009C Sample Midterm 1, Problem 1 - Revision history

Kayla Murray at 05:51, 12 November 2017

2017-11-12T05:51:12Z

Kayla Murray at 01:00, 5 November 2017

2017-11-05T01:00:06Z

Kayla Murray at 00:58, 5 November 2017

2017-11-05T00:58:16Z

Kayla Murray at 16:56, 16 April 2017

2017-04-16T16:56:38Z

Kayla Murray at 18:07, 27 March 2017

2017-03-27T18:07:49Z

Kayla Murray at 01:48, 27 February 2017

2017-02-27T01:48:16Z

Kayla Murray at 00:16, 19 February 2017

2017-02-19T00:16:11Z

Kayla Murray at 17:23, 14 February 2017

2017-02-14T17:23:28Z

Kayla Murray at 16:25, 14 February 2017

2017-02-14T16:25:31Z

Kayla Murray at 23:06, 12 February 2017

2017-02-12T23:06:32Z

@@ Line 7: / Line 7: @@
 ::<math>a_n=\frac{\ln n}{n}</math>
 <hr>
-(insert picture of handwritten solution)
+[[009C Sample Midterm 1, Problem 1 Solution|'''<u>Solution</u>''']]
 [[009C_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]

@@ Line 6: / Line 6: @@
 ::<math>a_n=\frac{\ln n}{n}</math>
+<hr>
 (insert picture of handwritten solution)

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 ::<math>a_n=\frac{\ln n}{n}</math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+[[009C Sample Midterm 1, Problem 1 Detailed Solution|'''<u>Detailed Solution for this Problem</u>''']]
 [[009C_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]

@@ Line 11: / Line 11: @@
 !Foundations: &nbsp;
 |-
-|'''L'Hôpital's Rule'''
+|'''L'Hôpital's Rule, Part 2'''
 |-
+|
-&nbsp; &nbsp; &nbsp; &nbsp; Suppose that &nbsp;<math style="vertical-align: -11px">\lim_{x\rightarrow \infty} f(x)</math>&nbsp; and &nbsp;<math style="vertical-align: -11px">\lim_{x\rightarrow \infty} g(x)</math>&nbsp; are both zero or both &nbsp;<math style="vertical-align: -1px">\pm \infty .</math>
+&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math style="vertical-align: -5px">f</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">g</math>&nbsp; be differentiable functions on the open interval &nbsp;<math style="vertical-align: -5px">(a,\infty)</math>&nbsp; for some value &nbsp;<math style="vertical-align: -4px">a,</math>&nbsp;
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; where &nbsp;<math style="vertical-align: -5px">g'(x)\ne 0</math>&nbsp; on &nbsp;<math style="vertical-align: -5px">(a,\infty)</math>&nbsp; and &nbsp;<math style="vertical-align: -18px">\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}</math>&nbsp; returns either &nbsp;<math style="vertical-align: -15px">\frac{0}{0}</math>&nbsp; or &nbsp;<math style="vertical-align: -15px">\frac{\infty}{\infty}.</math>&nbsp;
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp;Then, &nbsp; <math style="vertical-align: -18px">\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}=\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}.</math>
 |}

@@ Line 62: / Line 62: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>0</math>
+|&nbsp; &nbsp; &nbsp; &nbsp; The sequence converges. The limit of the sequence is &nbsp;<math style="vertical-align: 0px">0.</math>
 |}
 [[009C_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]

@@ Line 14: / Line 14: @@
 |-
+|
-&nbsp; &nbsp; &nbsp; &nbsp; Suppose that <math style="vertical-align: -11px">\lim_{x\rightarrow \infty} f(x)</math>&thinsp; and <math style="vertical-align: -11px">\lim_{x\rightarrow \infty} g(x)</math>&thinsp; are both zero or both <math style="vertical-align: -1px">\pm \infty .</math>
+&nbsp; &nbsp; &nbsp; &nbsp; Suppose that &nbsp;<math style="vertical-align: -11px">\lim_{x\rightarrow \infty} f(x)</math>&nbsp; and &nbsp;<math style="vertical-align: -11px">\lim_{x\rightarrow \infty} g(x)</math>&nbsp; are both zero or both &nbsp;<math style="vertical-align: -1px">\pm \infty .</math>
 |-
+|
-&nbsp; &nbsp; &nbsp; &nbsp; If <math style="vertical-align: -19px">\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}</math>&thinsp; is finite or&thinsp; <math style="vertical-align: -4px">\pm \infty ,</math>
+&nbsp; &nbsp; &nbsp; &nbsp; If &nbsp;<math style="vertical-align: -19px">\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}</math>&nbsp; is finite or &nbsp;<math style="vertical-align: -4px">\pm \infty ,</math>
 |-
+|
-&nbsp; &nbsp; &nbsp; &nbsp; then <math style="vertical-align: -19px">\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}\,=\,\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}.</math>
+&nbsp; &nbsp; &nbsp; &nbsp; then &nbsp;<math style="vertical-align: -19px">\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}\,=\,\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}.</math>
 |}
@@ Line 37: / Line 37: @@
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{n\rightarrow \infty} n=\infty.</math>
 |-
-|Therefore, the limit has the form <math style="vertical-align: -11px">\frac{\infty}{\infty},</math>
+|Therefore, the limit has the form &nbsp;<math style="vertical-align: -11px">\frac{\infty}{\infty},</math>
 |-
 |which means that we can use L'Hopital's Rule to calculate this limit.
@@ Line 45: / Line 45: @@
 !Step 2: &nbsp;
 |-
-|First, switch to the variable <math style="vertical-align: 0px">x</math> so that we have functions and
+|First, switch to the variable &nbsp;<math style="vertical-align: 0px">x</math> &nbsp; so that we have functions and
 |-
 |can take derivatives. Thus, using L'Hopital's Rule, we have

← Older revision		Revision as of 00:16, 19 February 2017
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	<span class="exam"> Be sure to jusify your answers!		<span class="exam"> Be sure to jusify your answers!

−	~~::::~~::<math>a_n=\frac{\ln n}{n}</math>	+	::<math>a_n=\frac{\ln n}{n}</math>

@@ Line 29: / Line 29: @@
 !Step 1: &nbsp;
 |-
-|First, we notice that
+|First, notice that
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{n\rightarrow \infty} \ln n =\infty</math>
@@ Line 39: / Line 39: @@
 |Therefore, the limit has the form <math style="vertical-align: -11px">\frac{\infty}{\infty},</math>
 |-
-|which means we can use L'Hopital's Rule to calculate this limit.
+|which means that we can use L'Hopital's Rule to calculate this limit.
 |}
@@ Line 45: / Line 45: @@
 !Step 2: &nbsp;
 |-
-|First, we switch to the variable <math style="vertical-align: 0px">x</math> so we have functions and
+|First, switch to the variable <math style="vertical-align: 0px">x</math> so that we have functions and
 |-
 |can take derivatives. Thus, using L'Hopital's Rule, we have

@@ Line 22: / Line 22: @@
 &nbsp; &nbsp; &nbsp; &nbsp; then <math style="vertical-align: -19px">\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}\,=\,\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}.</math>
 |}
 '''Solution:'''