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Tutor profile: Jason C.

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Jason C.
Tutor from MIT
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Questions

Subject:Python Programming

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Question:

Write a function that takes two integers and returns their greatest common divisor.

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Jason C.

The greatest common divisor is at most the smallest of the two inputs. Therefore, we can just loop up to the minimum of the two inputs and check: $$\verb|def f(a, b):|$$ $$\verb| common_divisors = []|$$ $$\verb| for d in range(1, min(a, b) + 1):|$$ $$\verb| if a%d == 0 and b%d == 0:|$$ $$\verb| common_divisors.append(d)|$$ $$\verb| return max(common_divisors)|$$

Subject:Pre-Calculus

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Question:

Evaluate $$(1+i\sqrt 3) e^{-i \pi/3}$$.

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Jason C.

First, we will convert $$e^{-i \pi/3}$$ to rectangular form: $( e^{-i \pi/3} = \cos (-\pi/3) + i \sin (-\pi/3) = \frac{1}{2} - \frac{\sqrt 3}{2}.$) Then, we take the product of the two terms: (\begin{align*} (1 + i \sqrt 3) \left( \frac{1}{2} - i \frac{\sqrt 3}{2} \right) &= \left( \frac{1}{2} - i \frac{\sqrt 3}{2} \right) + i \sqrt 3 \left( \frac{1}{2} - i \frac{\sqrt 3}{2} \right) \\ &= \frac{1}{2} - i \frac{\sqrt 3}{2} + i \frac{\sqrt 3}{2} + \frac{3}{2} \\ &= 2. \end{align*})

Subject:Calculus

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Question:

Evaluate $( \oint_C [(x^2 - y^2)\,dx + (x^2 + y^2)\,dy],$) where $$C$$ is the boundary of the square with vertices $$(0, 0)$$, $$(1, 0)$$, $$(1, 1)$$, and $$(0, 1)$$, oriented counterclockwise

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Jason C.

By Green's theorem, we know that the integral about the boundary of region is equal to the integral of the divergence over the region. (\begin{align*}\oint_C [(x^2 - y^2)\,dx + (x^2 + y^2)\,dy] &= \iint_{[0, 1]^2} \left[\frac{d}{dx}(x^2 + y^2) - \frac{d}{dy}(x^2 - y^2)\right] \,dA \\ &= \int_0^1 \int_0^1 [2x + 2y] \,dx\,dy = 2.\end{align*})

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