Write and simplify an expression for the time needed for the round trip as a function of the boat's speed. Express the team's time on the upstream leg as a function of the speed of the current. Write a function for the team's time on the downstream leg. Write and simplify an expression for the total time for the training run as a function of the current's speed.
Two pilots for the Flying Express parcel service receive packages simultaneously. The prevailing winds blow from east to west. Express Orville's flying time as a function of the windspeed. Write a function for Wilbur's flying time. Who reaches his destination first?
The higher degree terms are going to grow much faster than the lesser degree terms. And so, we could say that for large x, for large x, and when I say "large" I mean high absolute value. High absolute value. And if we're going to negative infinity, that's high absolute value. So, f of x is going to be approximately equal to the highest degree term on the top, which is 7x-squared, divided by the highest degree term on the bottom.
So, as this becomes larger and larger and larger, this is going to matter a lot, lot less. So, it's going to be approximately that. Which is equal to 7x over Well, even here, if you think about what happens when x becomes very, very negative here.
Well, you're just gonna get larger, you're gonna get more and more and more negative values for f of x. So, once again, f of x itself is going to approach, is going to go to, negative infinity as x goes to negative infinity. Let's do another one of these. So, here they're telling us to find the horizontal asymptote of q.
A horizontal asymptote, you can think about it as what is the function approaching as x becomes, as x approaches infinity, or as x approaches negative infinity. And just as a couple of examples here. It's not necessarily the q of x that we're focused on. But you could imagine a function, let's say it has a horizontal asymptote at y is equal to two, so that's y is equal to two there.
Let me draw that line. So, let's say it has a horizontal asymptote like that. Well then the graph could look something like this. It could look, let me draw a couple of them that have horizontal asymptotes. So, maybe it's over here, it does some stuff, but as x gets really large, it starts approaching, the function starts approaching that y equals two without ever quite getting there. And it could do that on this side as well. As x becomes more and more negative.
As it gets more negative, it approaches it without ever getting there. Or, it could do something like this. You could have, if it has a vertical asymptote, too, it could look something like this. Where it approaches the horizontal asymptote from below, as x becomes more negative, and from above, as x becomes more positive.
Or vice versa. So, this is just a sense of what a horizontal asymptote is. It'll show you what's the behavior, what value is this function approaching, as x becomes really positive or x becomes really negative. Well, let's just think about it. We could essentially do what we just did in that last example. What happens if we were to, if we were to divide all of these terms by the highest degree term in the denominator?
Well, if we divide, so q of x is going to be equal to, the highest degree term in the denominator is x to the ninth power. So, we could say six, 6x to the fifth divided by x to the ninth is going to be six over x to the fourth.
And then minus two times x to the ninth. All of that over three over, I'm gonna divide this by x to the ninth, x to the seventh, plus one. Well, if x approaches positive or negative infinity, six divided by arbitrarily large numbers, that's gonna go to zero. Two divided by arbitrarily large numbers, whether they are positive or negative, that's going to go to zero. So your numerator's clearly gonna go to zero. This term of the denominator, three divided by arbitrarily large numbers, whether we're going in the positive or the negative direction, it is gonna approach zero.
It'll approach zero from the negative direction, or we could say from below. If we're dealing with very negative x's. If we're dealing with very positive x's, then we're going to approach zero from above. We're gonna get smaller and smaller positive values. So, all of these things go to zero and this right over here is going to be, would stay at, one. And so if you're approaching zero in your numerator and approaching one in your denominator, the whole thing is going to approach zero.
So, in the case of q of x, you have a horizontal asymptote at y is equal to zero. I don't know exactly what the graph looks like but we could draw a horizontal line at y equals zero and it would approach it. It would approach it from above or below. Let's do one more. What does f of x approach as x approaches negative infinity?
INVESTING ADVICE APPS
Solution Start by factoring the numerator and denominator, if possible. In this example, there are no factors that cancel. To find the vertical asymptotes, set the denominator equal to zero and solve for x. Removable Discontinuities Sometimes a graph of a rational function will contain a hole. A hole is a single point where the graph is not defined and is indicated by an open circle. These holes come from the factors of the denominator that cancel with a factor of the numerator.
When the function is simplified, the hole disappears. Factor the numerator and denominator. If any factors are common to both the numerator and denominator, set it equal to zero and solve. This is the location of the removable discontinuity. Solution Factor the numerator and the denominator. Horizontal asymptotes can be found by substituting a large number like 1,, for x and estimating y.
There are three possibilities for horizontal asymptotes. Let N be the degree of the numerator and D be the degree of the denominator. Substitute in a large number for x and estimate y. Figure 8: No horizontal asymptote when the degree of the numerator is greater than the degree of the denominator.
Slant Asmptotes Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. To find the equation of the slant asymptote, divide the fraction and ignore the remainder. Figure 9: Slant Asymptote when the degree of the numerator is 1 more than the degree of the denominator.
Notice that a graph of a rational function will never cross a vertical asymptote, but the graph may cross a horizontal or slant asymptote. Also, the graph of a rational function may have several vertical asymptotes, but the graph will have at most one horizontal or slant asymptote. In general, if the degree of the numerator is larger than the degree of the denominator, the end behavior of the graph will be the same as the end behavior of the quotient of the rational fraction.
The location of the horizontal asymptote is determined by looking at the degrees of the numerator n and denominator m. That is, the ratio of the leading coefficients. Holes Sometimes, a factor will appear in the numerator and in the denominator. Let's assume the factor x-k is in the numerator and denominator. This means that one of two things can happen. Let's look at what will happen in each of these cases.
There are more x-k factors in the denominator. After dividing out all duplicate factors, the x-k is still in the denominator. Factors in the denominator result in vertical asymptotes. There are more x-k factors in the numerator. After dividing out all the duplicate factors, the x-k is still in the numerator. Factors in the numerator result in x-intercepts. There are equal numbers of x-k factors in the numerator and denominator.
After dividing out all the factors because there are equal amounts , there is no x-k left at all. There is just a hole in the graph, someplace other than on the x-axis. Oblique Asymptotes When the degree of the numerator is exactly one more than the degree of the denominator, the graph of the rational function will have an oblique asymptote.
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