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="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>&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>.
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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>.
  
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 
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|A differential is a method of approximating a change,  
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|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
 
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::<math>\int x^n dn = \frac{x^{n+1}}{n+1} + C</math>
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::<math>m\,=\,\frac{\Delta y}{\Delta x},</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.
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|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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::<math>f'(x_0)\,=\,m\,=\,\frac{dy}{dx}.</math>
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|We can then rearrange this to find <math>dy=f'(x_0)\cdot dx.</math>
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Revision as of 19:12, 15 May 2015

Use differentials to approximate the change in profit given   units and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx = 0.2}   units, where profit is given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x) = -4x^2 + 90x - 128} .

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} as the slope of a line, and use the standard relation
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\,=\,\frac{\Delta y}{\Delta x},}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta y} represents the change in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} values, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x} represents the change in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} values. Due to the use of the derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'\left(x_0\right)} as the slope, we usually rewrite this using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dy} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx} to indicate the relative changes. Thus,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x_0)\,=\,m\,=\,\frac{dy}{dx}.}
We can then rearrange this to find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dy=f'(x_0)\cdot dx.}

 Solution:

Step 1:  
Step 2:  

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