1,2,2,4,8,32,? what comes next?
2 answers:
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Answer:
Step-by-step explanation:
The area of a rectangle can is defined as where w is the width and l is the length. The width in this case is 12 and the length is 15. Also it doesn't really matter what order you put it in. This gives you an area of
It depends on the value of b
Exponential functions are defined for positive values of the base [ŧex] b [/tex]. Also, they're not defined if , or at least let's say that this is a trivial case, since it is the constant function 1.
Case 0<b<1:
If the base sits between 0 and 1, the exponential function is constantly decreasing. The explanation is quite intuitive, assume for the sake of simpleness that b is rational. If it sits between 0 and 1, it means that it can be written as a fraction p/q, with p<q. So, if you give a large exponent to b, you obtain
On the other hand, if you consider negative exponents, you switch numerator and denominator and then raise to the same exponent the fraction q/p, which gets larger and larger.
So, if 0<b<1, we have
and thus 0 is a horizontal asymptote as x tends to (positive) infinity.
Case b>1:
This case is very similar, except all roles are inverted. Now you start with a fraction p/q where p>q. So, with positive, large exponents you get
And as before, negative exponents switch numerator and denominator, so the fraction becomes q/p and thus you have
So, again, 0 is a horizontal asymptote, but this time for x tending towards negative infinite.