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Tutor profile: Michael S.

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Michael S.
Recent college grad tutoring in computer science, Spanish language, and various maths
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Questions

Subject: Spanish

TutorMe
Question:

¿Cuál es el lugar más interesante que has visitado?

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Michael S.
Answer:

The question reads "what is the most interesting place you've ever visited?"

Subject: Computer Science (General)

TutorMe
Question:

How would you find the number missing from an array that originally contained the integers from 1 to 100 (inclusive)?

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Michael S.
Answer:

Calculate the sum of the integers in the array and subtract this sum from the formula for the sum of consecutive integers from 1 to n: $$\frac{1}{2} \times n(n+1)$$. The difference is the number missing from the array.

Subject: Calculus

TutorMe
Question:

Which is larger: $$e^{\pi}$$ or $$\pi^{e}$$?

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Michael S.
Answer:

Consider the Taylor expansion for $$e^{x}$$: $$e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots $$ The partial sum $$1+x$$ is less than the infinite expansion of $$e^{x}$$, so $$e^{x} > 1+x$$. Choose an $$x$$ to simplify the terms. Let $$x = \frac{\pi}{e} - 1$$. Then $$e^{\frac{\pi}{e} - 1} > 1 + \frac{\pi}{e} - 1$$ holds. Simplifying gives $$e^{\frac{\pi}{e}} > \pi$$. Raise both sides to exponent $$e$$ to obtain the result: $$e^{\pi} > \pi^{e} \blacksquare$$

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