Difference between revisions of "022 Exam 1 Sample A"

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'''This is a sample, and is meant to represent the material usually covered in Math 22 up to the first exam. An actual test may or may not be similar. Click on the <span style="color:blue;">blue problem numbers</span> to go to a solution.'''
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'''This is a sample, and is meant to represent the material usually covered in Math 22 up to the first exam. 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.'''
  
 
== Definition of the Derivative ==
 
== Definition of the Derivative ==
  
<span style="font-size:135%"><font face=Times Roman>[[022_Exam_1_Sample_A,_Problem_1|Problem 1.]] Use the definition of derivative to find the derivative
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<span class="exam">[[022_Exam_1_Sample_A,_Problem_1|<span class="biglink">&nbsp;Problem 1.&nbsp;</span>]] Use the definition of derivative to find the derivative
 
of <math style="vertical-align: -15%">f(x)=\sqrt{x-5}</math>.
 
of <math style="vertical-align: -15%">f(x)=\sqrt{x-5}</math>.
  
 
== Implicit Differentiation ==
 
== Implicit Differentiation ==
  
<span style="font-size:135%">[[022_Exam_1_Sample_A,_Problem_2 |'''Problem 2.''']] Use implicit differentiation to find <math style="vertical-align: -16%;">dy/dx</math> at the
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<span class="exam">[[022_Exam_1_Sample_A,_Problem_2 |<span class="biglink">&nbsp;Problem 2.&nbsp;</span>]] Use implicit differentiation to find <math style="vertical-align: -16%;">dy/dx</math> at the
 
point <math style="vertical-align: -17%;">(1,0)</math> on the curve defined by <math style="vertical-align: -12%;">x^{3}-y^{3}-y=x</math>.
 
point <math style="vertical-align: -17%;">(1,0)</math> on the curve defined by <math style="vertical-align: -12%;">x^{3}-y^{3}-y=x</math>.
  
 
== Continuity and Limits ==
 
== Continuity and Limits ==
  
<span style="font-size:135%"><font face=Times Roman>Problem 3. Given a function <math style="vertical-align: -41%;">g(x)=\frac{x+5}{x^{2}-25}</math>&thinsp;,
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<span class="exam">[[022_Exam_1_Sample_A,_Problem_3 |<span class="biglink">&nbsp;Problem 3.&nbsp;</span>]] Given a function <math style="vertical-align: -40%;">g(x)=\frac{x+5}{x^{2}-25}</math>&thinsp;,
  
&nbsp;&nbsp; <span style="font-size:135%"><font face=Times Roman>(a) Find the intervals where <math style="vertical-align: -14%;">g(x)</math> is continuous.
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:<span class="exam">(a) Find the intervals where <math style="vertical-align: -20%;">g(x)</math> is continuous.
  
&nbsp;&nbsp; <span style="font-size:135%"><font face=Times Roman>(b). Find <math style="vertical-align: -40%;">\lim_{x\rightarrow5}g(x)</math>.
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:<span class="exam">(b). Find <math style="vertical-align: -55%;">\lim_{x\rightarrow5}g(x)</math>.
  
 
== Increasing and Decreasing ==
 
== Increasing and Decreasing ==
  
<span style="font-size:135%">Problem 4. Determine the intervals where the function&thinsp; <math style="vertical-align: -16%">h(x)=2x^{4}-x^{2}</math>
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<span class="exam">[[022_Exam_1_Sample_A,_Problem_4 |<span class="biglink">&nbsp;Problem 4.&nbsp;</span>]] Determine the intervals where the function&thinsp; <math style="vertical-align: -16%">h(x)=2x^{4}-x^{2}</math>
 
is increasing or decreasing.
 
is increasing or decreasing.
  
 
== Marginal Revenue and Profit ==
 
== Marginal Revenue and Profit ==
  
<span style="font-size:135%">Problem 5. Find the marginal revenue and marginal profit at <math style="vertical-align: -3%">x=4</math>, given the demand function  
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<span class="exam">[[022_Exam_1_Sample_A,_Problem_5 |<span class="biglink">&nbsp;Problem 5.&nbsp;</span>]] Find the marginal revenue and marginal profit at <math style="vertical-align: -3%">x=4</math>, given the demand function  
  
<math>p=\frac{200}{\sqrt{x}}</math>  
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::<math>p=\frac{200}{\sqrt{x}}</math>  
  
<span style="font-size:135%">and the cost function  
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<span class="exam">and the cost function  
  
<math>C=100+15x+3x^{2}.</math>
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::<math>C(x)=100+15x+3x^{2}.</math>
  
<span style="font-size:135%">Should the firm produce one more item under these conditions? Justify
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<span class="exam">Should the firm produce one more item under these conditions? Justify
 
your answer.
 
your answer.
  
 
== Related Rates (Word Problem) ==
 
== Related Rates (Word Problem) ==
  
<span style="font-size:135%">Problem 6. A 15-foot ladder is leaning against a house. The base of
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<span class="exam">[[022_Exam_1_Sample_A,_Problem_6 |<span class="biglink">&nbsp;Problem 6.&nbsp;</span>]] A 15-foot ladder is leaning against a house. The base of
 
the ladder is pulled away from the house at a rate of 2 feet per second.
 
the ladder is pulled away from the house at a rate of 2 feet per second.
 
How fast is the top of the ladder moving down the wall when the base
 
How fast is the top of the ladder moving down the wall when the base
of the ladder is 9 feet from the house.
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of the ladder is 9 feet from the house?
  
 
== Slope of Tangent Line ==
 
== Slope of Tangent Line ==
  
<span style="font-size:135%">Problem 7. Find the slope of the tangent line to the graph of <math style="vertical-align: -14%">f(x)=x^{3}-3x^{2}-5x+7</math>
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<span class="exam">[[022_Exam_1_Sample_A,_Problem_7 |<span class="biglink">&nbsp;Problem 7.&nbsp;</span>]] Find the slope of the tangent line to the graph of <math style="vertical-align: -14%">f(x)=x^{3}-3x^{2}-5x+7</math>
 
at the point <math style="vertical-align: -14%">(3,-8)</math>.
 
at the point <math style="vertical-align: -14%">(3,-8)</math>.
  
 
== Quotient and Chain Rule ==
 
== Quotient and Chain Rule ==
  
<span style="font-size:135%">Problem 8. Find the derivative of the function <math style="vertical-align: -43%">f(x)=\frac{(3x-1)^{2}}{x^{3}-7}</math>.
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<span class="exam">[[022_Exam_1_Sample_A,_Problem_8|<span class="biglink">&nbsp;Problem 8.&nbsp;</span>]] Find the derivative of the function <math style="vertical-align: -43%">f(x)=\frac{(3x-1)^{2}}{x^{3}-7}</math>.
 
You do not need to simplify your answer.
 
You do not need to simplify your answer.
  
 
== Marginal Cost ==
 
== Marginal Cost ==
  
<span style="font-size:135%">Problem 9. Find the marginal cost to produce one more item if the
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<span class="exam">[[022_Exam_1_Sample_A,_Problem_9 |<span class="biglink">&nbsp;Problem 9.&nbsp;</span>]] Find the marginal cost to produce one more item if the
 
fixed cost is $400, the variable cost formula is <math style="vertical-align: -5%">x^{2}+30x</math>,
 
fixed cost is $400, the variable cost formula is <math style="vertical-align: -5%">x^{2}+30x</math>,
 
and the current production quantity is 9 units.
 
and the current production quantity is 9 units.
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'''Contributions to this page were made by [[Contributors|John Simanyi]]'''

Latest revision as of 11:40, 28 July 2015

This is a sample, and is meant to represent the material usually covered in Math 22 up to the first exam. An actual test may or may not be similar. Click on the  boxed problem numbers  to go to a solution.

Definition of the Derivative

 Problem 1.  Use the definition of derivative to find the derivative of .

Implicit Differentiation

 Problem 2.  Use implicit differentiation to find at the point on the curve defined by .

Continuity and Limits

 Problem 3.  Given a function  ,

(a) Find the intervals where is continuous.
(b). Find .

Increasing and Decreasing

 Problem 4.  Determine the intervals where the function  is increasing or decreasing.

Marginal Revenue and Profit

 Problem 5.  Find the marginal revenue and marginal profit at , given the demand function

and the cost function

Should the firm produce one more item under these conditions? Justify your answer.

Related Rates (Word Problem)

 Problem 6.  A 15-foot ladder is leaning against a house. The base of the ladder is pulled away from the house at a rate of 2 feet per second. How fast is the top of the ladder moving down the wall when the base of the ladder is 9 feet from the house?

Slope of Tangent Line

 Problem 7.  Find the slope of the tangent line to the graph of at the point .

Quotient and Chain Rule

 Problem 8.  Find the derivative of the function . You do not need to simplify your answer.

Marginal Cost

 Problem 9.  Find the marginal cost to produce one more item if the fixed cost is $400, the variable cost formula is , and the current production quantity is 9 units.


Contributions to this page were made by John Simanyi