Difference between revisions of "022 Exam 2 Sample A, Problem 8"

From Math Wiki
Jump to navigation Jump to search
(Created page with "Use differentials to approximate the change in profit given <math style="vertical-align: -5%">x = 10</math>  units and <math style="vertical-align: 0%">dx = 0.2</math>&...")
 
Line 1: Line 1:
Use differentials to approximate the change in profit given <math style="vertical-align: -5%">x = 10</math>&thinsp; units and <math style="vertical-align: 0%">dx = 0.2</math>&thinsp; units, where profit is given by  <math style="vertical-align: -23%">P(x) = -4x^2 + 90x - 128</math>.
+
<span class="exam">Use differentials to approximate the change in profit given <math style="vertical-align: -5%">x = 10</math>&thinsp; units and <math style="vertical-align: 0%">dx = 0.2</math>&thinsp; units, where profit is given by  <math style="vertical-align: -23%">P(x) = -4x^2 + 90x - 128</math>.
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 
|-
 
|-
|A differential is a method of approximating a change,  
+
|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="vertical-align: 0%">x_0</math> as the slope of a line, and use the standard relation
 
|-
 
|-
 
|
 
|
::<math>\int x^n dn = \frac{x^{n+1}}{n+1} + C</math>
+
::<math>m\,=\,\frac{\Delta y}{\Delta x},</math>
 
|-
 
|-
|For setup of the problem we need to integrate the region between the x - axis, the curve, x = 1, and x = 4.
+
|where <math style="vertical-align: -20%">\Delta y</math> represents the change in <math style="vertical-align: -20%">y</math> values, and <math style="vertical-align: 0%">\Delta x</math> represents the change in <math style="vertical-align: 0%">x</math> values. Due to the use of the derivative <math style="vertical-align: -22%">f'\left(x_0\right)</math> as the slope, we usually rewrite this using <math>dy</math> and <math style="vertical-align: 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:  

Return to Sample Exam