Difference between revisions of "007A Sample Final 1"

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== [[007A_Sample Final 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
 
== [[007A_Sample Final 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
text
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<span class="exam">The equation of motion of a particle is
 +
 
 +
::<math>s=2t^3-7t^2+4t+1</math>
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<span class="exam">where &nbsp;<math style="vertical-align: 0px">s</math>&nbsp; is in meters and &nbsp;<math style="vertical-align: 0px">t</math>&nbsp; is in seconds.
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<span class="exam">(a) Find the velocity and acceleration as functions of &nbsp;<math style="vertical-align: 0px">t.</math>
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<span class="exam">(b) Find the acceleration after 1 second.
  
 
== [[007A_Sample Final 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
 
== [[007A_Sample Final 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
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== [[007A_Sample Final 1,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
 
== [[007A_Sample Final 1,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
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<span class="exam">A curve is defined implicitly by the equation
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::<math>x^3+y^3=6xy.</math>
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<span class="exam">(a) Using implicit differentiation, compute &nbsp;<math style="vertical-align: -12px">\frac{dy}{dx}</math>.
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<span class="exam">(b) Find an equation of the tangent line to the curve &nbsp;<math style="vertical-align: -4px">x^3+y^3=6xy</math>&nbsp; at the point &nbsp;<math style="vertical-align: -5px">(3,3)</math>.
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== [[007A_Sample Final 1,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
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<span class="exam">Consider the following continuous function:
  
<span class="exam">Let
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::<math>f(x)=x^{\frac{1}{3}}(x-8)</math>
  
::<math>y=x^3.</math>
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<span class="exam">defined on the closed, bounded interval &nbsp;<math style="vertical-align: -5px">[-8,8].</math>
  
<span class="exam">(a) Find the differential &nbsp;<math style="vertical-align: -4px">dy</math>&nbsp; of &nbsp;<math style="vertical-align: -4px">y=x^3</math>&nbsp; at &nbsp;<math style="vertical-align: 0px">x=2</math>.
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<span class="exam">(a) Find all of the critical points for &nbsp;<math style="vertical-align: -5px">f(x).</math>
  
<span class="exam">(b) Use differentials to find an approximate value for &nbsp;<math style="vertical-align: -1px">1.9^3</math>.
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<span class="exam">(b) Determine the absolute maximum and absolute minimum values for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; on
  
== [[007A_Sample Final 1,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
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<span class="exam">the interval &nbsp;<math style="vertical-align: -5px">[-8,8].</math>
  
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== [[007A_Sample Final 1,_Problem_10|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 10&nbsp;</span>]] ==
 
<span class="exam">Given the function &nbsp;<math style="vertical-align: -5px">f(x)=x^3-6x^2+5</math>,  
 
<span class="exam">Given the function &nbsp;<math style="vertical-align: -5px">f(x)=x^3-6x^2+5</math>,  
  
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<span class="exam">(e) Use the above information (a) to (d) to sketch the graph of &nbsp;<math style="vertical-align: -5px">y=f(x)</math>.
 
<span class="exam">(e) Use the above information (a) to (d) to sketch the graph of &nbsp;<math style="vertical-align: -5px">y=f(x)</math>.
 
== [[007A_Sample Final 1,_Problem_10|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 10&nbsp;</span>]] ==
 
 
<span class="exam">If a resistor of &nbsp;<math style="vertical-align: 0px">R</math>&nbsp; ohms is connected across a battery of &nbsp;<math style="vertical-align: 0px">E</math>&nbsp; volts with internal resistance &nbsp;<math style="vertical-align: 0px">r</math>&nbsp; ohms, then the power (in watts) in the external resistor is
 
 
::<math>P=\frac{E^2R}{(R+r)^2}.</math>
 
 
<span class="exam">If &nbsp;<math style="vertical-align: 0px">E</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">r</math>&nbsp; are fixed but &nbsp;<math style="vertical-align: 0px">R</math>&nbsp; varies, what is the maximum value of the power?
 

Latest revision as of 23:01, 2 December 2017

This is a sample, and is meant to represent the material usually covered in Math 7A for the final. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity.

(a)  

(b)  

(c)  

 Problem 2 

Consider the following piecewise defined function:

(a) Show that    is continuous at  

(b) Using the limit definition of the derivative, and computing the limits from both sides, show that    is differentiable at  .

 Problem 3 

Find the derivatives of the following functions.

(a)  

(b)  

 Problem 4 

The equation of motion of a particle is

where    is in meters and    is in seconds.

(a) Find the velocity and acceleration as functions of  

(b) Find the acceleration after 1 second.

 Problem 5 

If   compute    and find the equation for the tangent line at  

You may leave your answers in point-slope form.

 Problem 6 

A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing when 50 (meters) of the string has been let out?

 Problem 7 

Consider the following function:

(a) Use the Intermediate Value Theorem to show that    has at least one zero.

(b) Use the Mean Value Theorem to show that    has at most one zero.

 Problem 8 

A curve is defined implicitly by the equation

(a) Using implicit differentiation, compute  .

(b) Find an equation of the tangent line to the curve    at the point  .

 Problem 9 

Consider the following continuous function:

defined on the closed, bounded interval  

(a) Find all of the critical points for  

(b) Determine the absolute maximum and absolute minimum values for    on

the interval  

 Problem 10 

Given the function  ,

(a) Find the intervals in which the function increases or decreases.

(b) Find the local maximum and local minimum values.

(c) Find the intervals in which the function concaves upward or concaves downward.

(d) Find the inflection point(s).

(e) Use the above information (a) to (d) to sketch the graph of  .