If a resistor of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R}
ohms is connected across a battery of 
volts with internal resistance 
ohms, then the power (in watts) in the external resistor is
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P={\frac {E^{2}R}{(R+r)^{2}}}.}
If and are fixed but Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R}
varies, what is the maximum value of the power?
| Background Information:
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| Recall
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Solution:
| Step 1:
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| Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)={\frac {1+x}{3x}}.}
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| Using the limit definition of derivative, we have
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {({\frac {1+(x+h)}{3(x+h)}})-({\frac {1+x}{3x}})}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {({\frac {1+x+h}{3x+3h}})-({\frac {1+x}{h}})}{h}}.}\end{array}}}
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| Step 2:
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| Now, we get a common denominator for the fractions in the numerator.
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| Hence, we have
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {{\frac {(1+x+h)3x}{(3x+3h)(3x)}}-{\frac {(1+x)(3x+3h)}{(3x+3h)(3x)}}}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {\frac {3x+3x^{2}+3xh-(3x+3h+3x^{2}+3hx)}{(3x+3h)(3x)}}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {-3h}{h(3x+3h)(3x)}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {-3}{(3x+3h)(3x)}}}\\&&\\&=&\displaystyle {\frac {-3}{(3x)(3x)}}\\&=&\displaystyle {-{\frac {1}{3x^{2}}}.}\end{array}}}
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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 \frac{dy}{dx}=-\frac{1}{3x^2}}
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