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Each of these functions can be computed using the stack, or using algebraic entry, whichever way you prefer:
Computing sin(x) / x:
Using the stack: C-x ( RET S TAB / C-x ).
Using algebraic entry: C-x ( ' sin($)/$ RET C-x ).
Computing the logarithm:
Using the stack: C-x ( TAB B C-x )
Using algebraic entry: C-x ( ' log($,$$) RET C-x ).
Computing the vector of integers:
Using the stack: C-x ( 1 RET 1 C-u v x C-x ). (Recall that C-u v x takes the vector size, starting value, and increment from the stack.)
Alternatively: C-x ( ~ v x C-x ). (The ~ key pops a number from the stack and uses it as the prefix argument for the next command.)
Using algebraic entry: C-x ( ' index($) RET C-x ).
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