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Nima M.
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Calculus
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Question:

Integrate $$\int ln(x)dx$$.

Nima M.

Let $$dv = dx$$ and $$u = ln(x)$$. By integration by parts $$uv - \int vdu$$. So we get: $$xln(x) - \int x \times \frac{1}{x}dx$$ = $$xln(x) - \intdx$$ which is consequently $$xln(x) - x + C$$ where C is an arbitrary constant

Statistics
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Question:

The probability of getting an 100% on a test is 3 % , 90 % on the test is 20% , 80 % on the test is 60% and a 70% on the test is 17%. What is the average grade of the test?

Nima M.

Using the definition of average = $$\Sigma_{i} x_{i}p_{i}$$ which results in $$100\times.03 + 90\times.2 + 80\times.6 + 70\times.17 = 80.9$$

Physics
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Question:

Explain why mathematically the gravitational pull of a black hole(the mass of the earth) 2cm in diameter is stronger than the Earth's pull at their respective surface.

Nima M.

Using Newton's Law of Gravitation is $$F = \frac{GM_{1}M_{2}}{r^2}$$ Assume inline LaTeX $$M_{1}$$ is 1 Kg and $$M_{2} = 5.972 \times 10^{24} kg$$ where the radii of the Earth and the black hole are $$r = 6.371 \times 10^{6}m, 10^{-2}m$$ Plugging in the formula, Force of attraction of Earth is $$9.81646 N$$ Force of attraction of the black hole is approximately $$3.98 \times 10^{18}$$

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