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updated mathjax, added calc iii notes

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Lucas Rufkahr
2026-03-02 21:12:34 -06:00
parent c3dd0c0dd8
commit 96ef87b663
9 changed files with 261 additions and 15 deletions
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content/calculus/Function of Several Variables.md
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date = '2026-03-02T19:00:00-06:00'
draft = false
title = 'Calculus Notes'
layout = 'chapter'
type = 'book'
tags = 'calculus'
chapterno = 15
+++
Definition:

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+++
date = '2026-03-02T19:00:00-06:00'
draft = false
title = 'Function of Several Variables'
layout = 'chapter'
type = 'book'
tags = 'calculus'
chapterno = 15
+++
## Max/Min Problems
-------------------
Definition:
\[
\text{Suppose } (a,b) \text{ is a point in region} R \text{ on which } f \text{ is defined and there is an open disk centered at } (a,b) \text{.}\\\\
\text{If: } f(x,y) \leq f(a,b) \text{, then } f(a,b) \text{ is a local max.}\\
\text{If: } f(x,y) \geq f(a,b) \text{, then } f(a,b) \text{ is a local min.}
\]
Theorem:
\[
\text{If } f \text{ has a local max or min value at } (a,b) \text{ and the partial derivatives } f_x \text{ and } f_y \text{ exist at } (a,b) \text{, then } f_{x}(a,b) = f_{y}(a,b) = 0 \text{.}
\]
If you are at \((a,b)\) and every where you look, the value next to you is higher, you are at a local minimum position. If everywhere you look, the value is lower, you are at a local maximum position.
Critical points are identified by finding where the partial derivatives of each variable in the multi-variate function equals zero.
Definition:
\[
\text{ A critical point in } f \text{ is located as an interior point } (a,b) \text{ if either:}\\\\
f_{x}(a,b) = f_{y}(a,b) = 0\\
f_{x} \text{ or } f{y} \text{ does not exist at } (a,b)
\]
If you are able to find the critical points of \(f\) then you can use the _Second Partial Derivative Test_ to determine the maximum and minimum values.
Theorem:
\[
\text{If the second partial derivatives of } f \text{ are continuous throughout an open disk centered at } (a,b) \text{. Let } D(x,y) = f_{xx}(x,y)f_{yy}(x,y) - (f_{xy}(x,y))^{2} \text{.}\\\\
\text{If } D(a,b) > 0 \text{ and } f_{xx}(a,b) < 0 \text{, there exists a local max at } (a,b) \text{.}\\
\text{If } D(a,b) > 0 \text{ and } f_{xx}(a,b) < 0 \text{, there exists a local min at } (a,b) \text{.}\\
\text{If } D(a,b) < 0 \text{, there exists a saddle point at }(a,b) \text{.}\\
\text{If } D(a,b) = 0 \text{ the test is inconclusive.}
\]

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[markup.goldmark.extensions.passthrough.delimiters]
block = [['\[', '\]'], ['$$', '$$']]
inline = [['\(', '\)']]
[params]
math = true

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<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
{{ if .Param "math" }}
{{ partialCached "math.html" . }}
{{ end }}
<title></title>
</head>
<body>
<body style="width: 60%; margin: 0 auto;">
<header>{{ partial "header.html" }}</header>
<main>
{{ block "main" . }}

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<script>
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tex: {
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load: ['ui/safe']
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displayAlign: 'left',
displayIndent: '1em'
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