Difference between revisions of "009A Sample Final A"

From Math Wiki
Jump to navigation Jump to search
 
(53 intermediate revisions by the same user not shown)
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'''.
+
'''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;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.'''  
  
  
 
== Limits ==
 
== Limits ==
  
<span style="font-size:135%"><font face=Times Roman>
+
<span class="exam">
1. Find the following limits:<br>&nbsp;&nbsp;(a) &nbsp; <math style="vertical-align: -45%;">\lim_{x\rightarrow0}\frac{\tan(3x)}{x^{3}}.</math>
+
[[009A_Sample_Final_A,_Problem_1| <span class="biglink">&nbsp;Problem 1.&nbsp;</span>]] &nbsp;Find the following limits:<br>&nbsp;&nbsp; (a) &nbsp; <math style="vertical-align: -45%;">\lim_{x\rightarrow0}\frac{\tan(3x)}{x^{3}}.</math>
 
<br><br>
 
<br><br>
&nbsp;&nbsp;(b)  <math style="vertical-align: -55%;">\lim_{x\rightarrow-\infty}\frac{\sqrt{x^{6}+6x^{2}+2}}{x^{3}+x-1}.</math>
+
&nbsp;&nbsp; (b)  <math style="vertical-align: -52%;">\lim_{x\rightarrow-\infty}\frac{\sqrt{x^{6}+6x^{2}+2}}{x^{3}+x-1}.</math>
 
<br><br>
 
<br><br>
&nbsp;&nbsp;(c) &nbsp; <math style="vertical-align: -65%;">\lim_{x\rightarrow3}\frac{x-3}{\sqrt{x+1}-2}.</math>
+
&nbsp;&nbsp; (c) &nbsp; <math style="vertical-align: -65%;">\lim_{x\rightarrow3}\frac{x-3}{\sqrt{x+1}-2}.</math>
 
<br><br>
 
<br><br>
&nbsp;&nbsp;(d) &nbsp; <math style="vertical-align: -65%;">\lim_{x\rightarrow3}\frac{x-1}{\sqrt{x+1}-1}.</math>
+
&nbsp;&nbsp; (d) &nbsp; <math style="vertical-align: -65%;">\lim_{x\rightarrow3}\frac{x-1}{\sqrt{x+1}-1}.</math>
 
<br><br>
 
<br><br>
&nbsp;&nbsp;(e)&nbsp; <math style="vertical-align: -50%;">\lim_{x\rightarrow\infty}\frac{5x^{2}-2x+3}{1-3x^{2}}.</math>
+
&nbsp;&nbsp; (e)&nbsp; <math style="vertical-align: -50%;">\lim_{x\rightarrow\infty}\frac{5x^{2}-2x+3}{1-3x^{2}}.</math>
</font face=Times Roman> </span>
+
</span>
  
 
== Derivatives ==
 
== Derivatives ==
<span style="font-size:135%"><font face=Times Roman>2. 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>
 
<br>
 
&nbsp;&nbsp; (a) &nbsp;<math style="vertical-align: -45%;">f(x)=\frac{3x^{2}-5}{x^{3}-9}.</math>
 
&nbsp;&nbsp; (a) &nbsp;<math style="vertical-align: -45%;">f(x)=\frac{3x^{2}-5}{x^{3}-9}.</math>
Line 23: Line 23:
 
&nbsp;&nbsp; (b) &nbsp;<math style="vertical-align: -15%;">g(x)=\pi+2\cos(\sqrt{x-2}).</math>
 
&nbsp;&nbsp; (b) &nbsp;<math style="vertical-align: -15%;">g(x)=\pi+2\cos(\sqrt{x-2}).</math>
 
<br><br>
 
<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>
 
<br>
  
 
== Continuity and Differentiability ==
 
== Continuity and Differentiability ==
  
<span style="font-size:135%"><font face=Times Roman>3. (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>
 
&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>
 
<br>
&nbsp;&nbsp;(a) Find a value of &nbsp;<math style="vertical-align: -0.1%;">C</math> which makes <math>f</math> continuous at <math style="vertical-align: -3%;">x=1.</math>  
+
&nbsp;&nbsp; (a) Find a value of &nbsp;<math style="vertical-align: -0.1%;">C</math> which makes <math>f</math> continuous at <math style="vertical-align: -3%;">x=1.</math>  
 
<br>
 
<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.
+
&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.
 
<br><br>  
 
<br><br>  
3. (Version II) Consider the following function:
+
[[009A_Sample_Final_A,_Problem_3|<span class="biglink">&nbsp;Problem 3.&nbsp;</span>]] &nbsp;(Version II) Consider the following function:
 
&nbsp;<math style="vertical-align: -80%;">g(x)=\begin{cases}
 
&nbsp;<math style="vertical-align: -80%;">g(x)=\begin{cases}
 
\sqrt{x^{2}+3}, & \quad\mbox{if } x\geq1\\
 
\sqrt{x^{2}+3}, & \quad\mbox{if } x\geq1\\
Line 41: Line 41:
 
\end{cases}</math>
 
\end{cases}</math>
 
<br>
 
<br>
&nbsp;&nbsp;(a) Find a value of &nbsp;<math style="vertical-align: -0.1%;">C</math> which makes <math>f</math> continuous at <math style="vertical-align: -2.95%;">x=1.</math>  
+
&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>
 
<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 ==
 
== Implicit Differentiation ==
<span style="font-size:135%"><font face=Times Roman>
+
<span class="exam">
4. Find an equation for the tangent
+
[[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 ==
 
== Derivatives and Graphing ==
  
<span style="font-size:135%"><font face=Times Roman>5. 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;
 
&nbsp;
 
<math style="vertical-align: -42%;">h(x)={\displaystyle \frac{x^{3}}{3}-2x^{2}-5x+\frac{35}{3}}.</math>
 
<math style="vertical-align: -42%;">h(x)={\displaystyle \frac{x^{3}}{3}-2x^{2}-5x+\frac{35}{3}}.</math>
 
<br>
 
<br>
&nbsp;&nbsp;(a) Find the intervals where the function is increasing and decreasing.
+
&nbsp;&nbsp; (a) Find the intervals where the function is increasing and decreasing.
 
<br>
 
<br>
&nbsp;&nbsp;(b) Find the local maxima and minima.
+
&nbsp;&nbsp; (b) Find the local maxima and minima.
 
<br>
 
<br>
&nbsp;&nbsp;(c) Find the intervals on which <math style="vertical-align: -14%;">f(x)</math> is concave upward and concave
+
&nbsp;&nbsp; (c) Find the intervals on which <math style="vertical-align: -14%;">f(x)</math> is concave upward and concave
 
downward.  
 
downward.  
 
<br>
 
<br>
&nbsp;&nbsp;(d) Find all inflection points.
+
&nbsp;&nbsp; (d) Find all inflection points.
 
<br>
 
<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>
 
<br>
  
 
== Asymptotes ==
 
== Asymptotes ==
<span style="font-size:135%"><font face=Times Roman>6. 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>
 
&nbsp;<math style="vertical-align: -60%;">f(x)=\frac{\sqrt{4x^{2}+3}}{10x-20}.</math>
 
<br>
 
<br>
Line 75: Line 75:
 
== Optimization ==
 
== Optimization ==
 
<br>
 
<br>
<span style="font-size:135%"><font face=Times Roman> 7. 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
 
500 sq. ft. of area. What dimensions for each pen will use the least
 
500 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]]
 
[[File:009A SF A 7 Pens.png|center|500px]]
Line 83: Line 83:
 
== Linear Approximation ==
 
== Linear Approximation ==
 
<br>
 
<br>
<span style="font-size:135%"> <font face=Times Roman>8. (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>
 
<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>
 
<br>
  
 
== Related Rates ==
 
== Related Rates ==
 
<br>
 
<br>
<span style="font-size:135%"> <font face=Times Roman> 9. 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
 
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>
 
<br>
  
 
== Two Important Theorems ==
 
== Two Important Theorems ==
<span style="font-size:135%"><font face=Times Roman>10. 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;
&nbsp;
 
 
<math style="vertical-align: -15%;">f(x)=2x^{3}+4x+\sqrt{2}.</math>
 
<math style="vertical-align: -15%;">f(x)=2x^{3}+4x+\sqrt{2}.</math>
 
<br>
 
<br>
Line 103: Line 102:
 
least one zero.  
 
least one zero.  
 
<br>
 
<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.
 +
 
 +
 
 +
'''Contributions to this page were made by [[Contributors|John Simanyi]]'''

Latest revision as of 11:38, 28 July 2015

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  boxed problem numbers  to go to a solution.


Limits

 Problem 1.   Find the following limits:
   (a)  

   (b)

   (c)  

   (d)  

   (e) 

Derivatives

 Problem 2.   Find the derivatives of the following functions:
   (a)  

   (b)  

   (c)  

Continuity and Differentiability

 Problem 3.   (Version I) Consider the following function:  
   (a) Find a value of   which makes continuous at
   (b) With your choice of  , is differentiable at ?  Use the definition of the derivative to motivate your answer.

 Problem 3.   (Version II) Consider the following function:  
   (a) Find a value of   which makes continuous at
   (b) With your choice of  , is differentiable at ?  Use the definition of the derivative to motivate your answer.

Implicit Differentiation

 Problem 4.   Find an equation for the tangent line to the function   at the point .

Derivatives and Graphing

 Problem 5.   Consider the function  
   (a) Find the intervals where the function is increasing and decreasing.
   (b) Find the local maxima and minima.
   (c) Find the intervals on which is concave upward and concave downward.
   (d) Find all inflection points.
   (e) Use the information in the above to sketch the graph of .

Asymptotes


 Problem 6.   Find the vertical and horizontal asymptotes of the function  

Optimization


 Problem 7.   A farmer wishes to make 4 identical rectangular pens, each with 500 sq. ft. of area. What dimensions for each pen will use the least amount of total fencing?

009A SF A 7 Pens.png

Linear Approximation


 Problem 8.   (a) Find the linear approximation to the function at the point .
    (b) Use to estimate the value of .

Related Rates


 Problem 9.   A bug is crawling along the -axis at a constant speed of   . How fast is the distance between the bug and the point changing when the bug is at the origin? (Note that if the distance is decreasing, then you should have a negative answer).

Two Important Theorems

 Problem 10.   Consider the function 
   (a) Use the Intermediate Value Theorem to show that has at least one zero.
   (b) Use Rolle's Theorem to show that has exactly one zero.


Contributions to this page were made by John Simanyi