Calculus_微积分_Graphing polynomials using technology

Calculus_微积分_Graphing polynomials using technology

You might ask: Why spend all this time to graph a polynomial by hand when technology can do it for us much more quickly? One answer is that the experience of producing a few such graphs by hand does wonders for one’s ability to process visually the results given by technology, and cements the connection between the visual and the algebraic. A second answer is that practicing the process by hand helps provide the skills needed to use technology to find all the features of a function’s graph even when they are initially hidden or outside the bounds of naively chosen plot boxes.

Example 5 Use a computer algebra system (CAS) to graph the function $f(x)=x^5-181 x^4+6514 x^3-4390 x^2+327 x+273$, showing all its features (use more than one picture if necessary).
Solution We still begin by finding the critical numbers and possible inflection locations so that we know on what interval we should graph the function. Using a CAS to solve the equation $f^{\prime}(x)=0$ results in the critical numbers $x=0.040975, x=0.415378, x=35.2089$, and $x=109.135$ (decimal approximations used; the exact values are not necessary for the purpose of graphing the function). Using the CAS to solve the equation $f^{\prime \prime}(x)=0$ results in the possible inflection locations $x=0.227515, x=22.4592$, and $x=85.9133$. Using the CAS to produce a plot of the function on an interval containing all these values of $x$, and playing with that interval to find something that looks good, results in figure 20 . We were asked to show all the features of the graph. Because it is somewhat difficult to tell from figure 20 whether there are extrema at the left-most critical numbers and possible inflection locations, we can produce additional graphs near these locations, as in figure 21.

Calculus_微积分_Horizontal asymptotes and limits at infinity

Consider the graph of $y=4+\frac{1}{x}$. You should be able to form a mental image of this graph fairly quickly because we can start with the graph of $y=\frac{1}{x}$, which should be on instant recall, and shift up 4 units (figure 1).
The dotted line, which is not part of the graph of the function, represents a horizontal asymptote with equation $y=4$. Just as with a vertical asymptote, the idea is that the function approaches the asymptote, and the dotted line serves as an aid in sketching the graph of the function. How do we interpret a horizontal asymptote? The idea for $y=4+\frac{1}{x}$ is that as $x$ gets “large,” $y$ gets “close” to 4 . This idea can be explored numerically by calculating values of $y$ as our values of $x$ get larger:

It appears to be the case that as $x$ gets larger, $y$ gets ever closer to 4 . This seems evident in this case, but what about more complicated functions? How do we know that something different doesn’t happen as $x$ gets even larger? The problem is that large and close are fairly ambiguous terms. Here is where the hyperreal numbers come to our rescue again. Instead of just using a “large” real number, we use an infinite hyperreal number such as $\Omega$ :

Now we see that if $x$ is infinitely large, then $y$ is infinitely close to 4 . This gives precise meanings to large and close!
The same phenomenon happens when we look at the left side of the graph, where the $x$-coordinates are negative. We still choose an infinite number, but it needs to be negative, such as $-\Omega$ :

Therefore, if $x$ is infinitely large but negative, then $y$ is still infinitely close to 4.
It appears that evaluating a function at infinite hyperreals gives us information about the behavior of the function on the far right and far left of the graph. We have previously used the term limit for calculations that help us determine similar behaviors, and we use this term again here.

微积分代考

Calculus_微积分_Horizontal asymptotes and limits at infinity

∖ begin arrayc∣c×&y∖∖h line-IOmega \& $4+\backslash{r a c{1}{-\backslash \mid Omega}=4–lomega\backslash因此，如果lap因此，如果x无限大但为负，则无限大但为负，则y$ 仍然无限接近4。

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