-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathweek11.html
382 lines (302 loc) · 36.2 KB
/
week11.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
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
<!doctype html>
<head>
<meta http-equiv="Cache-Control" content="no-cache, no-store, must-revalidate" />
<meta http-equiv="Pragma" content="no-cache" />
<meta http-equiv="Expires" content="0" />
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.css" integrity="sha384-Um5gpz1odJg5Z4HAmzPtgZKdTBHZdw8S29IecapCSB31ligYPhHQZMIlWLYQGVoc" crossorigin="anonymous">
<script defer src="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.js" integrity="sha384-YNHdsYkH6gMx9y3mRkmcJ2mFUjTd0qNQQvY9VYZgQd7DcN7env35GzlmFaZ23JGp" crossorigin="anonymous"></script>
<script>
document.addEventListener("DOMContentLoaded", function(){
var tex = document.querySelectorAll('script[type^="math/tex"]');
for(var i = 0; i < tex.length; ++i)
{
var display = tex[i].getAttribute('type').indexOf('mode=display') > -1;
var math = tex[i].previousSibling;
math.className = 'katex-render';
var content = tex[i].textContent;
katex.render(content, math, {
/* output: html, */
displayMode: display,
throwOnError: false,
trust: true
});
tex[i].parentNode.removeChild(tex[i]);
}
});
</script>
<script src="js/sidebar.js"></script>
<script defer src="js/search.js"></script>
<script async defer src="https://hypothes.is/embed.js"></script>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.1.0/jquery.min.js"></script>
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css">
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js"></script>
<script src="js/openclosebuttons.js"></script>
<link rel="stylesheet" type="text/css" href="css/iLaTeXen.css">
<script defer src="js/quiz.js"></script>
<script defer src="js/orderquiz.js"></script>
<script defer src="js/formatquiz.js"></script>
<link rel="stylesheet" type="text/css" href="https://tikzjax.com/v1/fonts.css">
<script src="https://tikzjax.com/v1/tikzjax.js"></script>
<script defer src="js/bubble.js"></script>
<script src="https://sagecell.sagemath.org/static/embedded_sagecell.js"></script>
<script>
window.MathJax = 1 /* Disable MathJax */
sagecell.makeSagecell({
inputLocation: ".sage",
autoeval: false,
evalButtonText: "Start",
linked: true,
languages : ["sage", "python", "macaulay2", "r"],
hide : ["editor", "language", "permalink"]
});
sagecell.makeSagecell({
inputLocation: ".sageM2",
autoeval: false,
evalButtonText: "Start",
linked: true,
languages : ["macaulay2", "sage", "python", "r"],
hide : ["editor", "language", "permalink"]
});
sagecell.makeSagecell({
inputLocation: ".sagepython",
autoeval: false,
evalButtonText: "Start",
linked: true,
languages : ["python", "sage", "macaulay2", "r"],
hide : ["editor", "language", "permalink"]
});
sagecell.makeSagecell({
inputLocation: ".sageR",
autoeval: false,
quietMode: true,
replaceOutput: true,
evalButtonText: "Start",
linked: true,
languages : ["r", "python", "sage", "macaulay2"],
hide : ["editor", "language", "permalink"]
});
</script>
</head>
<body><div class="sidenav normalwidth"><button style="border:none; background-color: Transparent;" onclick="showhidemenu()" title="Toggle toc"><span style="font-size: 30px;">☰</span></button><button class="openbs" title="Open buttons">+</button><button class="closebs" title="Close buttons">-</button><ul class="leftmenu" style="display: none;"><li><a class="kap" href="introduction.html#159ca076-dc4e-4bb9-83f5-a5ddb8ff2299"><b>1</b> Introduction</a></li><a href="#kap1" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap1" class="collapse"><li><a href="introduction.html#ae495219-240d-4cf1-9a9b-29918804ef56">1.1 Readings</a></li><li><a href="introduction.html#26671802-fb3b-4273-8e82-12d04b3b35a4">1.2 Notes</a></li><li><a href="introduction.html#3fc447d4-9ac7-439e-afc9-6c5ad35b2e24">1.3 Exercises</a></li></ul><li><a class="kap" href="week1.html#3a6133ac-4a7b-47d1-9ea5-47ca81f168f2"><b>2</b> Course week 1</a></li><a href="#kap2" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap2" class="collapse"><li><a href="week1.html#cbb4e3bf-894c-49ad-90c5-58d0ed1e7cc3">2.1 Readings</a></li><li><a href="week1.html#d018418e-a17c-4b70-b799-f34e0298333b">2.2 Notes</a></li><li><a href="week1.html#e010e5dd-5121-4b83-8e89-d32234dee55e">2.3 Exercises</a></li></ul><li><a class="kap" href="week2.html#5c85fc37-321e-4707-9e48-bdb3f9e4a5bd"><b>3</b> Course week 2</a></li><a href="#kap3" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap3" class="collapse"><li><a href="week2.html#8ec6af3d-7c28-4353-89f8-1992db17e7e8">3.1 Readings</a></li><li><a href="week2.html#b18e048c-25be-4300-9ed0-4407e46d540c">3.2 Notes</a></li><li><a href="week2.html#cca75863-ec45-459f-a0f4-3a965a60422e">3.3 Exercises</a></li><li><a href="week2.html#ee2537cc-717b-4463-8904-87085c60123a">3.4 Feedback</a></li></ul><li><a class="kap" href="week3.html#9beeae3e-d65c-4020-b00a-c526ceadf899"><b>4</b> Course week 3</a></li><a href="#kap4" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap4" class="collapse"><li><a href="week3.html#6bef6f20-56d9-4f57-803f-0ba61b2540ac">4.1 Readings</a></li><li><a href="week3.html#7a92877c-98c7-4bdd-9e2f-762cb37563e8">4.2 Notes</a></li><li><a href="week3.html#b4fcf09c-686b-41fc-8a20-854b9c21bf31">4.3 Exercises</a></li><li><a href="week3.html#fe6dc55b-aaac-49aa-9a44-55212b41273d">4.4 Feedback</a></li></ul><li><a class="kap" href="week4.html#2402c1cb-8568-4fb6-bdbe-b6485e20d1ca"><b>5</b> Course week 4</a></li><a href="#kap5" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap5" class="collapse"><li><a href="week4.html#1e2ed3d8-2100-4005-899d-ebde86500961">5.1 Readings</a></li><li><a href="week4.html#34298628-e47f-4a13-a2e8-ee6d069fc8b5">5.2 Notes</a></li><li><a href="week4.html#23a8a6e0-68cc-464c-a101-71cdfbbe4353">5.3 Problems</a></li><li><a href="week4.html#497d6801-a7db-445d-b745-3c9c166483b5">5.4 Feedback</a></li></ul><li><a class="kap" href="week5.html#881f0ad5-1f5b-406f-b1f3-cf97397cdfe3"><b>6</b> Course week 5</a></li><a href="#kap6" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap6" class="collapse"><li><a href="week5.html#f420a408-715e-462b-b2c1-59a979e8b14e">6.1 Readings</a></li><li><a href="week5.html#718a53d7-d8cf-4f5e-8ecf-6561f6f394b8">6.2 Notes</a></li><li><a href="week5.html#08659956-822b-4a08-9e5f-f727bd863256">6.3 Problems</a></li><li><a href="week5.html#74ef3f61-c1c2-42f7-b3c5-dbf69578da08">6.4 Feedback</a></li></ul><li><a class="kap" href="week6.html#b028fe91-09aa-484e-af70-e887bdccbf90"><b>7</b> Course week 6</a></li><a href="#kap7" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap7" class="collapse"><li><a href="week6.html#5f5a6489-a309-4ba4-9cc2-e37611e8e350">7.1 Readings</a></li><li><a href="week6.html#bebc3711-369c-44c2-a9a1-23c8bc9900b0">7.2 Notes</a></li><li><a href="week6.html#2619640f-84f5-4826-8212-f383e4451cb3">7.3 Problems</a></li><li><a href="week6.html#8626687f-527d-42cc-94eb-5f19d20ba029">7.4 Feedback</a></li></ul><li><a class="kap" href="week7.html#c4dc25ad-24a4-45e7-b4c5-e59016fd3dcc"><b>8</b> Course week 7</a></li><a href="#kap8" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap8" class="collapse"><li><a href="week7.html#ee3b5046-f23c-41e8-beb7-08e80a78ebca">8.1 Readings</a></li><li><a href="week7.html#98cafeeb-e279-4a89-a571-d5fc17e6824b">8.2 Notes</a></li><li><a href="week7.html#fdd931f4-395e-40ca-81b8-c3186c52d21d">8.3 Problems</a></li><li><a href="week7.html#4de6c567-85d9-4936-92e4-4f323d10a16b">8.4 Feedback</a></li></ul><li><a class="kap" href="week8.html#a6129057-0dca-47e2-a908-acd3fc725bca"><b>9</b> Course week 8</a></li><a href="#kap9" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap9" class="collapse"><li><a href="week8.html#91a7f924-2cc3-46e0-99d5-84b72cd25982">9.1 Readings</a></li><li><a href="week8.html#2227f711-3610-4e0e-b835-539a1c67559b">9.2 Notes</a></li><li><a href="week8.html#a580698f-3a0d-489f-a786-09490c52da6a">9.3 Problems</a></li><li><a href="week8.html#f636cddc-14ea-46dc-921a-fc8de482f125">9.4 Feedback</a></li></ul><li><a class="kap" href="week9.html#1673477a-155b-4ee5-8c3c-524c4da8f0da"><b>10</b> Course week 9</a></li><a href="#kap10" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap10" class="collapse"><li><a href="week9.html#8d4c9c5d-c542-4cd9-8401-e1a52df4286f">10.1 Readings</a></li><li><a href="week9.html#4dec455a-332a-4ae1-99a8-9c55b704483d">10.2 Notes</a></li><li><a href="week9.html#af8ac1d5-5cc3-4c7d-8eeb-bd6086a2beb9">10.3 Problems</a></li><li><a href="week9.html#9fb03b93-76bb-4df9-ba3b-3076a33a3c84">10.4 Feedback</a></li></ul><li><a class="kap" href="week10.html#9a44ce9a-039b-43ea-bab2-a8ca6d6ad271"><b>11</b> Course week 10</a></li><a href="#kap11" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap11" class="collapse"><li><a href="week10.html#0a130e84-47b6-42a1-b4a0-c045425900f4">11.1 Readings</a></li><li><a href="week10.html#e0c9a4d0-1c3b-4411-9a42-5eac771e0265">11.2 Notes</a></li><li><a href="week10.html#13ab74d2-e187-4b74-a77c-c9599ae7c0ed">11.3 Problems</a></li><li><a href="week10.html#7b803349-1c7b-40d6-9855-33a3e7b15c17">11.4 Feedback</a></li></ul><li><a class="kap" href="week11.html#bad3f0c8-0554-443d-bd04-1392582ea095"><b>12</b> Course week 11</a></li><a href="#kap12" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap12" class="collapse"><li><a href="week11.html#8a687e45-dd04-4a19-a682-b8573340d40e">12.1 Readings</a></li><li><a href="week11.html#03a71c8c-2af2-4746-bbb6-8efe3d3e5e6b">12.2 Notes</a></li><li><a href="week11.html#da7ad12d-8b44-4cfe-a32f-9f8954d6f1bb">12.3 Problems</a></li><li><a href="week11.html#f729eb4e-feb9-4840-a28e-4524cefeda12">12.4 Feedback</a></li></ul><li><a class="kap" href="week12.html#4edeeb01-8694-4c0d-bab3-cea029123630"><b>13</b> Course week 12</a></li><a href="#kap13" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap13" class="collapse"><li><a href="week12.html#3a9c26cd-489b-4a83-84e0-b7ce17925a99">13.1 Readings</a></li><li><a href="week12.html#ae4d4c1f-1399-464e-af22-d908c76bb143">13.2 Notes</a></li><li><a href="week12.html#caf55b0e-7ea3-4ccd-84fb-0c2dfaa25472">13.3 Problems</a></li><li><a href="week12.html#ac0eb5fa-eb33-436e-95b9-d3bf33460ad2">13.4 Feedback</a></li></ul><li><a class="kap" href="week13.html#f0e1b4c2-d02a-4740-bc27-8b653fedf832"><b>14</b> Course week 13</a></li><a href="#kap14" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap14" class="collapse"><li><a href="week13.html#36ebe4f7-f0c7-46e4-8404-b0bb54f694c9">14.1 Readings</a></li><li><a href="week13.html#b6ef6597-50c4-47d2-a2d0-15334207b7d0">14.2 Notes</a></li><li><a href="week13.html#7b891666-2d5c-4f9a-80f9-adfff76ce1f4">14.3 Problems</a></li><li><a href="week13.html#988cdc9c-6611-42fc-b4eb-370b70fb44c1">14.4 Feedback</a></li></ul><li><a class="kap" href="week14.html#96e609e7-8f1b-4f46-8653-af953c4e9823"><b>15</b> Course week 14</a></li><a href="#kap15" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap15" class="collapse"><li><a href="week14.html#90ef85b9-1287-4e2c-9a13-5b8f4bca1584">15.1 Readings</a></li><li><a href="week14.html#3350aecb-60a7-4229-97a1-c063c9336eeb">15.2 Notes</a></li><li><a href="week14.html#aeadc6cf-2833-4923-b0f5-14a9ab76ce1e">15.3 Problems</a></li></ul></ul></div><div class="main normalmargin"><div style="margin-top:20px"></div><div style="margin-top:20px"></div><h1 id="bad3f0c8-0554-443d-bd04-1392582ea095">12<span style="float:right;">Course week 11</span></h1><div style="margin-top:20px"></div><span id="sec12.1"></span><h2 id="8a687e45-dd04-4a19-a682-b8573340d40e">12.1 Readings</h2>
Read sections 10.5-10.6 in the textbook.<div style="margin-top:20px"></div><span id="sec12.2"></span><h2 id="03a71c8c-2af2-4746-bbb6-8efe3d3e5e6b">12.2 Notes</h2>
<b>Integration of a sum or difference of two functions</b>
<div class="math"></div><script type="math/tex; mode=display">
\int f(x) \pm g(x) \, {\mathrm{d}}x = \int f(x) \pm \int g(x) \, {\mathrm{d}}x
</script><div style="margin-top:20px"></div><b>Integration by parts (indefinite integral)</b>
<div class="math"></div><script type="math/tex; mode=display">
\int f(x)g'(x) \, {\mathrm{d}}x = f(x)g(x) - \int f'(x)g(x) \, {\mathrm{d}}x
</script><div style="margin-top:20px"></div><b>Integration by parts (definite integral)</b>
<div class="math"></div><script type="math/tex; mode=display">
\int_a^b f(x)g'(x) \, {\mathrm{d}}x = \bigg|_a^b f(x)g(x) - \int_a^b f'(x)g(x) \, {\mathrm{d}}x
</script>
<b>Integration by substitution (indefinite integral)</b><div style="margin-top:20px"></div>Let <span class="math"></span><script type="math/tex">u=g(x)</script>, then:
<div class="math"></div><script type="math/tex; mode=display">
\int f(g(x)) g'(x) \, {\mathrm{d}}x = \int f(u) \, {\mathrm{d}}u
</script>
<b>Integration by substitution (definite integral)</b><div style="margin-top:20px"></div>Let <span class="math"></span><script type="math/tex">u=g(x)</script>, then:
<div class="math"></div><script type="math/tex; mode=display">
\int_a^b f(g(x)) g'(x) \, {\mathrm{d}}x = \int_{g(a)}^{g(b)} f(u) \, {\mathrm{d}}u
</script><div style="margin-top:20px"></div><span id="sec12.3"></span><h2 id="da7ad12d-8b44-4cfe-a32f-9f8954d6f1bb">12.3 Problems</h2><div style="margin-top:20px"></div><span id="env12.1"></span><div class="exercise" data-count="12.1">
<div id="quiz1" style="width:100%;"><div class="row quizquestion">
Below are some potential steps to apply in integration by parts for an indefinite integral. Arrange
them in order by dragging and dropping from the left to the right to give a correct recipe for integration by parts for the integral
<div class="math"></div><script type="math/tex; mode=display">
\int f(x) g'(x) \, \mathrm{d} x
</script><div style="margin-top:20px"></div></div><div class="row"><div class="col-xs-5"><div class="row"><div class="orderquizanswer" id="dq1_left1" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"> Check that the new integrand is simpler than the original.<div style="margin-top:20px"></div></div><div class="orderquizanswer" id="dq1_left2" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"> Compute <span class="math"></span><script type="math/tex">\mathrm{d} u = g'(x) \mathrm{d} x</script>.<div style="margin-top:20px"></div></div><div class="orderquizanswer" id="dq1_left3" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"> Identify the inner function in a composite function as <span class="math"></span><script type="math/tex">u</script>.<div style="margin-top:20px"></div>
</div><div class="orderquizanswer" id="dq1_left4" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"> Identify <span class="math"></span><script type="math/tex">f(x)</script> as a function whose indefinite integral is known.<div style="margin-top:20px"></div>
</div><div class="orderquizanswer" id="dq1_left5" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"> Identify <span class="math"></span><script type="math/tex">g'(x)</script> as a function whose indefinite integral is known.<div style="margin-top:20px"></div></div><div class="orderquizanswer" id="dq1_left6" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"> The first term is <span class="math"></span><script type="math/tex">f(x) g(x)</script>, where <span class="math"></span><script type="math/tex">g'(x)</script> has been integrated.<div style="margin-top:20px"></div>
</div><div class="orderquizanswer" id="dq1_left7" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"> The second term is <span class="math"></span><script type="math/tex">-\displaystyle\int f'(x) g(x) \, \mathrm{d} x</script>.<div style="margin-top:20px"></div></div><div class="orderquizanswer" id="dq1_left8" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"> Identify <span class="math"></span><script type="math/tex">f(x)</script> as a function whose derivative can be computed.<div style="margin-top:20px"></div></div><div class="orderquizanswer" id="dq1_left9" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"> Solve the integral <span class="math"></span><script type="math/tex">-\displaystyle\int f'(x) g(x) \, \mathrm{d} x</script>.<div style="margin-top:20px"></div></div></div></div><div class="col-xs-5"><div class="row"><div class="orderquizanswer empty" id="dq1_right1" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"></div><div class="orderquizanswer empty" id="dq1_right2" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"></div><div class="orderquizanswer empty" id="dq1_right3" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"></div><div class="orderquizanswer empty" id="dq1_right4" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"></div><div class="orderquizanswer empty" id="dq1_right5" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"></div><div class="orderquizanswer empty" id="dq1_right6" draggable="true" ondragstart="orderQuizStartDrag(1, event)" ondragover="orderQuizAllowDrop(1, event)" ondrop="orderQuizDrop(1, event)"></div></div></div></div><span class="orderquizresult" id="dq1_result"><span class="orderquizfeedback hidden" datav="(or (is 586719) (is 856719))"><b class="correct">Correct!</b>
<span style="color:green;"><b>Correct!</b></span><div style="margin-top:20px"></div></span><span class="orderquizfeedback hidden" datav="(or (contains 2) (contains 3))"><b class="wrong">Wrong.</b>
<span style="color:red;"><b>Wrong. Are you thinking of integration by substitution?</b></span><div style="margin-top:20px"></div></span><span class="orderquizfeedback hidden" datav="default"><b class="wrong">Wrong.</b>
<span style="color:red;"><b>Wrong. Try again!</b></span>
</span></span></div>
</div><div style="margin-top:20px"></div><span id="env12.2"></span><div class="exercise" data-count="12.2">
Use integration by parts to evaluate the following indefinite integrals:
<ol class="number"><li id="ite12.1"> <span class="math"></span><script type="math/tex">\displaystyle\int x \ln(x)\, {\mathrm{d}}x </script>
</li><li id="ite12.2"> <span class="math"></span><script type="math/tex">\displaystyle\int 4x e^{4x}\, {\mathrm{d}}x </script>
</li><li id="ite12.3"> <span class="math"></span><script type="math/tex">\displaystyle\int \frac{4x+2}{e^x}\, {\mathrm{d}}x </script>
</li></ol>
</div><div style="margin-top:20px"></div><span id="env12.3"></span><div class="exercise" data-count="12.3">
Use integration by parts to evaluate the following definite integrals:
<ol class="number"><li id="ite12.4"> <span class="math"></span><script type="math/tex">\displaystyle\int_{-1}^1 x \ln(x+2)\, {\mathrm{d}}x </script>
</li><li id="ite12.5"> <span class="math"></span><script type="math/tex">\displaystyle\int_0^1 x^2e^x\, {\mathrm{d}}x </script>
</li><li id="ite12.6"> <span class="math"></span><script type="math/tex">\displaystyle\int_0^2 (x-2)e^{x/2}\, {\mathrm{d}}x </script>
</li></ol>
</div><div style="margin-top:20px"></div><span id="env12.4"></span><div class="exercise" data-count="12.4">
Last week you found the following definite integral:
<div class="math"></div><script type="math/tex; mode=display">
\int_{-3}^{3}(x-1)^2\, {\mathrm{d}}x
</script>
This time find the integral by using integration by substitution.
</div><div style="margin-top:20px"></div><span id="env12.5"></span><div class="exercise" data-count="12.5">
Use integration by substitution to evaluate the following indefinite integral:
<div class="math"></div><script type="math/tex; mode=display">
\int 2x(x^2+1)^2\, {\mathrm{d}}x
</script>
</div><div style="margin-top:20px"></div><span id="env12.6"></span><div class="exercise" data-count="12.6">
Use integration by substitution to evaluate the following definite integral:
<div class="math"></div><script type="math/tex; mode=display">
\int_5^8\frac{x}{x-4}\, {\mathrm{d}}x
</script>
</div><div style="margin-top:20px"></div><span id="env12.7"></span><div class="exercise" data-count="12.7">
(Previous exam problem)<div style="margin-top:20px"></div>An integral is given by:
<div class="math"></div><script type="math/tex; mode=display">
\int_{0}^{1} x(1+x)^3\, {\mathrm{d}}x
</script>
<ol class="number"><li id="ite12.7"> Calculate the integral.
</li></ol>
</div><div style="margin-top:20px"></div><span id="env12.8"></span><div class="exercise" data-count="12.8">
(Previous exam problem)<div style="margin-top:20px"></div>Calculate the integral:
<div class="math"></div><script type="math/tex; mode=display">
\int x\sqrt{1+x^2}\, {\mathrm{d}}x
</script>
</div><div style="margin-top:20px"></div><span id="env12.9"></span><div class="exercise" data-count="12.9">
(Previous exam problem)<div style="margin-top:20px"></div>Calculate the integral:
<div class="math"></div><script type="math/tex; mode=display">
\int_0^1 \frac{e^x}{(x+1)^{-2}}\, {\mathrm{d}}x
</script>
</div><div style="margin-top:20px"></div><span id="env12.10"></span><div class="exercise" data-count="12.10">
(Previous exam problem)<div style="margin-top:20px"></div><ol class="number"><li id="ite12.8"> Evaluate the following integral:
<span class="math"></span><script type="math/tex">\displaystyle \int_0^1 (e^x+e^{-x})^2\, {\mathrm{d}}x </script>
</li><li id="ite12.9"> Evaluate the following integral:
<span class="math"></span><script type="math/tex">\displaystyle \int \frac{w}{(1-w)^3}\, {\mathrm{d}}w </script>
</li></ol>
</div><div style="margin-top:20px"></div><span id="env12.11"></span><div class="exercise" data-count="12.11">
This exercise is a little longer than usual, so click the button below to open it.<div style="margin-top:20px"></div><a href="#58c04942-7776-471c-b743-a836c3636ef0" class ="btn btn-default" data-toggle="collapse">Open/Close</a><div id=58c04942-7776-471c-b743-a836c3636ef0 class="collapse">
In 2022 the Danish broadcaster, DR, featured an <a href="https://www.dr.dk/nyheder/indland/ferievaner-fra-tiden-med-corona-bider-sig-fast-ny-vandrerute-paa-djursland-skal" target="_blank">article</a>
with the news that the Danes' vacation habits had changed as a consequence of the covid-19 pandemic,
and that, according to numbers from Statistics Denmark, there had been 27 % fewer bookings of vacation homes
than the corresponding period in 2019 (before the pandemic). This is illustrated in the figure below.
<div style="margin-top:20px"></div><center>
<iframe src="./img/week11/p1.html" title="plot" width=900 height=600 style="border:none;" ></iframe><div style="margin-top:20px"></div></center><div style="margin-top:20px"></div>An economist who is interested in modelling the market for vacation houses, constructs
a function <span class="math"></span><script type="math/tex">f(t)</script> that seems to describe the number of booked house-weeks in 2019.
Here, <span class="math"></span><script type="math/tex">t</script> is time as a continuous variable that can assume values in the interval <span class="math"></span><script type="math/tex">[0, 12]</script>,
where the value 0 corresponds the beginning of the year and the value 12 corresponds to the
end of the year. Below is shown a plot of this function along with the data points for 2019
from the figure above.<div style="margin-top:20px"></div><center>
<div class="centerimg"><img src="img/week11/booked_2019.png" width=50%></div>
</center><div style="margin-top:20px"></div><span id="{labelname}"></span><div id="quiz2" style="width:100%;"><div class="row quizquestion">
<b>a)</b> How can the total number of booked house-weeks from the end of May to the end of December be expressed in terms of <span class="math"></span><script type="math/tex">f(t)</script>?
</div><div class="row quizboxes"><div class="col-xs-5 quizanswer" datav="F">
<div class="math"></div><script type="math/tex; mode=display">\int_{0}^{12} f(t) \, \mathrm{d}t</script><div style="margin-top:20px"></div></div><div class="col-xs-5 quizanswer" datav="F">
<div class="math"></div><script type="math/tex; mode=display">f(12) - f(5)</script><div style="margin-top:20px"></div></div><div class="col-xs-5 quizanswer" datav="T">
<div class="math"></div><script type="math/tex; mode=display">\int_{5}^{12} f(t) \, \mathrm{d}t</script><div style="margin-top:20px"></div></div><div class="col-xs-5 quizanswer" datav="F">
<div class="math"></div><script type="math/tex; mode=display">\lim_{b\to\infty}\int_{5}^{b} f(t) \, \mathrm{d}t</script><div style="margin-top:20px"></div></div></div></div><button id="quizbutton1" class = "btn btn-default">Check</button><script>$("#quizbutton1").click(function(){quizshowanswers(quiz2, quizbutton1);});</script><div style="margin-top:20px"></div>Below is a plot of the function <span class="math"></span><script type="math/tex">g(t)</script> which describes the number of booked house-weeks in 2022.<div style="margin-top:20px"></div><center>
<div class="centerimg"><img src="img/week11/booked_2022.png" width=50%></div>
</center><div style="margin-top:20px"></div><b>b)</b> Create an expression for the total number of booked house-weeks in the period from May to December 2022 minus the corresponding number for 2019 using the functions <span class="math"></span><script type="math/tex">f(t)</script> and <span class="math"></span><script type="math/tex">g(t)</script>. Explain why it corresponds to the coloured area in the figure below.<div style="margin-top:20px"></div><center>
<div class="centerimg"><img src="img/week11/booked_diff.png" width=50%></div>
</center>'<div style="margin-top:20px"></div><b>c)</b> Formulate the following statement mathematically by using definite integrals and the functions <span class="math"></span><script type="math/tex">f(t)</script> and <span class="math"></span><script type="math/tex">g(t)</script>:
<i>From May to December 2022 there were 27 percent fewer bookings than for the corresponding period in 2019</i>.<div style="margin-top:20px"></div>
<b>d)</b> <em>Voluntary extra question</em>. Try to construct a function that describes the observed data from 2019. An initial (but bad) guess has been given,
and you can change the expression for <span class="math"></span><script type="math/tex">f(t)</script> in the input form below. First you have to press Start in order to start the interactive plot.
What is the best function you can find? You can use powers (type <tt>t^2</tt> for <span class="math"></span><script type="math/tex">t^2</script>), exponentials (type <tt>exp(t)</tt> for <span class="math"></span><script type="math/tex">e^t</script>), logarithms (type <tt>log(t)</tt> for <span class="math"></span><script type="math/tex">\ln t</script>) etc. Remember to use an asterisk (<tt>*</tt>) for multiplication.
To help you find the best function the sum of squared errors (SSE) are printed, and the goal is to find the smallest possible value of SSE.<div style="margin-top:20px"></div>If you want to see the data behind the plot, click here:
<a href="#b325dd2d-3995-4263-bc97-78ca4b446c40" class ="btn btn-default" data-toggle="collapse">Data</a><div id=b325dd2d-3995-4263-bc97-78ca4b446c40 class="collapse">
<link href="https://unpkg.com/[email protected]/dist/bootstrap-table.min.css" rel="stylesheet">
<script src="https://cdn.jsdelivr.net/npm/[email protected]/tableExport.min.js"></script>
<script src="https://cdn.jsdelivr.net/npm/[email protected]/libs/jsPDF/jspdf.min.js"></script>
<script src="https://cdn.jsdelivr.net/npm/[email protected]/libs/jsPDF-AutoTable/jspdf.plugin.autotable.js"></script>
<script src="https://unpkg.com/[email protected]/dist/bootstrap-table.min.js"></script>
<script src="https://unpkg.com/[email protected]/dist/extensions/export/bootstrap-table-export.min.js"></script>
<style>
#toolbar {
margin: 0;
}
</style>
<div id="toolbar" class="select">
<select class="form-control">
<option value="">Export Basic</option>
<option value="all">Export All</option>
<option value="selected">Export Selected</option>
</select>
</div>
</table>
<table
id="table"
data-show-export="true"
data-search="true"
data-pagination="true"
data-side-pagination="client"
data-url="./tabledata/booked_table.json">
<thead>
<tr>
<th data-field="year" data-sortable="true">Year</th>
<th data-field="month" data-sortable="false">Month</th>
<th data-field="booked" data-sortable="false">Booked weeks</th>
</tr>
</thead>
</table>
<script>
var $table = $('#table')
$(function() {
$('#toolbar').find('select').change(function () {
$table.bootstrapTable('destroy').bootstrapTable({
exportDataType: $(this).val(),
exportTypes: ['json', 'xml', 'csv', 'txt', 'sql', 'excel', 'pdf'],
columns: [
{
field: 'state',
checkbox: true,
visible: $(this).val() === 'selected'
},
{
field: 'year',
title: 'Year'
}, {
field: 'month',
title: 'Month'
}, {
field: 'booked',
title: 'Booked weeks'
}
]
})
}).trigger('change')
})
</script>
</div><div style="margin-top:20px"></div><p><div style="float:right;"><a href="#0f02f8ff-ccea-44ae-8e9b-caf95936930f" class="btn btn-default"data-toggle="collapse"><span class="glyphicon glyphicon-eye-open"></span><span class="glyphicon glyphicon-eye-open"></span></a></div><div style="clear: both;"></div></p><div id="0f02f8ff-ccea-44ae-8e9b-caf95936930f" class="collapse">
The sum of squared errors (SSE) is defined as
<div class="math"></div><script type="math/tex; mode=display">\mathrm{SSE} = \sum_{t = 5}^{12} \left(y_t - f(t)\right)^2,</script>
where <span class="math"></span><script type="math/tex">y_t</script> is the actual number of booked house-weeks in month <span class="math"></span><script type="math/tex">t</script> and <span class="math"></span><script type="math/tex">f(t)</script> is the number predicted by the model function <span class="math"></span><script type="math/tex">f(t)</script>.<div style="margin-top:20px"></div>The function <span class="math"></span><script type="math/tex">f(t)</script> found by the economist gives <span class="math"></span><script type="math/tex">\text{SSE} = 2.58\cdot 10^6</script>.
</div>
<div class=sage><script type="text/x-sage">
t = var('t')
booked_2019 =[(5, 9596), (6, 11141), (7, 28172), (8, 8968), (9, 3316), (10, 1696), (11, 505), (12, 447)]
def frexp10(x):
expo = int(math.floor(math.log10(abs(x))))
return x / 10**expo, expo
@interact
def booked_function(expression=input_box('25000 - 2000 * t', 'f(t) =', str)):
if expression:
try:
SSE = n(sum([(SR(expression)(t = x) - y)**2 for x, y in booked_2019]))
SSE_string = 'SSE\u0024={0:.2f}\\cdot 10^{{{1}}}\u0024'.format(*frexp10(SSE))
# SSE_string = 'SSE = {:.2g}'.format(SSE)
p1 = plot(SR(expression), (t, 5, 12))
p2 = point(booked_2019)
p3 = text(SSE_string, (9,2.5e4), horizontal_alignment = 'left', fontsize = 14)
(p1 + p2 + p3).show()
pretty_print(html('Value you can copy: SSE = {}'.format(SSE)))
except TypeError:
print("There is a problem with your function expression.")
else:
print("Make sure your expression can be plotted.")
</script></div><div style="margin-top:20px"></div>To participate in the draw for a chocolate bar, copy your expression for <span class="math"></span><script type="math/tex">f(t)</script> and your value of SSE
and insert them in this <a href="https://survey.au.dk/LinkCollector?key=TY3NDKMXS6C6" target="_blank">form</a>.<div style="margin-top:20px"></div><p><div style="float:right;"><a href="#61d761dd-e1a7-480f-a6ab-57fb9edb91d4" class="btn btn-default"data-toggle="collapse"><span class="glyphicon glyphicon-eye-open"></span><span class="glyphicon glyphicon-eye-open"></span></a></div><div style="clear: both;"></div></p><div id="61d761dd-e1a7-480f-a6ab-57fb9edb91d4" class="collapse">
There are, of course, mathematical methods to find the function of a particular type that best describes data.
One well-known example is linear regression which finds the best line (linear function) for a data set. You will
learn much more about regression in the course <em>Quantitative Methods</em> in your second semester.<div style="margin-top:20px"></div>One has to take care not to succumb to over fitting, that is tailoring the function too much to the data at
hand. See the figure below from <a href="xkcd" target="_blank">https://xkcd.com/2048</a>.
<center>
<div class="centerimg"><img src="img/week11/curve_fitting.png" ></div>
</center>
</div>
</div>
</div><div style="margin-top:20px"></div><span id="sec12.4"></span><h2 id="f729eb4e-feb9-4840-a28e-4524cefeda12">12.4 Feedback</h2>
We would like to know which exercises helped you the most with <em>understanding</em> the mathematics and with your mathematical <em>skills</em>.<div style="margin-top:20px"></div>Examples of mathematical <em>understanding</em> might be an aha! experience of insight, understanding a concept by seeing it in different ways, or by thinking about why an answer was correct or incorrect.
Examples of mathematical <em>skills</em> might be the ability to make a new kind of calculation or analysis, or to become faster and more confident in applying a method.<div style="margin-top:20px"></div>Your answers are anonymous.<div style="margin-top:20px"></div>
<div class="strawpoll-embed" id="strawpoll_XOgOJ2EYon3" style="height: 1156px; max-width: 640px; width: 100%; margin: 0 auto; display: flex; flex-direction: column;"><iframe title="StrawPoll Embed" id="strawpoll_iframe_XOgOJ2EYon3" src="https://strawpoll.com/embed/polls/XOgOJ2EYon3" style="position: static; visibility: visible; display: block; width: 100%; flex-grow: 1;" frameborder="0" allowfullscreen allowtransparency>Loading...</iframe><script async src="https://cdn.strawpoll.com/dist/widgets.js" charset="utf-8"></script></div>
<div class="strawpoll-embed" id="strawpoll_XOgOJ2EYGn3" style="height: 1156px; max-width: 640px; width: 100%; margin: 0 auto; display: flex; flex-direction: column;"><iframe title="StrawPoll Embed" id="strawpoll_iframe_XOgOJ2EYGn3" src="https://strawpoll.com/embed/polls/XOgOJ2EYGn3" style="position: static; visibility: visible; display: block; width: 100%; flex-grow: 1;" frameborder="0" allowfullscreen allowtransparency>Loading...</iframe><script async src="https://cdn.strawpoll.com/dist/widgets.js" charset="utf-8"></script></div>
<div style="margin-top:20px"></div></div></body><div style="margin-top:20px"></div>