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Which function "grows" faster, a linear function or an exponential function and why?

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    $\begingroup$ Did you computed its derivative? $\endgroup$ – Sigur Nov 12 '13 at 0:11
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    $\begingroup$ What are your thoughts on the matter? Also this has nothing to do with linear algebra. $\endgroup$ – Cameron Williams Nov 12 '13 at 0:11
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Take derivatives:

$y_1 = mx + b \Longrightarrow y_1' = m$ where $m$ is a constant

OR

$y_2 = a^x \Longrightarrow y_2' = a^x ln(a)$

It should be clear that $y_1$ is changing at a constant pace whereas $y_2$ is changing with respect to a variable. Eventually it will surpass the rate of the linear function, no matter how big, and then the linear function itself. (But only for certain $a$, and you should verify this).

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For each unit of increase in its argument, a linear (increasing) function goes up by the same amount. For an (increasining) exponential function, this increase is by the same multiplicative factor.

So, what do you think now?

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