Anyone Else See A Problem With This Example From My Book?

In summary, the conversation discusses the theorem for parabolas and an example using the theorem. There is a discrepancy in the values for a and d, causing confusion about the correct ratio between them. The author of the book has been notified about the typo.
  • #1
Asphyxiated
264
0

Homework Statement



My book gives the theorem for parabolas as:

The graph of the equation:

[tex] y = ax^{2} [/tex]

(where [tex] a \neq 0 [/tex] ) is the parabola with focus [tex] F(0,\frac{1}{4}a) [/tex] and the directrix [tex] y = -(\frac{1}{4}a)[/tex]. Its vertex is [tex] (0,0) [/tex], and its axis is the y-axis.

It then goes on to use these equations in an example like so:

PROOF: Let us find the equation of the parabola with focus F(0,d) and directrix y= -d.

Where [tex] d= \frac{1}{4}a [/tex]

*Rest of proof omitted as it has nothing to do with what I am asking*

so then it moves on to an example where it asks:

Find the focus and directrix of the parabola:

[tex] y = -\frac{1}{2}x^{2} [/tex]

*straight from the book*=

Using Theorem 1 (the theorem posted above):

[tex] a = -\frac{1}{2} \;\;\;\; and \;\;\;\; d= \frac{1}{4}a [/tex]

so in this problem:

[tex] d = -\frac{1}{2} [/tex] ?

_________________________________
*End from book*

Shouldn't d = -1/8 not -1/2? If d = (1/4)a and a = -1/2, then isn't (-1/2)(1/4) = -1/8?

I ask this only because every other point in this chapter builds from this point and I want to make sure I am not just stupid and there is actually a problem here.
 
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  • #2
The problem is that

[tex]d = \frac{1}{4a}[/tex]

Perhaps you copied down out of the book incorrectly? Was it written like d = 1/4a? Without parentheses it's kind of ambiguous if the a is in the numerator or denominator.
 
  • #3
Well here is an image right out of the textbook:

[URL]http://images2e.snapfish.com/232323232%7Ffp53834%3Enu%3D52%3A%3A%3E379%3E256%3EWSNRCG%3D32%3B5328%3B%3C%3A347nu0mrj[/URL]

I am very confident when I say that it has been presenting it as:

[tex] d = \frac{1}{4}a [/tex]

In the image it looks, quite clearly to me, that the a is not in the denominator of the fraction.

I am not sure I did the image right, if it doesn't show here is a link:

http://www2.snapfish.com/snapfish/slideshow/AlbumID=2046768024/PictureID=67575340024/a=2078147024_2078147024/otsc=SHR/otsi=SPIClink/COBRAND_NAME=snapfish/"
 
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  • #4
That's definitely a typo in the book. If a = (1/4)d, then d = 4a.

I suspect the author really meant d to be the distance from the vertex of the parabola to the directrix (or from the vertex to the focus) and a is the distance across the parabola through the focus. The relation then would be a = 4d, or d = (1/4)a.

Some books define parabolas geometrically as the locus of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Who is the author of this book?
 
  • #5
Some books define parabolas geometrically as the locus of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

That is the definition of parabola they gave in this book, the book is:

Elementary Calculus: An Infinitesimal Approach
by H. Jerome Keisler
 
  • #6
I had his email on file so I sent him an email about it already, if that's why you asked...
 
  • #8
Yeah, that's where the image came from, although I downloaded it as pdf, not that it matters. As I said, H Jerome Keisler has been notified. All I can do is move on and not get mixed up by his typo.
 
  • #9
I am confused. When I look at EXAMPLE 2 form the link that I gave, I see

[tex]y = -2x^2[/tex]

In Theorem 1, [itex]a = -2[/itex] and [itex]d = \frac{1}{4} a = - \frac{1}{2}[/itex]
 
  • #10
Oh yeah, ok I didn't bother to look as I downloaded the full pdf from that same site. Evidently it has been updated and I have an older version of the book. I wouldn't just make up a false picture :) lol.
 
  • #11
Also if you look the problem with:

[tex] d = \frac {1}{4} a \;\;\;\;and\;\;\;\; a = \frac{1}{4}d [/tex]

is still there, right above EXAMPLE 2.
 
  • #12
Asphyxiated said:
Evidently it has been updated and I have an older version of the book.

Yes, this what I meant in my first post in this thread.

Good luck.
 

1. What is the problem with this example from your book?

The problem with this example is that it contains errors or inconsistencies that may affect the accuracy or validity of the information presented.

2. Can you provide more context about the example in question?

Absolutely. Providing more context about the example can help us better understand the specific issue that needs to be addressed.

3. How does this problem affect the overall message or findings in your book?

The problem with the example can potentially impact the overall message or findings in the book by introducing incorrect or misleading information that could alter the interpretation of the data.

4. What steps will you take to address this problem?

As a scientist, I will carefully review the example and work to correct any errors or inconsistencies. I will also consult with other experts in the field to ensure the accuracy of the information presented.

5. Is this problem a common occurrence in scientific literature?

Unfortunately, errors and inconsistencies can occur in any type of literature, including scientific literature. However, as scientists, it is our responsibility to ensure the accuracy and integrity of our work through careful review and collaboration with other experts.

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