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1 year ago

Compute the following without using a calculator. You must show your work to receive full credit.

Mathematics

1 answer:

Tomtit [17]1 year ago

3 0

Answer: 50,001²-49,999²=0,2.

Step-by-step explanation:

$50,001^2-49,999^2=(50,001+49,999)*(50,001-49,999)=100*0,002=0,2.$

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$\dfrac{\mathrm d}{\mathrm dx}(x^2e^x\log x)$

$=\dfrac{\mathrm d(x^2)}{\mathrm dx}e^x\log x+x^2\dfrac{\mathrm d(e^x)}{\mathrm dx}\log x+x^2e^x\dfrac{\mathrm d(\log x)}{\mathrm dx}$

Power rule:

$\dfrac{\mathrm d(x^2)}{\mathrm dx}=2x$

The exponential function is its own derivative:

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Assuming the base of $\log x$ is $e$ , its derivative is

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But if you mean a logarithm of arbitrary base $b$ , we have

$y=\log_bx\implies x=b^y=e^{y\ln b}\implies1=e^{y\ln b}\ln b\dfrac{\mathrm dy}{\mathrm dx}$

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$\implies\dfrac{\mathrm d(\log_bx)}{\mathrm dx}=\dfrac1{x\ln b}$

So we end up with

$2xe^x\log x+x^2e^x\log x+\dfrac{x^2e^x}x$

$=xe^x(2\log x+x\log x+1)$

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Answer:

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