Difference between revisions of "005 Sample Final A"

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'''This is a sample final, and is meant to represent the material usually covered in Math 8A.  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.'''  
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'''This is a sample final, and is meant to represent the material usually covered in Math 5.  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.'''  
  
 
<div class="noautonum">__TOC__</div>
 
<div class="noautonum">__TOC__</div>
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<span class = "exam">Find the domain of the following function. Your answer should be in interval notation <math> f(x) = \frac{1}{\sqrt{x^2-x-2}}</math> <br>
 
<span class = "exam">Find the domain of the following function. Your answer should be in interval notation <math> f(x) = \frac{1}{\sqrt{x^2-x-2}}</math> <br>
  
==[[005 Sample Final A, Question 3|<span class = "biglink"style="font-size:80%">&nbsp;Question 3&nbsp;</span>]]==
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==[[005 Sample Final A, Question 3|<span class = "biglink" style="font-size:80%">&nbsp;Question 3&nbsp;</span>]]==
 
<span class = "exam">Find f <math>\circ</math> g and its domain if <math>f(x) = x^2+1 \qquad g(x)=\sqrt{x-1}</math>
 
<span class = "exam">Find f <math>\circ</math> g and its domain if <math>f(x) = x^2+1 \qquad g(x)=\sqrt{x-1}</math>
  
==[[005 Sample Final A, Question 4|<span class = "biglink"style="font-size:80%">&nbsp;Question 4&nbsp;</span>]]==
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==[[005 Sample Final A, Question 4|<span class = "biglink" style="font-size:80%">&nbsp;Question 4&nbsp;</span>]]==
 
<span class = "exam">Find the inverse of the following function <math> f(x) = \frac{3x}{2x-1}</math>
 
<span class = "exam">Find the inverse of the following function <math> f(x) = \frac{3x}{2x-1}</math>
  
==[[005 Sample Final A, Question 5|<span class = "biglink"style="font-size:80%">&nbsp;Question 5&nbsp;</span>]]==
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==[[005 Sample Final A, Question 5|<span class = "biglink" style="font-size:80%">&nbsp;Question 5&nbsp;</span>]]==
 
<span class = "exam">Solve the following inequality. Your answer should be in interval notation. <math>\frac{3x+5}{x+2}\ge 2</math>
 
<span class = "exam">Solve the following inequality. Your answer should be in interval notation. <math>\frac{3x+5}{x+2}\ge 2</math>
  
==[[005 Sample Final A, Question 6|<span class = "biglink"style="font-size:80%">&nbsp;Question 6&nbsp;</span>]] ==
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==[[005 Sample Final A, Question 6|<span class = "biglink" style="font-size:80%">&nbsp;Question 6&nbsp;</span>]] ==
 
<span class = "exam"> Factor the following polynomial completely, &nbsp;&nbsp;&nbsp;&nbsp;<math>p(x) = x^4 + x^3 + 2x-4 </math>
 
<span class = "exam"> Factor the following polynomial completely, &nbsp;&nbsp;&nbsp;&nbsp;<math>p(x) = x^4 + x^3 + 2x-4 </math>
  
==[[005 Sample Final A, Question 7|<span class = "biglink"style="font-size:80%">&nbsp;Question 7&nbsp;</span>]] ==
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==[[005 Sample Final A, Question 7|<span class = "biglink" style="font-size:80%">&nbsp;Question 7&nbsp;</span>]] ==
 
<span class="exam"> Solve the following equation, &nbsp;&nbsp;&nbsp;&nbsp; <math> 2\log_5(x) = 3\log_5(4)</math>
 
<span class="exam"> Solve the following equation, &nbsp;&nbsp;&nbsp;&nbsp; <math> 2\log_5(x) = 3\log_5(4)</math>
  
==[[005 Sample Final A, Question 8|<span class = "biglink"style="font-size:80%">&nbsp;Question 8&nbsp;</span>]] ==
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==[[005 Sample Final A, Question 8|<span class = "biglink" style="font-size:80%">&nbsp;Question 8&nbsp;</span>]] ==
 
<span class="exam"> Solve the following equation, &nbsp;&nbsp;&nbsp;&nbsp; <math> 3^{2x} + 3^x -2 = 0 </math>
 
<span class="exam"> Solve the following equation, &nbsp;&nbsp;&nbsp;&nbsp; <math> 3^{2x} + 3^x -2 = 0 </math>
  
==[[005 Sample Final A, Question 9|<span class = "biglink"style="font-size:80%">&nbsp;Question 9&nbsp;</span>]]==
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==[[005 Sample Final A, Question 9|<span class = "biglink" style="font-size:80%">&nbsp;Question 9&nbsp;</span>]]==
 
<span class = "exam"> Solve the following system of equations <br>
 
<span class = "exam"> Solve the following system of equations <br>
 
<center><math>  \begin{align} 2x + 3y  &= & 1\\ -x + y & = & -3\end{align}</math></center>
 
<center><math>  \begin{align} 2x + 3y  &= & 1\\ -x + y & = & -3\end{align}</math></center>
  
==[[005 Sample Final A, Question 10|<span class = "biglink"style="font-size:80%">&nbsp;Question 10&nbsp;</span>]]==
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==[[005 Sample Final A, Question 10|<span class = "biglink" style="font-size:80%">&nbsp;Question 10&nbsp;</span>]]==
 
<span class="exam"> Write the partial fraction decomposition of the following, <center> <math> \frac{x+2}{x^3-2x^2+x}</math></center></span>
 
<span class="exam"> Write the partial fraction decomposition of the following, <center> <math> \frac{x+2}{x^3-2x^2+x}</math></center></span>
  
==[[005 Sample Final A, Question 11|<span class = "biglink"style="font-size:80%">&nbsp;Question 11&nbsp;</span>]] ==
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==[[005 Sample Final A, Question 11|<span class = "biglink" style="font-size:80%">&nbsp;Question 11&nbsp;</span>]] ==
 
<span class="exam">Solve the following equation in the interval <math> [0, 2\pi)</math> <br>
 
<span class="exam">Solve the following equation in the interval <math> [0, 2\pi)</math> <br>
 
<center><math> \sin^2(\theta) - \cos^2(\theta)=1+\cos(\theta)</math></center></span>
 
<center><math> \sin^2(\theta) - \cos^2(\theta)=1+\cos(\theta)</math></center></span>
  
==[[005 Sample Final A, Question 12|<span class = "biglink"style="font-size:80%">&nbsp;Question 12&nbsp;</span>]]==
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==[[005 Sample Final A, Question 12|<span class = "biglink" style="font-size:80%">&nbsp;Question 12&nbsp;</span>]]==
 
<span class="exam"> Given that <math>\sec(\theta) = -2</math> and <math>\tan(\theta) > 0 </math>, find the exact values of the remaining trig functions.
 
<span class="exam"> Given that <math>\sec(\theta) = -2</math> and <math>\tan(\theta) > 0 </math>, find the exact values of the remaining trig functions.
  
==[[005 Sample Final A, Question 13|<span class = "biglink"style="font-size:80%">&nbsp;Question 13&nbsp;</span>]]==
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==[[005 Sample Final A, Question 13|<span class = "biglink" style="font-size:80%">&nbsp;Question 13&nbsp;</span>]]==
 
<span class="exam"> Give the exact value of the following if its defined, otherwise, write undefined. <br>
 
<span class="exam"> Give the exact value of the following if its defined, otherwise, write undefined. <br>
 
<math>(a) \sin^{-1}(2) \qquad \qquad (b) \sin\left(\frac{-32\pi}{3}\right) \qquad \qquad (c)\sec\left(\frac{-17\pi}{6}\right)</math>
 
<math>(a) \sin^{-1}(2) \qquad \qquad (b) \sin\left(\frac{-32\pi}{3}\right) \qquad \qquad (c)\sec\left(\frac{-17\pi}{6}\right)</math>
  
==[[005 Sample Final A, Question 14|<span class = "biglink"style="font-size:80%">&nbsp;Question 14&nbsp;</span>]]==
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==[[005 Sample Final A, Question 14|<span class = "biglink" style="font-size:80%">&nbsp;Question 14&nbsp;</span>]]==
 
<span class="exam"> Prove the following identity, <br>
 
<span class="exam"> Prove the following identity, <br>
 
<center><math>\frac{1-\sin(\theta)}{\cos(\theta)}=\frac{\cos(\theta)}{1+\sin(\theta)}</math></center>
 
<center><math>\frac{1-\sin(\theta)}{\cos(\theta)}=\frac{\cos(\theta)}{1+\sin(\theta)}</math></center>
  
==[[005 Sample Final A, Question 15|<span class = "biglink"style="font-size:80%">&nbsp;Question 15&nbsp;</span>]]==
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==[[005 Sample Final A, Question 15|<span class = "biglink" style="font-size:80%">&nbsp;Question 15&nbsp;</span>]]==
 
<span class="exam"> Find an equivalent algebraic expression for the following, <center><math> \cos(\tan^{-1}(x))</math></center></span>
 
<span class="exam"> Find an equivalent algebraic expression for the following, <center><math> \cos(\tan^{-1}(x))</math></center></span>
  
==[[005 Sample Final A, Question 16|<span class = "biglink"style="font-size:80%">&nbsp;Question 16&nbsp;</span> ]]==
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==[[005 Sample Final A, Question 16|<span class = "biglink" style="font-size:80%">&nbsp;Question 16&nbsp;</span> ]]==
 
<span class="exam"> Graph the following, </span><center><math> -x^2+4y^2-2x-16y+11=0</math></center></span>
 
<span class="exam"> Graph the following, </span><center><math> -x^2+4y^2-2x-16y+11=0</math></center></span>
  
==[[005 Sample Final A, Question 17|<span class = "biglink"style="font-size:80%">&nbsp;Question 17&nbsp;</span>]]==
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==[[005 Sample Final A, Question 17|<span class = "biglink" style="font-size:80%">&nbsp;Question 17&nbsp;</span>]]==
 
<span class="exam"> Graph the following function, <center><math>f(x) = \log_2(x+1) + 2</math></center> </span> <br>
 
<span class="exam"> Graph the following function, <center><math>f(x) = \log_2(x+1) + 2</math></center> </span> <br>
 
<span class="exam">Make sure to label any asymptotes, and at least two points on the graph. </span>
 
<span class="exam">Make sure to label any asymptotes, and at least two points on the graph. </span>
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&nbsp;&nbsp;&nbsp;&nbsp; b. Find the sum <math> \displaystyle{\sum_{k=1}^\infty a_k}</math>
 
&nbsp;&nbsp;&nbsp;&nbsp; b. Find the sum <math> \displaystyle{\sum_{k=1}^\infty a_k}</math>
 
</span>
 
</span>
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'''Contributions to this page were made by [[Contributors|Matthew Lee]]'''

Latest revision as of 14:43, 24 September 2015

This is a sample final, and is meant to represent the material usually covered in Math 5. 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.

 Question 1 

Please circle either true or false,
    a. (True/False)In a geometric sequence, the common ratio is always positive.
    b. (True/False) A linear system of equations always has a solution.
    c. (True/False) Every function has an inverse.
    d. (True/False) Trigonometric equations do not always have unique solutions.
    e. (True/False) The domain of is all real numbers.
    f. (True/False) The function is defined for all real numbers.

 Question 2 

Find the domain of the following function. Your answer should be in interval notation

 Question 3 

Find f g and its domain if

 Question 4 

Find the inverse of the following function

 Question 5 

Solve the following inequality. Your answer should be in interval notation.

 Question 6 

Factor the following polynomial completely,     

 Question 7 

Solve the following equation,     

 Question 8 

Solve the following equation,     

 Question 9 

Solve the following system of equations

 Question 10 

Write the partial fraction decomposition of the following,

 Question 11 

Solve the following equation in the interval

 Question 12 

Given that and , find the exact values of the remaining trig functions.

 Question 13 

Give the exact value of the following if its defined, otherwise, write undefined.

 Question 14 

Prove the following identity,

 Question 15 

Find an equivalent algebraic expression for the following,

 Question 16 

Graph the following,

 Question 17 

Graph the following function,


Make sure to label any asymptotes, and at least two points on the graph.

 Question 18 

Graph the following function,


Make sure to label any asymptotes, and at least two points on the graph.

 Question 19 

Consider the following function,


     a. What is the amplitude?
     b. What is the period?
     c. What is the phase shift?
     d. What is the vertical shift?
     e. Graph one cycle of f(x). Make sure to label five key points.

 Question 20 

Consider the following rational function,


     a. What is the domain of f?
     b. What are the x and y-intercepts of f?
     c. What are the vertical and horizontal asymptotes of f, if any? Does f have any holes?
     d. Graph f(x). Make sure to include the information you found above.

 Question 21 

Find the sum

 Question 22 

Consider the following sequence,


     a. Determine a formula for , the n-th term of the sequence.
     b. Find the sum


Contributions to this page were made by Matthew Lee