forked from PSCCMATH/math1830old
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathu1s1.html
624 lines (440 loc) · 28.8 KB
/
u1s1.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
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
<!DOCTYPE html>
<html lang="en">
<head>
<title>MATH 1830 Notes</title>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1">
<link rel="stylesheet" href="css/notes.css" />
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.12.4/jquery.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.6/js/bootstrap.min.js"></script>
<script src="js/togglebuttons.js"></script>
<script src="js/loadmathjax.js"></script>
<script src="js/googlestats.js"></script>
<script src="https://www.desmos.com/api/v0.6/calculator.js?apiKey=dcb31709b452b1cf9dc26972add0fda6"></script>
<script src="js/desmosgraphs.js"></script>
<!-- <script>
unit1section1();
</script> -->
</head>
<body>
<nav class="navbar navbar-default navbar-fixed-top">
<div class="container-fluid">
<!-- Brand and toggle get grouped for better mobile display -->
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#bs-example-navbar-collapse-1" aria-expanded="false">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="https://psccmath.github.io/math1830/">MATH 1830</a>
</div>
<!-- Collect the nav links, forms, and other content for toggling -->
<div class="collapse navbar-collapse" id="bs-example-navbar-collapse-1">
<ul class="nav navbar-nav">
<!-- <li class="active"><a href="#">Link <span class="sr-only">(current)</span></a></li>
<li><a href="#">Link</a></li> -->
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-haspopup="true" aria-expanded="false">Unit 1<span class="caret"></span></a>
<ul class="dropdown-menu">
<li><a href="notesu1.html">Notes</a></li>
<li><a href="homeworku1.html">Homework</a></li>
</ul>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-haspopup="true" aria-expanded="false">Unit 2<span class="caret"></span></a>
<ul class="dropdown-menu">
<li><a href="notesu2.html">Notes</a></li>
<li><a href="homeworku2.html">Homework</a></li>
</ul>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-haspopup="true" aria-expanded="false">Unit 3<span class="caret"></span></a>
<ul class="dropdown-menu">
<li><a href="notesu3.html">Notes</a></li>
<li><a href="homeworku3.html">Homework</a></li>
</ul>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-haspopup="true" aria-expanded="false">Unit 4<span class="caret"></span></a>
<ul class="dropdown-menu">
<li><a href="notesu4.html">Notes</a></li>
<li><a href="homeworku4.html">Homework</a></li>
</ul>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-haspopup="true" aria-expanded="false">Toggle<span class="caret"></span></a>
<ul class="dropdown-menu">
<li><a href="#" onclick="togglesolutions();return false;">Solutions</a></li>
<li><a href="#" onclick="toggleanswers();return false;">Answers</a></li>
</ul>
</li>
</ul>
</div>
<!-- /.navbar-collapse -->
</div>
<!-- /.container-fluid -->
</nav>
<div class="container">
<h1 id="u1toc">MATH 1830 Notes</h1>
<p>Mary Monroe-Ellis</p>
<p>Susan Mosteller</p>
<h2 >Unit 1 Limits</h2>
<div class="m1830-toc">
<ul>
<li><a href="notesu1.html#u1sp">1.P Writing Equations of Lines</a></li>
<li><a href="notesu1.html#u1s1">1.1 Limits Graphically and Algebraically</a></li>
<li><a href="notesu1.html#u1s2">1.2 Infinite Limits and Asymptotes</a></li>
<li><a href="notesu1.html#u1s3">1.3 Continuity</a></li>
<li><a href="notesu1.html#u1s4">1.4 Definition of Derivatives</a></li>
<li><a href="notesu1.html#u1s5">1.5 Derivatives: The Power Rule</a></li>
<li><a href="notesu1.html#u1s6">1.6 Marginal Analysis</a></li>
</ul>
</div>
<div id="u1s1" class="m1830-section">
<h3>1.1 Limits Graphically and Algebraically</h3>
<h4>Introduction</h4>
<!-- <p>
sample desmos graph insert
</p>
<div class="graph" id="desu1s1">
</div>
<a href="https://www.desmos.com"><img class="desmosicon" alt="desmos logo" src="https://s3.amazonaws.com/desmos/img/calc_thumb.png"></a> -->
<ol>
<li class="m1830-problem">
<p>Box Office Receipts</p>
<p class="m1830-problem-text">
The total worldwide box-office receipts for a long running indie film are approximated by the function $$T(x) = \frac{{120{x^2}}}{{{x^2} + 4}}$$ where T(x) is measured in millions of dollars and x is the number of months since the movie’s release.
What are the total box-office receipts after:</p>
<ol>
<li>The first month?
<p class="m1830-solution">$T\left( 1 \right) = \frac{{120{{\left( 1 \right)}^2}}}{{{1^2} + 4}} = 24$</p>
<p class="m1830-answer">The total box-office receipts after 1 month are \$24 million.</p>
</li>
<li>The second?
<p class="m1830-solution">$T\left( 2 \right) = \frac{{120{{\left( 2 \right)}^2}}}{{{2^2} + 4}} = 60$ </p>
<p class="m1830-answer">The total box-office receipts after 2 months are \$60 million.</p>
</li>
<li>The third?
<p class="m1830-solution">$T\left( 3 \right) = \frac{{120{{\left( 3 \right)}^2}}}{{{3^2} + 4}} = 83$</p>
<p class="m1830-answer">The total box-office receipts after 3 months are \$83 million.</p>
</li>
<li>The hundredth?
<p class="m1830-solution">$T\left( {100} \right) = \frac{{120{{\left( {100} \right)}^2}}}{{{{100}^2} + 4}} = 119.95$</p>
<p class="m1830-answer">The total box-office receipts after 100 months are \$119.95 million.</p>
</li>
<li>Graph $T(x)$
<!-- TODO embed desmos graph -->
<div class="intrinsic-container">
<iframe id="myIframe" src="https://www.desmos.com/calculator/0ruefdia8u" allowfullscreen>
<p>Your browser does not support iframes.</p>
</iframe>
</div>
<!-- <a title="View with the Desmos Graphing Calculator" href="https://www.desmos.com/calculator/x0b2cgwueu" target="_blank"> <img src="https://s3.amazonaws.com/calc_thumbs/production/x0b2cgwueu.png" width="200" height="200" style="border:1px solid #ccc; border-radius:5px" />Link to Graph of Total Box-office Receipts</a> -->
</li>
<li>What will the movie gross in the long run? (When x is very large.)
<p class="m1830-answer">In the long run, the movie will gross approximately \$120 million.</p>
</li>
</ol>
</li>
<li class="m1830-problem">
<p>Driving Costs</p>
<p class="m1830-problem-text">
A study of driving costs of 1992 model subcompact cars found that the average cost (car payments, gas, insurance, upkeep, and depreciation), measured in cents/mile, is approximated by $$C(x) = \frac{{2010}}{{{x^{2.2}}}} + 17.80$$ where x denotes the number
of miles (in thousands of miles) the car is driven in a year. What is the average cost of driving a subcompact car:</p>
<ol>
<li>5,000 miles per year?
<p class="m1830-solution">$C\left( 5 \right) = \frac{{2010}}{{{5^{2.2}}}} + 17.80 = 76.10$</p>
<p class="m1830-answer">The average cost when driving 5,000 miles per year is 76.1 cents per mile.</p>
</li>
<li>10,000 miles per year?
<p class="m1830-solution">$C\left( {10} \right) = \frac{{2010}}{{{{10}^{2.2}}}} + 17.80 = 30.50$</p>
<p class="m1830-answer">The average cost when driving 10,000 miles per year is 30.5 cents per mile.</p>
</li>
<li>25,000 miles per year?
<p class="m1830-solution">$C\left( {25} \right) = \frac{{2010}}{{{{25}^{2.2}}}} + 17.80 = 19.50$</p>
<p class="m1830-answer">The average cost when driving 25,000 miles per year is 19.5 cents per mile.</p>
</li>
<li>50,000 miles per year?
<p class="m1830-solution">$C\left( {50} \right) = \frac{{2010}}{{{{50}^{2.2}}}} + 17.80 = 18.17$</p>
<p class="m1830-answer">The average cost when driving 50,000 miles per year is 18.2 cents per mile.</p>
</li>
<li>Graph $C(x)$ </li>
<!-- TODO embed desmos graph -->
<!-- <a title="View with the Desmos Graphing Calculator" href="https://www.desmos.com/calculator/mxmu0narn6" target="_blank"> <img src="https://s3.amazonaws.com/calc_thumbs/production/mxmu0narn6.png" width="200px" height="200px" style="border:1px solid #ccc; border-radius:5px" />Link to Graph of Driving Costs.</a>
<li>What happens to the average cost as the number of miles driven increases without bound?
<p class="m1830-answer">The average cost approaches 17.8 cents per mile when the car is driven “infinite” miles in a year.</p>
</li>
<li>Verify by evaluating the cost when the number of miles is 1,000,000 (or any large number)
<p class="m1830-solution">$C\left( {1000} \right) = \frac{{2010}}{{{{1000}^{2.2}}}} + 17.80$</p>
<p class="m1830-answer">The average cost per mile is approximately 17.80 cents per mile when the car is driven 1,000,000 miles per year.</p>
</li>
</ol>
<p><cite>Source <a href="https://domoremath.files.wordpress.com/2013/09/limits-word-prob-with-solns.pdf">https://domoremath.files.wordpress.com/2013/09/limits-word-prob-with-solns.pdf</a> </cite></p>
</li>
</ol>
<hr>
<div class="pagebreak"></div>
<h4>Notes</h4>
<p> <strong> Read It: </strong> <a href="http://cnx.org/contents/i4nRcikn@2.37:dKCfyV9u@3/The-Limit-of-a-Function" target="_blank" class="btn btn-primary">The Limit of a Function</a> </p>
<p> <strong> Watch It: </strong> <a href="https://www.khanacademy.org/math/calculus-home/ap-calculus-ab/limits-basics-ab#limits-from-graphs-ab" target="_blank" class="btn btn-primary">The Limit of a Function</a> </p> -->
<p> <strong> Try It: </strong><a href="http://curvebank.calstatela.edu/limit/limit.htm" target="_blank" class="btn btn-primary">Limits Graphically</a></p>
<p> <strong> Try It: </strong><a href="https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/limcondirectory/LimitConstant.html" target="_blank" class="btn btn-primary">Limits Algebraically</a></p>
</div>
</div>
<!-- TODO Watch It -->
<!-- TODO Check It -->
<p><strong>Limits: A Graphical Approach</strong> </p>
<img src="images/notes/u1s1p1.png" alt="Piece 1: y=-x+1 from negative infinity to (1,0) Hole at (1,0). Piece 2: y=3 on (1,2) domain. Defined point at (1,3). Hole at (2,3). Piece 3: y=-(x-3)^2+4 on domain (2,4). Hole in graph at x=2. No additional point at x=2. Hole in piece 3 x=4. Piece 4: y=-1 from 4 to infinity, including 4.">
<!-- TODO re-create graph in desmos -->
<p> $f(x)=
\begin{cases}
-x+1 & x \lt 1 \\
3 & 1\leq x\lt 2 \\
-\left(x-3\right)^2+4 & 2 \lt x \lt 4\\
-1 & x\geq 4\\
\end{cases}
$
</p>
<p>Use the piecewise function to answer the questions below.</p>
<p>Evaluate the limits graphically. If the limit does not exist, explain why.</p>
<ol>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to {0^ - }} f\left( x \right) =$
<span class="m1830-answer">$1$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to {0^ + }} f\left( x \right) =$
<span class="m1830-answer">$1$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to 0} f\left( x \right) =$
<span class="m1830-answer">$1$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$f\left( 0 \right) = $
<span class="m1830-answer">$1$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to {1^ - }} f\left( x \right) =$
<span class="m1830-answer">$0$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to {1^ + }} f\left( x \right) =$
<span class="m1830-answer">$3$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to 1} f\left( x \right)=$
<span class="m1830-answer">Does not exist</span></p>
<p class="m1830-answer">The limits from the left and right are not equal, so the limit does not exist. </p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$f\left( 1 \right) =$
<span class="m1830-answer">$3$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to {2^ - }} f\left( x \right) =$
<span class="m1830-answer">$3$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to {2^ + }} f\left( x \right) =$
<span class="m1830-answer">$3$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to 2} f\left( x \right) =$
<span class="m1830-answer">$3$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$f\left( 2 \right) =$
<span class="m1830-answer">undefined</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
Is it possible to define $f\left( 1 \right)$ so that $\mathop {\lim }\limits_{x\; \to 1} f\left( x \right) = f\left( 1 \right)$? Explain.</p>
<p class="m1830-answer">No. The limit does not exist. There is a jump in the graph.</p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
Is it possible to redefine $f\left( 2 \right)$ so that $\mathop {\lim }\limits_{x\; \to 2} f\left( x \right) = f\left( 2 \right)$? Explain.</p>
<p class="m1830-answer">Yes. Let $f\left( 2 \right) = \;3$ </p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to - 1} f\left( x \right) =$
<span class="m1830-answer">$2$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to 4} f\left( x \right) =$
<span class="m1830-answer">Does not exist</span></p>
<p class="m1830-answer">The limits from the left and right are not equal, so the limit does not exist. </p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to 2} f\left( x \right) =$
<span class="m1830-answer">$3$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to 3} f\left( x \right) =$
<span class="m1830-answer">$4$</span></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x\; \to - 2} f\left( x \right) =$
<span class="m1830-answer">$3$</span></p>
<p><strong>Limits: An Algebraic Approach</strong></p>
<!-- TODO Read It -->
<!-- TODO Watch It -->
<!-- TODO Check It -->
<p><strong>Find each indicated quantity, if it exists.</strong></p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x \to 4} \;{x^2} - 5x + 1 =$</p>
<p class="m1830-solution">${4^2} - 5\left( 4 \right) + 1 =$
<p class="m1830-answer">$- 3$</p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\mathop {\lim }\limits_{x \to - 5} \;2{x^2} + 10x + 7 =$</p>
<p class="m1830-solution">$2{\left( { - 5} \right)^2} + 10\left( { - 5} \right) + 7 =$
<p class="m1830-answer">$7$</p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
<p> $f(x)=
\begin{cases}
x+5 & x \lt -4 \\
\sqrt{x+4} & x\gt -4\\
\end{cases}
$
</p>
<ol>
<li>$\mathop {\lim }\limits_{x \to {-4^ + }} {\mkern 1mu} f(x)=$
<p class="m1830-solution">$\sqrt{-4+4}=\sqrt0$
<p class="m1830-answer">$=0$</p>
</li>
<li>$\mathop {\lim }\limits_{x \to {-4^ - }} {\mkern 1mu} f(x)$
<p class="m1830-solution">$=-4+5=$
<p class="m1830-answer">$1$</p>
</li>
<li>$\mathop {\lim }\limits_{x \to {-4}} {\mkern 1mu} f(x)=$
<span class="m1830-answer">Does not exist</span>
</li>
<li>$f(-4)=$
<span class="m1830-answer">undefined</span>
</li>
</ol>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$g\left( x \right) = \;\frac{{x - 2}}{{\left| {x - 2} \right|}}$</p>
<ol>
<li>$\mathop {\lim }\limits_{x\; \to {2^ + }} g\left( x \right) = \;$
<span class="m1830-answer">1</span>
</li>
<li>$\mathop {\lim }\limits_{x\; \to {2^ - }} g\left( x \right) = \;$
<span class="m1830-answer">-1</span>
</li>
<li>$\mathop {\lim }\limits_{x\; \to {2^{}}} g\left( x \right) = \;$
<span class="m1830-answer">Does not exist</span>
</li>
<li>$g\left( 2 \right) = $
<span class="m1830-answer">undefined</span>
</li>
</ol>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$f\left( x \right) = \;\frac{{3{x^2}\; + 2x - 1}}{{x{\;^2} + 3x + 2}}$</p>
<ol>
<li>$\underset{x\rightarrow -3}{lim}f(x)=$
<p class="m1830-solution">$\underset{x\rightarrow -3}{lim}f(x)=\frac{3{{\left( -3 \right)}^{2}}~+2\left( -3 \right)-1}{\left( -3 \right){{~}^{2}}+3\left( -3 \right)+2}=\frac{20}{2}$
<p class="m1830-answer">$10$</p>
</li>
<li>$\underset{x\rightarrow -1}{lim}f(x)=$
<p class="m1830-solution">$\frac{3{{\left( -1 \right)}^{2}}~+2\left( -1 \right)-1}{\left( -1 \right){{~}^{2}}+3\left( -1 \right)+2}=\frac{0}{0}$</p>
<p class="m1830-solution">Indeterminate form. Factor, Reduce, Try Again.</p>
<p class="m1830-solution">$\underset{x\rightarrow -1}{lim}f(x)=\underset{x\rightarrow -1}{lim}\frac{\left( 3x-1 \right)\left( x+1 \right)}{\left( x+1 \right)\left( x+2 \right)}=\underset{x\rightarrow -1}{lim}\frac{\left( 3x-1 \right)}{\left( x+2
\right)}=\frac{-4}{1}$
<p class="m1830-answer">$-4$</p>
</li>
<li>$\mathop {\lim }\limits_{x\; \to \;2} f\left( x \right)$
<span class="m1830-answer">$=\frac{5}{4}=1.25$</span>
<br> </li>
<li>$\mathop {\lim }\limits_{x\; \to \; - 2} f\left( x \right)$
<span class="m1830-answer">$=\frac{7}{0}\,$ Does not exist</span>
<br> </li>
</ol>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
$\underset{x\rightarrow 10}{lim} \frac{{{x^2}\; - 15x + 50}}{{{{\left( {x - 10} \right)}^2}}}$</p>
<p class="m1830-solution">$=\frac{0}{0}$</p>
<p class="m1830-solution">Indeterminate form. Factor, reduce, try again.</p>
<p class="m1830-solution">$\underset{x\rightarrow 10}{lim}\,\frac{\left( x-5 \right)\left( x-10 \right)}{{{\left( x-10 \right)}^{2}}}=\underset{x\rightarrow 10}{lim}\,\frac{\left( x-5 \right)}{\left( x-10 \right)}=\frac{5}{0}$</p>
<p class="m1830-answer">The limit does not exist. Vertical asymptote at x=10. On inspection of the graph, the limit as x approaches 10 from the left and the limit as x approaches 10 from the right are not equal.</p>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
Compute the following limit for the function: $\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}$ $$f\left( x \right) = {x^2}\; + 5x - 1$$</p>
<ol>
<li>Define $f\left( {x + h} \right)$ and $f\left( x \right)$
<p class="m1830-solution">$f\left( x+h \right)={{\left( x+h \right)}^{2}}+5\left( x+h \right)-1$</p>
<p class="m1830-solution">$f\left( x+h \right)={{x}^{2}}+2xh+{{h}^{2}}+5x+5h-1$</p>
<p class="m1830-answer">$f\left( x \right)={{x}^{2}}+5x-1$</p>
</li>
<li>Calculate $f\left( {x + h} \right) - f\left( x \right)$
<p class="m1830-solution">$ = \left( {{x^2} + 2xh + {h^2} + 5x + 5h - 1} \right) - \left( {{x^2} + 5x - 1} \right)$</p>
<p class="m1830-answer">$ = 2xh + {h^2} + 5h$</p>
</li>
<li>Divide by h
<p class="m1830-solution">$\frac{{2xh + {h^2} + 5h}}{h} =$</p>
<p class="m1830-answer">$ 2x + h + 5$</p>
</li>
<li>Evaluate the limit
<p class="m1830-solution">$\mathop {\lim }\limits_{h \to 0} \left ( { 2x + h + 5} \right) = 2x + 0 + 5 =$</p>
<p class="m1830-answer">$ 2x + 5$</p>
</li>
</ol>
</li>
<li class="m1830-problem">
<p class="m1830-problem-text">
A taxi service charges \$3.00 per mile for the first 10 miles. If the trip is over 10 miles, they charge \$5.00 per mile for every mile. Write a piecewise definition of the charge G(x) for taxi fares of x miles.</p>
<p>Graph G(x) for $0 < x\; \le 25.$
<p><img src="images/notes/u1spp1.png" alt="Blank Cartesian Graph">
<p class="m1830-answer">
$f(x)=
\begin{cases}
3x & 0\leq x\leq 10 \\
5x & x > 10\\
\end{cases}
$
</p>
<p>Find:</p>
<p>$\underset{x\rightarrow10^-}{lim}G(x)=$
<span class="m1830-answer">$30$</span></p>
<p>$\underset{x\rightarrow10^+}{lim}G(x)=$
<span class="m1830-answer">$50$</span></p>
<p>$\underset{x\rightarrow10}{lim}G(x)=$
<span class="m1830-answer">Does not exist.</span></p>
<p class="m1830-solution"><img src="images/notes/u1s1p27aa.png" alt="Graph of f(x) on domain (0,25]. f(x)=3x for domain (0,10]. f(x)=5x for domain (10,25]"></p>
</li>
</ol>
</div>
<!-- ending of body container -->
</body>
</html>