009A Sample Final A - Revision history

MathAdmin at 18:38, 28 July 2015

2015-07-28T18:38:08Z

MathAdmin at 18:36, 28 July 2015

2015-07-28T18:36:41Z

MathAdmin at 18:33, 28 July 2015

2015-07-28T18:33:00Z

MathAdmin at 18:29, 28 July 2015

2015-07-28T18:29:10Z

MathAdmin: /* Derivatives and Graphing */

2015-05-06T20:19:17Z

Derivatives and Graphing

MathAdmin: /* Derivatives */

2015-04-12T04:49:49Z

Derivatives

MathAdmin at 04:01, 12 April 2015

2015-04-12T04:01:31Z

MathAdmin at 03:12, 12 April 2015

2015-04-12T03:12:03Z

MathAdmin: /* Two Important Theorems */

2015-04-12T03:07:02Z

Two Important Theorems

MathAdmin: /* Related Rates */

2015-04-12T03:06:42Z

Related Rates

← Older revision		Revision as of 18:38, 28 July 2015
Line 105:		Line 105:


−	'''Contributions to this page were made by [[Contributors~~: '''~~John Simanyi~~'''~~]]'''	+	'''Contributions to this page were made by [[Contributors\|John Simanyi]]'''

← Older revision		Revision as of 18:36, 28 July 2015
Line 105:		Line 105:


−	Contributions to this page were made by [[~~http://wiki.math.ucr.edu/index.php/~~Contributors: '''John Simanyi''']]	+	'''Contributions to this page were made by [[Contributors: '''John Simanyi''']]'''

← Older revision		Revision as of 18:33, 28 July 2015
Line 105:		Line 105:


−	Contributions to this page were made by [[http://wiki.math.ucr.edu/index.php/Contributors]]	+	Contributions to this page were made by [[http://wiki.math.ucr.edu/index.php/Contributors: '''John Simanyi''']]

← Older revision		Revision as of 18:29, 28 July 2015
Line 103:		Line 103:
	<br>		<br>
	(b) Use Rolle's Theorem to show that <math style="vertical-align: -14%;">f(x)</math> has exactly one zero.		(b) Use Rolle's Theorem to show that <math style="vertical-align: -14%;">f(x)</math> has exactly one zero.
		+
		+
		+	Contributions to this page were made by [[http://wiki.math.ucr.edu/index.php/Contributors]]

@@ Line 65: / Line 65: @@
 &nbsp;&nbsp; (d) Find all inflection points.
 <br>
-&nbsp;&nbsp; (e) Use the information in the above to sketch the graph of <math style="vertical-align: -14%;">f(x)</math>. </font face=Times Roman> </span>
+&nbsp;&nbsp; (e) Use the information in the above to sketch the graph of <math style="vertical-align: -14%;">f(x)</math>. </span>
 <br>

@@ Line 23: / Line 23: @@
 &nbsp;&nbsp; (b) &nbsp;<math style="vertical-align: -15%;">g(x)=\pi+2\cos(\sqrt{x-2}).</math>
 <br><br>
-&nbsp;&nbsp; (c)</font face=Times Roman> </span>&nbsp;<math style="vertical-align: -25%;">h(x)=4x\sin(x)+e(x^{2}+2)^{2}.</math>
+&nbsp;&nbsp; (c) &nbsp;<math style="vertical-align: -25%;">h(x)=4x\sin(x)+e(x^{2}+2)^{2}.</math>
 <br>

← Older revision		Revision as of 04:01, 12 April 2015
Line 1:		Line 1:
−	'''This is a sample final, and is meant to represent the material usually covered in Math 9A. Moreover, it contains enough questions to represent a three hour test. An actual test may or may not be similar. Click on the <span class="biglink" style="color:~~blue~~;"> boxed problem numbers </span> to go to a solution.'''	+	'''This is a sample final, and is meant to represent the material usually covered in Math 9A. Moreover, it contains enough questions to represent a three hour test. An actual test may or may not be similar. Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''

@@ Line 17: / Line 17: @@
 == Derivatives ==
-<span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_2|<span class="biglink">&nbsp;Problem 2.&nbsp;</span>]] &nbsp;Find the derivatives of the following functions:
+<span class="exam">[[009A_Sample_Final_A,_Problem_2|<span class="biglink">&nbsp;Problem 2.&nbsp;</span>]] &nbsp;Find the derivatives of the following functions:
 <br>
 &nbsp;&nbsp; (a) &nbsp;<math style="vertical-align: -45%;">f(x)=\frac{3x^{2}-5}{x^{3}-9}.</math>
@@ Line 28: / Line 28: @@
 == Continuity and Differentiability ==
-<span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_3|<span class="biglink">&nbsp;Problem 3.&nbsp;</span>]] &nbsp;(Version I) Consider the following function:
+<span class="exam">[[009A_Sample_Final_A,_Problem_3|<span class="biglink">&nbsp;Problem 3.&nbsp;</span>]] &nbsp;(Version I) Consider the following function:
 &nbsp;<math style="vertical-align: -80%;">f(x) = \begin{cases} \sqrt{x}, & \mbox{if }x\geq 1, \\ 4x^{2}+C, & \mbox{if }x<1. \end{cases}</math>
 <br>
@@ Line 43: / Line 43: @@
 &nbsp;&nbsp; (a) Find a value of &nbsp;<math style="vertical-align: 0%">C</math> which makes <math>f</math> continuous at <math style="vertical-align: -2.95%;">x=1.</math>
 <br>
-&nbsp;&nbsp; (b) With your choice of &nbsp;<math style="vertical-align: -0.1%;">C</math>, is <math>f</math> differentiable at <math style="vertical-align: -3%;">x=1</math>? &nbsp;Use the definition of the derivative to motivate your answer. </font face=Times Roman> </span>
+&nbsp;&nbsp; (b) With your choice of &nbsp;<math style="vertical-align: -0.1%;">C</math>, is <math>f</math> differentiable at <math style="vertical-align: -3%;">x=1</math>? &nbsp;Use the definition of the derivative to motivate your answer.
 == Implicit Differentiation ==
-<span style="font-size:135%"><font face=Times Roman>
+<span class="exam">
 [[009A_Sample_Final_A,_Problem_4 |<span class="biglink">&nbsp;Problem 4.&nbsp;</span>]] &nbsp;Find an equation for the tangent
-line to the function &nbsp;<math style="vertical-align: -13%;">-x^{3}-2xy+y^{3}=-1</math>  at the point <math style="vertical-align: -15%">(1,1)</math>. </font face=Times Roman> </span>
+line to the function &nbsp;<math style="vertical-align: -13%;">-x^{3}-2xy+y^{3}=-1</math>  at the point <math style="vertical-align: -15%">(1,1)</math>.
 == Derivatives and Graphing ==
-<span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_5 |<span class="biglink">&nbsp;Problem 5.&nbsp;</span>]] &nbsp;Consider the function
+<span class="exam">[[009A_Sample_Final_A,_Problem_5 |<span class="biglink">&nbsp;Problem 5.&nbsp;</span>]] &nbsp;Consider the function
 &nbsp;
 <math style="vertical-align: -42%;">h(x)={\displaystyle \frac{x^{3}}{3}-2x^{2}-5x+\frac{35}{3}}.</math>
@@ Line 69: / Line 69: @@
 == Asymptotes ==
-<br><span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_6 |<span class="biglink">&nbsp;Problem 6.&nbsp;</span>]] &nbsp;Find the vertical and horizontal asymptotes of the function</font face=Times Roman> </span>
+<br><span class="exam">[[009A_Sample_Final_A,_Problem_6 |<span class="biglink">&nbsp;Problem 6.&nbsp;</span>]] &nbsp;Find the vertical and horizontal asymptotes of the function
 &nbsp;<math style="vertical-align: -60%;">f(x)=\frac{\sqrt{4x^{2}+3}}{10x-20}.</math>
 <br>
@@ Line 75: / Line 75: @@
 == Optimization ==
 <br>
-<span style="font-size:135%"><font face=Times Roman>  [[009A_Sample_Final_A,_Problem_7 |<span class="biglink">&nbsp;Problem 7.&nbsp;</span>]] &nbsp;A farmer wishes to make 4 identical rectangular pens, each with
+<span class="exam">  [[009A_Sample_Final_A,_Problem_7 |<span class="biglink">&nbsp;Problem 7.&nbsp;</span>]] &nbsp;A farmer wishes to make 4 identical rectangular pens, each with
 sq. ft. of area. What dimensions for each pen will use the least
-amount of total fencing? </font face=Times Roman> </span>
+amount of total fencing?
 [[File:009A SF A 7 Pens.png|center|500px]]
@@ Line 83: / Line 83: @@
 == Linear Approximation ==
 <br>
-<span style="font-size:135%"> <font face=Times Roman>[[009A_Sample_Final_A,_Problem_8|<span class="biglink">&nbsp;Problem 8.&nbsp;</span>]] &nbsp;(a) Find the linear approximation <math style="vertical-align: -14%;">L(x)</math> to the function <math style="vertical-align: -14%;">f(x)=\sec x</math> at the point <math style="vertical-align: -14%;">x=\pi/3</math>.
+<span class="exam">[[009A_Sample_Final_A,_Problem_8|<span class="biglink">&nbsp;Problem 8.&nbsp;</span>]] &nbsp;(a) Find the linear approximation <math style="vertical-align: -14%;">L(x)</math> to the function <math style="vertical-align: -14%;">f(x)=\sec x</math> at the point <math style="vertical-align: -14%;">x=\pi/3</math>.
 <br>
-&nbsp;&nbsp;&nbsp;&nbsp;(b) Use <math style="vertical-align: -14%;">L(x)</math> to estimate the value of <math style="vertical-align: -14%;">\sec\,(3\pi/7)</math>. </font face=Times Roman> </span>
+&nbsp;&nbsp;&nbsp;&nbsp;(b) Use <math style="vertical-align: -14%;">L(x)</math> to estimate the value of <math style="vertical-align: -14%;">\sec\,(3\pi/7)</math>.
 <br>
 == Related Rates ==
 <br>
-<span style="font-size:135%"> <font face=Times Roman>  [[009A_Sample_Final_A,_Problem_9|<span class="biglink">&nbsp;Problem 9.&nbsp;</span>]] &nbsp;A bug is crawling along the <math style="vertical-align: 0%;">x</math>-axis at a constant speed of &nbsp; <math style="vertical-align: -42%;">\frac{dx}{dt}=30</math>.
+<span class="exam">  [[009A_Sample_Final_A,_Problem_9|<span class="biglink">&nbsp;Problem 9.&nbsp;</span>]] &nbsp;A bug is crawling along the <math style="vertical-align: 0%;">x</math>-axis at a constant speed of &nbsp; <math style="vertical-align: -42%;">\frac{dx}{dt}=30</math>.
 How fast is the distance between the bug and the point <math style="vertical-align: -14%;">(3,4)</math> changing
-when the bug is at the origin? ''(Note that if the distance is decreasing, then you should have a negative answer)''.   </font face=Times Roman> </span>
+when the bug is at the origin? ''(Note that if the distance is decreasing, then you should have a negative answer)''.
 <br>
 == Two Important Theorems ==
-<span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_10|<span class="biglink">&nbsp;Problem 10.&nbsp;</span>]] &nbsp;Consider the function
+<span class="exam">[[009A_Sample_Final_A,_Problem_10|<span class="biglink">&nbsp;Problem 10.&nbsp;</span>]] &nbsp;Consider the function&nbsp;
 <math style="vertical-align: -15%;">f(x)=2x^{3}+4x+\sqrt{2}.</math>
 <br>
@@ Line 103: / Line 102: @@
 least one zero.
 <br>
-&nbsp;&nbsp; (b) Use Rolle's Theorem to show that <math style="vertical-align: -14%;">f(x)</math> has exactly one zero. </font face=Times Roman> </span>
+&nbsp;&nbsp; (b) Use Rolle's Theorem to show that <math style="vertical-align: -14%;">f(x)</math> has exactly one zero.