrema – the highlighting changes. This is because $f'$ is changing sign as the function changes from increasing to decreasing or vice versa.
This function has a critical point at $0$, as can be seen. It corresponds to a point where the derivative does not exist. It is still identified through `find_zeros`, which picks up zeros and in case of discontinuous functions, like `f'`, zero crossings:
@@ -493,8 +494,8 @@ Concave up functions are "opening" up, and often clearly $U$-shaped, though that
> The **second derivative test**: If $c$ is a critical point of $f(x)$ with $f''(c)$ existing in a neighborhood of $c$, then
>
> * The value $f(c)$ will be a relative maximum if $f''(c) > 0$,
> * The value $f(c)$ will be a relative minimum if $f''(c) < 0$, and
> * $f$ will have a relative maximum at the critical point $c$ if $f''(c) > 0$,
> * $f$ will have a relative minimum at the critical point $c$ if $f''(c) < 0$, and
> * *if* $f''(c) = 0$ the test is *inconclusive*.