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Moving to the next question prevents changes to this answer:Question 2Let fx)ex_ +x2 and g(x) = 3x2 +2x Which statement is tne for the 2 functions? g(x) grows faste...

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Moving to the next question prevents changes to this answer:Question 2Let fx)ex_ +x2 and g(x) = 3x2 +2x Which statement is tne for the 2 functions? g(x) grows faster fOI x >f(x) grows faster for x > 2 g(x) grows faster for x > 2grows faster for x > [Movingto the Dert question prevents changes t0 this answer;PSC80Ooo%/

Moving to the next question prevents changes to this answer: Question 2 Let fx)ex_ +x2 and g(x) = 3x2 +2x Which statement is tne for the 2 functions? g(x) grows faster fOI x > f(x) grows faster for x > 2 g(x) grows faster for x > 2 grows faster for x > [ Movingto the Dert question prevents changes t0 this answer; PSC 80 Ooo %/



Answers

Which of the following functions grow faster than $x^{2}$ as $x \rightarrow \infty ?$ Which grow at the same rate as $x^{2} ?$ Which grow slower?
$$
\begin{array}{ll}{\text { a. } x^{2}+4 x} & {\text { b. } x^{5}-x^{2}} \\ {\text { c. } \sqrt{x^{4}+x^{3}}} & {\text { d. }(x+3)^{2}} \\ {\text { e. } x \ln x} & {\text { f. } 2^{x}} \\ {\text { g. } x^{3} e^{-x}} & {\text { h. } 8 x^{2}}\end{array}
$$

In this example, we have eight different functions are provided, and we're going to be comparing each one of these functions to f of X equals X squared as X goes to infinity. Well, there's three situations for the rates of growth. Will have functions that grow slower, will have functions that have the same rate of growth. And we'll also have functions that have a higher rate of growth. Let's begin categorizing these, starting with the first function X squared plus four x Well, this one has the same degree as a provide function f of X, which is to and so in this situation. It's more likely than not that the function will grow slower. But to be sure, let's look at the limit. As X goes to infinity, we will place X squared in the numerator divide by X squared plus four x in the denominator. So for this limit, we have a rational function, and so we know it will have a horizontal Assam tote with the equation Y equals one. And that came from the coefficients of one here and one here in the numerator and denominator. This then tells us that the limit and Infinity is one itself, which is a constant. But then that tells us that the growth rate is in fact, the same rate of growth for that function, so we can place why equals X squared plus four x into this column here. Let's color code this blue and place X squared plus four x here. Now let's go off to the next function for the function expire five minus X squared. We have a different situation where this time the degree is five, which soundly beats the function f of X here, and that tells us that this will certainly have a higher rate of growth. So Y equals X power. Five Mice X squared will belong to this column. Now for the third function with the Square root here, we have a different kind of situation where this is not a polynomial. If we consider the square root together, the power for that dominant term acts much like X squared, just like over here. That strongly suggests we should analyze this one by taking a limit. So let's look at the limit as X goes to infinity of all. Place the square root in the numerator x four plus X cubed and then the given function all place in the denominator, which is X squared. This limit could be evaluated by considering limit as X goes to infinity of X power for plus X, cubed the next squared converts to the square root of X power. For then, we could simplify inside the radical sign and obtain. After we divide, explore 41 plus one over X, the one over X tends to zero. So this goes to one and the limit. And that tells us with certainty that Y equals the square root of X Power. Four plus X Cubed does indeed have the same rate of growth now for the next function, which is X plus three to the power of to we can expand it and obtain altogether ex word plus six x plus nine. And in this situation, it's a degree to polynomial, which matches ours. So right off the bat, we know that Y equals expose. Three to the power of two will have the same rate of growth that takes us to a death next function that involves a natural lock. The natural law grows faster, slower than any function on this list and it's multiplying X. So our hunch is that this one should have a slower rate of growth bliss. Be extra sure by considering this limit as X goes to infinity, we can place X natural log X in the numerator divide by our function, which is X squared, and we know that this is going to have a limit of zero. Once we take an application, the low petals rule. Let's take a look at this. The numerator hence tends towards infinity, as does the denominator. So we have ah indeterminant form where low petals rule will apply. And the rule says that we can consider the limit as X goes to infinity, where we take the derivative of the numerator cell that will be one times natural log X plus x times, the derivative of natural log X, which is one over X and take the derivative of X squared in the denominator and we obtain a two X so this limit becomes limit as X goes to infinity of the natural log of X plus one. Divide by two X, it's still indeterminant form and so we can take another application of low petals rule detained the limit as X goes to infinity of derivative of natural Log X is one over x derivative of the two X, and the denominator is too and we ignore the constant since it goes to zero in the derivative. Now the one over X in the numerator goes to zero. So this overall has a limited zero. And that does tell us because X natural log X was placed in the numerator. That X squared has a higher rate of growth. And so why equals ex natural Log X will grow slower than f of X. Now we're at the third to the last function, which is to to the X. This one is special. This is an exponential function with base too. And the two here and the two here really do enter into the picture at all as soon as we have an exponential function, y equals two to the power of X will have a higher rate of growth than any polynomial. Even if the degree here was something ridiculous, say, like two million to the power of two million, we can still find a number X such that y calls to the X will eventually have a higher rate of growth, so let's now go to X cubed times e to the power of negative X. This one's a bit of a trick, since you know the power of negative X has the following graph. It tends towards zero as X goes to infinity and multiplying by X cubed does not change that fact. So that tells us that why equals X cubed times. Either the power of negative X will grow slower, then our degree to polynomial, and that's strictly due to the negative sign that's here and really nothing else. If we convert that to positive, it would have grown faster. Now we have one last function to consider, which is eight times X squared. But for this function, it is just a constant multiple of F of X and multiplying by Constant does not change the rate of growth. So this is another example where we have the same rate of growth y equals eight x squared, and that completes our analysis of these eight functions

In this example we have a set of functions there provided above that we see listed out here intercourse to compare this set of functions to the provided function f of X and specifically we're going to categorise them by their rate of growth. So let's work left to right, starting with X squared, plus the square to vex. Here we have a function that is involves a first term which is degree to which it matches are given function f of x. So if we look at the limit as X goes to infinity of X squared plus the root of X divide by are given function X squared will be able to compare their rates of growth first. This is an indeterminant form. Since both the numerator and denominator tend towards infinity. Sophie apply low petals rule. This will become the limit as X goes to infinity of first. The derivative of the numerator will be two x plus one over twice a square root of X, and the derivative in the denominator will be a two X as well. Now, this is still well, not quite in indeterminant form at this stage because as X goes to infinity. This portion goes toward zero, leaving the two exes. And so this time we have that the limit is equal to one. Exactly what this tells us then is at the functions, because we have a constant will have the same rate of growth so we can place why equals X squared, plus a squared of X into this column for the next comparison. Let's look at 10 to the power of X squared. It involves really the same function we have here for f of X butts been multiplied by a constant 10. And multiplication is by a constant. In fact, do not change the growth rate so we can place y equals 10 x squared into the same rate of growth column as well. The third function that we're going to consider is tricky, and that's due to the power eat negative X, The negative X power. Rather, if we were to graft that function, it's craft would look like this for y equals just e power Negative X. So the limit of this particular function is going to be zero, and we could verify that by low petals rule. This tells us that y equals X squared times e to the power Negative X will have a slower rate of growth. Now, if we go to a logger of them a log rhythm, Teoh, any base will have a very slow rate of growth. Really. The only thing that can help us make this get a high rate of growth is if we place the right exponential or type of function along those lines inside. But all together we could rewrite this function as to log based 10 of X, and we see that the X power to really doesn't have any magical effect. It's just a constant multiple and constant multiples like we saw before will not change the growth, growth rates and our definitions. So this four function y equals log base 10 of X squared. We'll also have a slower rate of growth that takes us to function five. Where the Keith about this function is it is a degree three polynomial as compared to our decree to polynomial F of X. If we find a polynomial of higher degree, it will automatically grow faster, so y equals X cubed minus X squared belongs to this column. Now, for the next function, we have an exponential function, but we can't get too excited about growth potential because the base be is 1/10 which is less than one. This means that a graph of this particular exponential function will look much like y calls either the negative X. It has a limit of zero as X goes to infinity. So this is for y equals 1/10 to the power of X, and that says this function also has a slower rate of growth. Now 1.1 to the power of X is a different situation. The base in this case is 1.1. And because the base is larger than one and because this is an exponential, it automatically will grow faster than f of X equals X squared. In fact, the power to is irrelevant here. This power to convey be any large, positive integer or real number, and still the exponential will grow faster. So why equals 1.1 to the power of X belongs in this column and recall that this is the key to knowing that the growth rate is faster. The base is strictly larger than one for the last cult. For the last function. Rather, we have degree too, and ah 100 times X, which sounds impressive. But I claim that this still has the same rate of growth because this 100 times X is of a degree one as a single term. Let's verify this. Let's take the limit as X goes off to infinity off that function X squared plus 100 X divide by our given function, which is X squared and calculate this limit. It's a indeterminant form as before, so it's hit it with low petals rule we have limit as X goes to infinity of two X plus 100 in the numerator. When we take its derivative divide by two X and the denominator, the still is in determinant of type infinity over infinity. So we'll just hit it with low P tells rule again. This is now the limit as X goes to infinity of in the numerator, we have just two and the denominator we have to as well. So we have a limit of one because a limit results in one which is a constant. We can say that the function y equals X squared plus 100 x has the same rate of growth, and that completes our analysis for these given functions

We'll compare the rates of group of these functions with the function X squared by looking at the limits of the quotients. So the limit is X approaches infinity of X squared, plus the square root of ax all over X squared that can be broken down by simplifying the fraction too. Just the limit is that's a purchase Anthony of one plus one over X to the three house. Since the limit is expressions ability of one of her exit three houses zero, we see that this limit is one which means that these two functions grow at the same rate. Now, if we're looking at the limit is except purchase infinity of 10 X squared over X weird. Well, the experts cancel and so are limited dressed 10 which means these functions grow at the same rate. If we're looking at the limit as X approaches infinity of X squared e to the negative X over X squared one scan. The X squared is cancelled out. And so we're actually looking at the limit as X approaches infinity of one over E to the X that will it zero. So these functions are sorry. The function X squared need to. The night of Ex grows at a slower rate than the function X squared. Looking at the limit is X approaches infinity of log Base tan of X squared over X squared. We can use a change of based formula to write the numerator in base eat. So we get the natural log of ax squared over the natural log of 10 all over X, where we can pull out the constant one over the natural log of 10. And so we will have the limit is extra purchase affinity of to Ellen of X. That's by using a rule of law algorithms that lets you bring the power down So this squared becomes a two out first all over X squared. Now he's lobbied house rule, and so we get to over the natural log of 10 times. The limit is X approaches infinity of one over X, divided by two ex. So we can just simplify this into two over the natural log of 10 ten's. The limit is X approaches infinity of one over two m X ray. Now that limit is going to go to zero, which means that log base 10 of X squared grows at a slower rate than the function X squared. Here we're looking at the limit This exit purchase infinity of X cubed minus X squared over X square so we can simplify the function. We're taking the limit of two just X minus one. So the limit this exit purchase and cindy of X minus one is infinity, which means that the function execute minus X squared grows at a faster rate than the function X squared. If we're looking at the limit is exit purchase infinity of one over 10 to the X power over X squared. We can rewrite this as the limit is X approaches infinity of one over 10 to the x Times X when that limit is zero, which means that the function wanna pretend to be X grows at a slower rate than the function X. Where if we're looking at the limit is X approaches infinity of 1.1 to the X power over X wed. We can use logic cultural here to evaluate the limit. So the derivative of the new Reiter becomes Ellen of 1.1 times 1.1 to the X power and our denominator becomes a to X. We'll use lobby casual a second time, so that we're looking at the limit is except purchase divinity of the natural log of 1.1 squared times 1.12 The acts all over too. So this limit goes to infinity, which means that 1.1 to the X power grows at a faster rate than the function ex win. Finally, the limit is X approaches infinity of X squared plus 100 acts all over X where can be reduced down to by using algebra. The limit is X approaches infinity of one plus 100 over axe Well, since 100 over exposed to zero as X goes to infinity This limit this one which means that the function X squared plus 100 ex grows at the same rate as the function X, where

Now, the question of the hell is asking you that has the value of X increases with the function effect equal to to wake all the function G s equal to to raise, to depart, eggs group faster, which among this will increase faster. And why so, simply as you can see we're here if you just take up, uh, say suppose when X is a coping, make a table of values over here, right? And then we find out the cash foreign values of the function epics as well as G s. That would be more convenient trade. You'll be able to compare it more visibly. So now let us save an excess called to to Then the function efforts will be equated to times where that is full and well. Here it will be too straight out is full. Okay, That means over here we are getting the same value. But what if you make your value of excess three over you, then the function F X will be equal to six, and the function gets will be quick to queue. That is Eek, isn't it? Let us take one more example. Let's say X is equal to four. Then the function F X will be equal to four times to that is eight, and the function jacks will be to rest. To depart for the rest of the powerful is what simply two times two times two times two That is 16. So this is from this table is really very clear that the function G X well group foster as compared to the function epics, isn't it? And what's the reason? The reason was just in front of you that, uh, for values of it's written and two X is a victim and two two ratio. The barracks is greater than two x.


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