Question

does a^x increase faster than x^a? Well, generally?

Answer 1

yes exponential function is the boss

Answer 2

Yes, exponential function increases exponentially so on a graph it is approaching infinity faster than any other function

Answer 3

not if \(a<1\)

Answer 4

consider below functions :

\(f(x) = x^{100}\)

\(g(x) = 100^x\)

Answer 5

Yes, assuming a is your constant and x is your increasing variable.

Answer 6

@satellite73 good point

Answer 7

there will be some "x" value at which the g(x) overtakes f(x) and after that g(x) > f(x) forever

Answer 8

@satellite73 how come?

Answer 9

The smaller you get, the more it decreases if a < 1

Answer 10

what are u talking about fractions less than 1, or negative numbers?

Answer 11

Negative numbers

Answer 12

wait, if (-a)? or if -(a)?

Answer 13

may be we can split it into cases for clarity :
1) \(a \gt 1\)
2) \(a = 1\)
3) \(0 \lt a \lt 1\)

Answer 14

Look at ratio (a^x) / (x^a) or
x log a - a log x
x log a will dominate for large x

Answer 15

@ganeshie8 ...so..
1) a>1....+
2) a=1....1
3) 0<a<1......0

Answer 16

Also graph both functions on the calculator to see how fast they go up

Answer 17

@douglaswinslowcooper what hold on waht do u mean by the logs?

Answer 18

Even faster, put the functions in google search and google will show you the graphs

Answer 19

@jtryon uhm...and then I tried the -a thing, and I'n confused on whether @satellite73 meant (-a)^x or -(a)^x?

Answer 20

If a < 1 then a is negative so any negative numbers and it depends on the constant and variable

Answer 21

?

Answer 22

\(a^x\) is not defined for \(a < 0\), dont wry about it for now..

Answer 23

Don't worry about it lucy, just know that an exponential equation increases much faster than a standard equation

Answer 24

okay then~ ^^

Answer 25

thnx

Answer 26

\(f(x) = x^a\)
\(g(x) = a^x\)

\(\color{red}{a = 1}\) is a boring case :
\(f(x) = x^1 = x\)
\(g(x) = 1^x = 1\)

Answer 27

clearly the polynomial dominates here as `y = x` grows, where as `y = 1` is stuck at 1 always

Answer 28

\(f(x) = x^a\)
\(g(x) = a^x\)

\(\color{red}{a \gt 1}\) :
\(f(x) = x^2 = x^2\)
\(g(x) = 2^x = 2^x\)

graph them both and get a feel of how they're increasing

Answer 29

0-0 woahhhhh ic~

Answer 30

I think the underlying assumption is that we ar to predict behavior as x gests very large.
I went to logs because if f(x) dominates g(x), then so does log f dominate log g
and sometimes the behavior is clearer.

Answer 31

@ganeshie8 wait but x^2 overtakes 2^x

Answer 32

in the long run, it never happens

Answer 33

oh wait nvm then it overtakes x^2 again

Answer 34

exactly ! i think x^2 is large between 2 and 4 ?
but the exponential function is always high for x > 4

Answer 35

you can always find such "x" after which exponential function is higher than the polynomial

Answer 36

@douglaswinslowcooper ahhh ic~ nice!

Answer 37

its not just for x^a and a^x case,

consider below :
1) \(f(x) = x^{1000000000}\)
2) \(g(x) = 2^x\)

Answer 38

who wins ?

Answer 39

\[\lim \limits_{x \to \infty} \dfrac{x^{1000000000}}{2^x}\]

Answer 40

hmmm my graphing thingies arent relly doing so well with x^1000 lol

Answer 41

use use calculus knowledge

Answer 42

take the limit of that ratio

Answer 43

\[\lim \limits_{x \to \infty} \dfrac{x^{1000000000}}{2^x} = 0\]

Answer 44

0-0 watttttt

Answer 45

if u graph both functions,
and go right far enough, u wil see that exponent function is much much greater than the polynomial that the ratio is almost 0

Answer 46

for ex :

\(\dfrac{10}{100000000000} \approx 0\)

Answer 47

ahh ok that makes sense ^^ It's just that I got confused when I wrote it down and was liek...they both go to 100 TT^TT

Answer 48

srry not 100. i meant infinity LOL

Answer 49

:) they both increase fast, but exponent function increases faster than polynomial

Answer 50

so for larger values of x, exponential function is always higher than the polynomial

Answer 51

aweseom ^^ thnx
I might be back tonight perhaps for integration stuff sooooo ^^ cya!

Answer 52

this may help :

1) \(f(x) = 100000000000000000000000\)
2) \(g(x) = x\)

Answer 53

which function grows fast ?

Answer 54

ahhhh x

Answer 55

yes, and if u go right side far enough ,u will see "x" overtaking the other million dollar funciton

Answer 56

I love limits so much hahaha XD hating integration... a bit confused as to riemann sums, but i suppose that comes with practice... also confused on derivative table problems... but i suppose that comes with practice too...I have a calc final tomorrow and the ap test on wednesday LOL hope I survive

Answer 57

no one in their right mind computes a riemann sum, except via a computer
that is why the fundamental theorem of calculus is ... fundamental

Answer 58

XD ikr??? i dont even know y the ap test would make us do that...

Answer 59

hahah true that xD

u may like watching this debate.. when u have time :
https://www.youtube.com/watch?v=iNtMLGvzFHA

Answer 60

awesome ^^ thanks