Use differentials to approximate the change in profit given Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=10}
units and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dx=0.2}
units, where profit is given by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P(x)=-4x^{2}+90x-128}
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| Foundations:
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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  as the slope of a line, and use the standard relation
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- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m\,=\,{\frac {\Delta y}{\Delta x}},}
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where  represents the change in  values, and  represents the change in  values. Due to the use of the derivative Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'\left(x_{0}\right)}
as the slope, we usually rewrite this using  and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dx}
to indicate the relative changes. Thus,
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- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x_{0})\,=\,m\,=\,{\frac {dy}{dx}}.}
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| We can then rearrange this to find Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dy=f'(x_{0})\cdot dx.}
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Solution:
| Step 1:
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| First, we must find the derivative. We have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P'(x)=-8x+90}
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| Step 2:
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| We need the derivative at our initial point, or 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 = 10}
. This is
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- 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_0)\,=\,P'(10) \,=\, -8(10) + 90\,=\,10.}
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| Step 3:
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| Finally, we plug in the values to find
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- 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\,=\,P'(x_0)\cdot dx\,=\,10\cdot 0.2\,=\,2.}
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| Note that if a teacher gives you units (thousands of dollars, dollars, cubits...), you should include them in your answer.
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| Final Answer:
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- 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\,=\,2.}
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