[Math] What does the hyp checkbox do on calculators

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What does the hyp checkbox do on calculators (for example on the Microsoft Windows calculator)?

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It's used to select hyperbolic functions (i.e. hyperbolic sine, cosine).

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Calculator – How Pocket Calculators Calculate Exponents

I would be surprised if they actually used Taylor series. For example, the basic 80x87 "exponential" instruction is F2XM1 which computes $2^x-1$ for $0 \le x \le 0.5$. I don't think the implementation is documented, but if I were programming this, I might use a minimax polynomial or rational approximation: the following polynomial of degree $9$ approximates $2^x - 1$ on this interval with maximum error approximately $1.57 \times 10^{-17}$:

-0.15639e-16+(.69314718055995154416+(.24022650695869054994+
(0.55504108675285271942e-1+(0.96181289721472527028e-2
+(0.13333568212100120656e-2+(0.15403075872034576440e-3
+(0.15265399313676342402e-4+(0.13003468358428470167e-5
+0.12113766044841794408e-6*x)*x)*x)*x)*x)*x)*x)*x)*x

By contrast, the Maclaurin polynomial of the same degree has maximum error about $7.11 \times 10^{-12}$ on this interval.

Why cant algebra calculators without graphing involved solve this equation $2^x=3x-1$

Polynomials and exponentials do not play nicely together, so a routine algebraic solution will not work. People are good at finding clever solutions, but calculators are not. People are biased toward toward integer solutions, so can see that $1$ is a solution (somewhat) easily. Calculators do not have the same bias, so $1$ is no different from $0.998$ to them.

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Calculator – How Pocket Calculators Calculate Exponents

Why cant algebra calculators without graphing involved solve this equation $2^x=3x-1$

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