# Calculus in Games

Calculus Level 3

Max is a virtual character in a particular 2D game. He has the choice of $$2$$ functions (curve) to walk on to collect coins.

• Road 1 $$\frac{x^3}{6} + \frac{1}{4x}$$ from $$0 \leq x \leq 5$$ where there are $$20$$ coins evenly distributed along this arc.
• Road 2 $$\ln {(\sec {x}})$$ from $$0 \leq x \leq \frac{\pi}{3}$$ where there is $$1$$ coin along this arc.

Which function is the most coin-to-distance efficient for him to take? (Meaning which function should he walk to gain more coin in a unit distance?)

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