Difference between revisions of "022 Exam 2 Sample A, Problem 8"
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(Created page with "Use differentials to approximate the change in profit given <math style="verticalalign: 5%">x = 10</math> units and <math style="verticalalign: 0%">dx = 0.2</math>&...") 

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−  Use differentials to approximate the change in profit given <math style="verticalalign: 5%">x = 10</math>  units and <math style="verticalalign: 0%">dx = 0.2</math>  units, where profit is given by <math style="verticalalign: 23%">P(x) = 4x^2 + 90x  128</math>.  +  <span class="exam">Use differentials to approximate the change in profit given <math style="verticalalign: 5%">x = 10</math>  units and <math style="verticalalign: 0%">dx = 0.2</math>  units, where profit is given by <math style="verticalalign: 23%">P(x) = 4x^2 + 90x  128</math>. 
{ class="mwcollapsible mwcollapsed" style = "textalign:left;"  { class="mwcollapsible mwcollapsed" style = "textalign:left;"  
!Foundations:  !Foundations:  
    
−  A differential is a method of approximating a  +  A differential is a method of linearly approximating the change of a function. We use the derivative of the function at an initial point <math style="verticalalign: 0%">x_0</math> as the slope of a line, and use the standard relation 
    
    
−  ::<math>\  +  ::<math>m\,=\,\frac{\Delta y}{\Delta x},</math> 
    
−    +  where <math style="verticalalign: 20%">\Delta y</math> represents the change in <math style="verticalalign: 20%">y</math> values, and <math style="verticalalign: 0%">\Delta x</math> represents the change in <math style="verticalalign: 0%">x</math> values. Due to the use of the derivative <math style="verticalalign: 22%">f'\left(x_0\right)</math> as the slope, we usually rewrite this using <math>dy</math> and <math style="verticalalign: 0%">dx</math> to indicate the relative changes. Thus, 
+    
+    
+  ::<math>f'(x_0)\,=\,m\,=\,\frac{dy}{dx}.</math>  
+    
+  We can then rearrange this to find <math>dy=f'(x_0)\cdot dx.</math>  
+    
+  
}  }  
Revision as of 20:12, 15 May 2015
Use differentials to approximate the change in profit given units and units, where profit is given by .
Foundations: 

A differential is a method of linearly approximating the change of a function. We use the derivative of the function at an initial point as the slope of a line, and use the standard relation 

where represents the change in values, and represents the change in values. Due to the use of the derivative as the slope, we usually rewrite this using and to indicate the relative changes. Thus, 

We can then rearrange this to find 
Solution:
Step 1: 

Step 2: 
