82–89. Comparing growth rates Determine which of the two funct...

Question

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

x¹⸍² and x¹⸍³

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. In comparing the growth rates of the functions, Y equals 5x the power of 2/3 and Y equals 4x the power of 1/8 for large values of X, which function grows faster. So that's our angle. Our final answer that we're ultimately trying to solve first we're trying to determine which of these two functions grows faster. And that is what we're ultimately trying to solve for. So with that in mind, let's read off our multiple choice answers to see what our final answer might be. A is Y equals 5x the power of 2/3 or 2 divided by 3. B is Y equals 4 x 1/8 or 1 divided by 8. C is both functions grow at the same rate, and D is the growth rate cannot be compared. OK. So first off, as we should recall and note, in order to compare the growth rates of Y equals 5 x 2/3 and Y equals 4 x 1/8. Using, we can use the so the the method that we could use in order to compare these two functions is we can recall and use Wahapatal's rule. So we need to examine the limit of their ratios as X approaches infinity. So let's make a note of that. So X approaches infinity. So once again, so we can compare the growth rates of our two functions using Lajapatal's rule, and we will also need to examine the limit of their ratio as X approaches infinity. So we need to compute the limit of the ratio. So in order to do that, we can go ahead and write that the limit as X approaches infinity of 5 X to the power of 2/3, divided by 4 X to the power of 1/8. And that will be our ratio. And now our first step to this that we need to take for the overall solving process is we need to set up the limit expression. So we need to note that we have that the limit as X approaches infinity of 5 X to the power of 2/3 divided by 4x the power of 1/8. It's going to be equal to 5/4 multiplied by the limit as X approaches infinity of X to the power of 2/3 minus 1/8. Awesome. So now we need to simplify the exponent. So let us note that 2/3 minus 1/8 is going to be equal to 16/24 minus 32 3 divided by 24, so 324 or 16 divided by 24 minus 3 divided by 24, which simplifies 2, 13 divided by 24. So now that we've simplified our exponent, our limit will simply become 5/4s multiplied by the limit as X approaches infinity of X to the power of 13 divided by 24 or 13 24th. So now, our second step is we need to evaluate the limit directly, and we need to note that since X to the power of 13 divided by 24 grows without bound as X approaches infinity, the limit of X to the power of 13 divided by 24, as X approaches infinity is infinite. So, thus the limit also tends to infinity. So therefore, without a doubt, we could go ahead and write that the limit as X approaches infinity of. 5 x 2/3 divided by 4 X 1/8. We can say that this is equal to infinity. So in conclusion, since the limit tends to infinity, that will mean that Y equals 5x the power of 2/3 will grow faster than Y equals 4x the power of 1/8 as X approaches infinity. So that means that our final answer. Without a doubt, has to be multiple choice answer letter A, which once again is Y equals 5x the power of 2/3 or 5 X to the power of 2 divided by 3. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

x¹⸍² and x¹⸍³

Key Concepts

Growth Rates of Functions

Limits and Asymptotic Behavior

Power Functions

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