022 Exam 2 Sample A, Problem 8 - Revision history

MathAdmin at 04:13, 16 May 2015

2015-05-16T04:13:26Z

MathAdmin at 03:12, 16 May 2015

2015-05-16T03:12:29Z

MathAdmin: Created page with "Use differentials to approximate the change in profit given $x = 10$ units and $dx = 0.2$ &..."

2015-05-16T02:52:20Z

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>&..."

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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>  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;"
!Foundations:  
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|A differential is a method of approximating a change,
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::<math>\int x^n dn = \frac{x^{n+1}}{n+1} + C</math>
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|For setup of the problem we need to integrate the region between the x - axis, the curve, x = 1, and x = 4.
|}

 '''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
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[[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]

@@ 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;"
 !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,
 |}

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 <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>.
@@ Line 24: / Line 25: @@
 !Step 1: &nbsp;
 |-
+|First, we must find the derivative.  We have <math style="vertical-align: -23%">P'(x) = -8x + 90</math>.
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 2: &nbsp;
 |-
+|
 |}
 [[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]

022 Exam 2 Sample A, Problem 8 - Revision history

MathAdmin at 04:13, 16 May 2015

MathAdmin at 03:12, 16 May 2015

MathAdmin: Created page with "Use differentials to approximate the change in profit given x = 10 units and dx = 0.2&..."

MathAdmin: Created page with "Use differentials to approximate the change in profit given $x = 10$ units and $dx = 0.2$ &..."