Difference between revisions of "009A Sample Final A"

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<span style="font-size:135%"><font face=Times Roman>
 
<span style="font-size:135%"><font face=Times Roman>
[[009A_Sample_Final_A,_Problem_1|'''1.''']] Find the following limits:<br>&nbsp;&nbsp; (a) &nbsp; <math style="vertical-align: -45%;">\lim_{x\rightarrow0}\frac{\tan(3x)}{x^{3}}.</math>
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[[009A_Sample_Final_A,_Problem_1|'''Problem 1.''']] 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: -52%;">\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>
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== Derivatives ==
 
== Derivatives ==
<span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_2|'''2.''']] Find the derivatives of the following functions:
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<span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_2|'''Problem 2.''']] 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>
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== Continuity and Differentiability ==
 
== Continuity and Differentiability ==
  
<span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_3|'''3.''']] (Version I) Consider the following function:
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<span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_3|'''Problem 3.''']] (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>
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&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:
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[[009A_Sample_Final_A,_Problem_3|'''Problem 3.''']] (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\\
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== Implicit Differentiation ==
 
== Implicit Differentiation ==
 
<span style="font-size:135%"><font face=Times Roman>
 
<span style="font-size:135%"><font face=Times Roman>
[[009A_Sample_Final_A,_Problem_4 |'''4.''']] Find an equation for the tangent
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[[009A_Sample_Final_A,_Problem_4 |'''Problem 4.''']] 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>. </font face=Times Roman> </span>
  
 
== Derivatives and Graphing ==
 
== Derivatives and Graphing ==
  
<span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_5 |'''5.''']] Consider the function
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<span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_5 |'''Problem 5.''']] 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>
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== Asymptotes ==
 
== Asymptotes ==
<br><span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_6 |'''6.''']] Find the vertical and horizontal asymptotes of the function</font face=Times Roman> </span>
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<br><span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_6 |'''Problem 6.''']] Find the vertical and horizontal asymptotes of the function</font face=Times Roman> </span>
 
&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>
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== Optimization ==
 
== Optimization ==
 
<br>
 
<br>
<span style="font-size:135%"><font face=Times Roman>  [[009A_Sample_Final_A,_Problem_7 |'''7.''']] A farmer wishes to make 4 identical rectangular pens, each with
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<span style="font-size:135%"><font face=Times Roman>  [[009A_Sample_Final_A,_Problem_7 |'''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
 
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? </font face=Times Roman> </span>
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== Linear Approximation ==
 
== Linear Approximation ==
 
<br>
 
<br>
<span style="font-size:135%"> <font face=Times Roman>[[009A_Sample_Final_A,_Problem_8|'''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 style="font-size:135%"> <font face=Times Roman>[[009A_Sample_Final_A,_Problem_8|'''Problem 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>.  
 
<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>. </font face=Times Roman> </span>
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== Related Rates ==
 
== Related Rates ==
 
<br>
 
<br>
<span style="font-size:135%"> <font face=Times Roman>  [[009A_Sample_Final_A,_Problem_9|'''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>.
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<span style="font-size:135%"> <font face=Times Roman>  [[009A_Sample_Final_A,_Problem_9|'''Problem 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>.
 
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)''.  </font face=Times Roman> </span>
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== Two Important Theorems ==
 
== Two Important Theorems ==
<span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_10|'''10.''']] Consider the function
+
<span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_10|'''Problem 10.''']] 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>

Revision as of 11:27, 30 March 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 blue 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.