-
Notifications
You must be signed in to change notification settings - Fork 0
/
calculus.html
319 lines (278 loc) · 7.87 KB
/
calculus.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2019-09-07 Sat 11:49 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Calculus Notes</title>
<meta name="generator" content="Org mode">
<meta name="author" content="George Kontsevich">
<meta name="description" content="Notes from studying Clojure"
>
<link rel="stylesheet" type="text/css" href="../web/worg.css" />
<link rel="shortcut icon" href="../web/panda.svg" type="image/x-icon">
</head>
<body>
<div id="org-div-home-and-up">
<a accesskey="h" href=".."> UP </a>
|
<a accesskey="H" href=".."> HOME </a>
</div><div id="content">
<h1 class="title">Calculus Notes</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org5e73d9c">5 Functions</a></li>
<li><a href="#org5846608">The step function</a></li>
<li><a href="#org02f6b18">Function composition</a></li>
<li><a href="#orgd711e10">l'Hospital's Rule</a></li>
<li><a href="#org48a4298">Fundamental Theorem of Calculus</a></li>
<li><a href="#orgfa50876">Mean Value Theorem</a></li>
<li><a href="#org033b56b">Taylor series</a></li>
<li><a href="#org5f41c80">Binomial Theorem</a></li>
<li><a href="#orgc1202de">Euler's equation</a></li>
</ul>
</div>
</div>
<p>
Random notes..
</p>
<div id="outline-container-org5e73d9c" class="outline-2">
<h2 id="org5e73d9c">5 Functions</h2>
<div class="outline-text-2" id="text-org5e73d9c">
<table>
<colgroup>
<col class="org-left">
<col class="org-left">
<col class="org-left">
<col class="org-left">
</colgroup>
<tbody>
<tr>
<td class="org-left"><b>Integral</b></td>
<td class="org-left"><b>Function</b></td>
<td class="org-left"><b>Derivative</b></td>
<td class="org-left"> </td>
</tr>
<tr>
<td class="org-left">x<sup>n+1</sup>/(n+1)</td>
<td class="org-left">x<sup>n</sup></td>
<td class="org-left">nx<sup>n</sup></td>
<td class="org-left"><code><- Nonsense integral for n=-1</code></td>
</tr>
<tr>
<td class="org-left">-cos(x)</td>
<td class="org-left">sin(x)</td>
<td class="org-left">cos(x)</td>
<td class="org-left"> </td>
</tr>
<tr>
<td class="org-left">sin(x)</td>
<td class="org-left">cos(x)</td>
<td class="org-left">sin(x)</td>
<td class="org-left"> </td>
</tr>
<tr>
<td class="org-left">e<sup>cx</sup>/c</td>
<td class="org-left">e<sup>cx</sup></td>
<td class="org-left">ce<sup>cx</sup></td>
<td class="org-left"> </td>
</tr>
<tr>
<td class="org-left">x ln(x)-x</td>
<td class="org-left">ln(x)</td>
<td class="org-left">1/x</td>
<td class="org-left"><code><- The n=-1 case</code></td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org5846608" class="outline-2">
<h2 id="org5846608">The step function</h2>
<div class="outline-text-2" id="text-org5846608">
<pre class="example">
Integral:
Ramp Function:
/
/
___________/
-------------------------->x
^slope=1
Function:
Step Function:
___________
___________|
-------------------------->x
^x=0
Derivative:
Delta Function:
|
|
____________|___________
-------------------------->x
</pre>
<p>
The <i>area</i> under the function should equal to the function above it.
Therefore the area under the <b>delta function</b> is 1
</p>
</div>
</div>
<div id="outline-container-org02f6b18" class="outline-2">
<h2 id="org02f6b18">Function composition</h2>
<div class="outline-text-2" id="text-org02f6b18">
<table>
<colgroup>
<col class="org-left">
<col class="org-left">
<col class="org-left">
</colgroup>
<tbody>
<tr>
<td class="org-left">Function</td>
<td class="org-left">Derivative</td>
<td class="org-left"> </td>
</tr>
<tr>
<td class="org-left">a * f(x) + b g(x)</td>
<td class="org-left">a * df/dx + b * df/dx</td>
<td class="org-left"> </td>
</tr>
<tr>
<td class="org-left">f(x) * g(x)</td>
<td class="org-left">f(x) * dg/dx + g(x) * df/dx</td>
<td class="org-left"><b>Product Rule</b></td>
</tr>
<tr>
<td class="org-left">f(x) / g(x)</td>
<td class="org-left">[g(x) * df/dx - f(x) * dg/dx]/g<sup>2</sup></td>
<td class="org-left"><b>Quotient Rule</b></td>
</tr>
<tr>
<td class="org-left">x = f(y)<sup>-1</sup></td>
<td class="org-left">dx/df = 1 / [dy/dx]</td>
<td class="org-left"> </td>
</tr>
<tr>
<td class="org-left">f(g(x))</td>
<td class="org-left">[df/dy] * [dy/dx] <i><- y = g(x)</i></td>
<td class="org-left"><b>Chain Rule</b></td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-orgd711e10" class="outline-2">
<h2 id="orgd711e10">l'Hospital's Rule</h2>
<div class="outline-text-2" id="text-orgd711e10">
<p>
Where f(x) -> 0 and g(x) -> 0 as x -> <i>a</i> (ie. both function go to zero at some point <i>a</i>)<br>
Then the fraction: <br>
f(x) / g(x) <br>
Goes to:<br>
[df/dx] / [dg/dx]
</p>
</div>
</div>
<div id="outline-container-org48a4298" class="outline-2">
<h2 id="org48a4298">Fundamental Theorem of Calculus</h2>
<div class="outline-text-2" id="text-org48a4298">
<p>
Integral( Derivative( f(<i>x</i>) )) = f(<i>x</i>) <br>
Derivative( Integral( f(<i>x</i>) )) = f(<i>x</i>) <br>
</p>
<p>
As long as they're continuious… (has a <code>max</code> and <code>min</code> and reaches all values in between)
</p>
</div>
</div>
<div id="outline-container-orgfa50876" class="outline-2">
<h2 id="orgfa50876">Mean Value Theorem</h2>
<div class="outline-text-2" id="text-orgfa50876">
<p>
If f(<i>x</i>) has a derivative between a and (ie. a <= x <= b)<br>
Then: (f(b) - f(a)) / (b-a) = [df/dx](c) will be true at some <i>c</i> between <i>a</i> and <i>b</i> <br>
This is "average speed" equation
</p>
</div>
</div>
<div id="outline-container-org033b56b" class="outline-2">
<h2 id="org033b56b">Taylor series</h2>
<div class="outline-text-2" id="text-org033b56b">
<p>
When you know f(<i>a</i>) and want to find a value close by f(<i>x</i>): <br>
f(<i>x</i>) = f(<i>a</i>) + f'(<i>a</i>)(<i>x</i>-<i>a</i>)+ (1/2!)f''(a)(x-a)<sup>2</sup> + … [1/(n+1)!] * f<sup>n+1</sup>(<i>a</i>)(x-a)<sup>n+1</sup>
</p>
<p>
In the traveling example:<br>
For some time <i>a</i> you get the location f(<i>a</i>) <br>
Then you get the speed at that time <i>a</i> and see how much further you'd go by time <i>x</i> and add that to f(<i>a</i>) <br>
Then you add a correction due to the acceleration <br>
etc…
</p>
</div>
</div>
<div id="outline-container-org5f41c80" class="outline-2">
<h2 id="org5f41c80">Binomial Theorem</h2>
<div class="outline-text-2" id="text-org5f41c80">
<table>
<colgroup>
<col class="org-left">
<col class="org-left">
</colgroup>
<tbody>
<tr>
<td class="org-left"><b>Function</b></td>
<td class="org-left"><b>Derivative</b></td>
</tr>
<tr>
<td class="org-left">(1+x)</td>
<td class="org-left">1 + 1x</td>
</tr>
<tr>
<td class="org-left">(1+x)<sup>2</sup></td>
<td class="org-left">1 + 2x +1x<sup>2</sup></td>
</tr>
<tr>
<td class="org-left">(1+x)<sup>3</sup></td>
<td class="org-left">! + 3x + 3x<sup>2</sup> +1x<sup>3</sup></td>
</tr>
<tr>
<td class="org-left">..</td>
<td class="org-left">..</td>
</tr>
</tbody>
</table>
<p>
Whole number powers give you Pascal's Triangle. And notice that each one can be differentiated <i>n</i> times
</p>
<p>
How about if powers aren't whole?:<br>
(1+x)<sup>p</sup> — <i>derivative</i> —> p(1+x)<sup>p-1</sup><br>
These can be differentiated indefinitely
</p>
<p>
We can do the Taylor expansion:<br>
(1+x)<sup>p</sup> = 1 + px + p(p-1)x<sup>2</sup>/2! + ..<br>
And the terms will go on forever
</p>
</div>
</div>
<div id="outline-container-orgc1202de" class="outline-2">
<h2 id="orgc1202de">Euler's equation</h2>
<div class="outline-text-2" id="text-orgc1202de">
<p>
Doing the Taylor's Series for <i>a</i> = 0: <br>
e<sup>x</sup> = e<sup>0</sup> + x + (1/2)x<sup>2</sup> (1/6)x<sup>3</sup> …
</p>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: George Kontsevich</p>
<p class="date">Created: 2019-09-07 Sat 11:49</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
</html>