Who wants to help Saaby with his Math

or rather, his calculator (I wanted to put that all in subject but it wouldn't let me.)

A year ago I could have been like all my peers and gotten a Ti 83 but (As I want to be an engineer and also I'm super nerd) I went with the HP 49G instead. So far I've been able to get along just fine. This was in part due to the fact that my teacher last year rarely used a calculator period.

Anyway I'm having a few troubles now figuring out the "HP Equivalent" of a few Ti things and wondered if you guys could help me. I've been looking up and down Google but hopefully somebody else will come up with better search terms or just knows this stuff.

We're graphic quadratic forumulas. I don't know what the proper name is to be honest but it's basically where you get 2 or 3 sqiggely lines stuck together a'la roller coaster.

Problem 1: Once you've graphed on the Ti you can hit {Calc} {Value} to which the calculator prompts "X=" you give it an X and it gives you a Y and jumps to that spot on the graph. I can trace my line but sometimes it does things like jump from (1.988 , 0.234) to (2.033 , 0.132) and what I really need is (2, Y). I realize I could just plug in 2 for x and solve the equation but if the Ti can automate this than the HP ought to be able to.

Problem 2: Minimum and Maximum. The Ti has another button that will tell you how high and how low the function goes on the graph. HP equivalent??

We've touched on how to do this stuff by hand, but the reality is that now days Math is taught using Calculators and if you can't drive your calculator you can get stuck in the dust. My sister has a Ti I could borrow but I've found just from borrowing other classmates calculators that my brain is stuck in RPN mode and I have a really hard time thinking Algebraically /ubbthreads/images/graemlins/wink.gif

{Edited by MicroE because my wife says that I am too wordy}
Don't fight City Hall.

{Edited by MicroE}
Saaby---Go buy a TI calculator and learn to use it.---Marc

Micro, I read the first post but you deleted the second one before I had a chance to read it. Bring it on....what'd ya say...

Saaby,

I have an HP 28S which I absolutely love, and I know that I can do the things you are talking about on it, but I don't have either the calculator or the manual in front of me, as I'm at work, and it's been a long time since I've had to graph anything on it. As I recall, there are cursor buttons and an (x,y) readout. The 49 is probably different. However, I'm sure if you look up "graphing" in your manual, it will tell you how to do these things. If you don't have the manual, GET ONE! You might be able to download it; I'm not sure.

On another note--

Saaby, please, for the love of God, learn how to do these things by hand. I'm sorry to be a petulant old schooler, but I'll wager a $20 bill that I can do these things faster by hand than any TI or HP user. Honestly, every science/engineering student should know how to find max's and min's and extrema of n-order polynomials. This stuff is BASIC, BASIC, BASIC! Or, rather, should become so. Take the time now to learn this stuff like the back of your hand, WITHOUT a calculator. I've encountered some of these modern calculator engineering students and they are handicaped by this dependence. It's sad. They don't even have a vague feeling for what the graph of any given function looks like. You NEED to KNOW this stuff, Saaby. Learn how to do it by hand. Work through the methods and reasoning until you know it inside and out, and then, AND ONLY THEN, use a calculator if you find it speeds things up. (In this case, you'll find it won't). The time you invest will repay you many times over. If nothing else, it gives the gray matter much needed exercise and elasticity.

Saaby,

I have a Ti-83 plus. What's wrong with the Ti if you already know how to use it?

Sidenote: Today in algebra 3/4 trig we learned how to solve linear systems of three dimentional equations. Oh man, is that ever fun! /ubbthreads/images/graemlins/wink.gif

You're right, js. Old-fashioned analysis should remain the core of mathematics classes, rather than just being "touched on." The sole advantage of computers is *speed*, and I've seen many whopping errors (both in homework and in research) that come out of people blindly trusting computer output without any logical way to double-check it.

IlluminatingBikr,

Yes, but did you learn how to solve them using matrices?
Then you can just solve them with a calculator. /ubbthreads/images/graemlins/smile.gif

Actually, in all seriousness, that is the one time that I found my HP to be a real aid: solving simultaneous linear equations with COMPLEX co-efficients. (Had to do this in second semeter Circuit Networks class). If you think something like this is fun:

3X+Y=7
-7X-Y=0

Try

(3+i)X+(0-i)Y=6+4i
(2+2i)X+(-7+i)Y=13-i

OUCH! Now that turns out to be a lot of scratching on paper. But since my HP could handle matrices with complex numbers, and since I could do it by hand already, I used the calculator after the first homework assignment (to show prof. I could do it by hand). After that I would simply write "Solution from calculator is . . .".

I do think that calculators have their place, but it's a terrible thing when they take the place of understanding.

Andrew,

Yeah. It reminds me of the Hubble Space Telescope and their mirror grinding problem. They had two instruments that both told them how to grind the parabolic miror. One of them (the digital one) was telling them that everything was peachy keen, good to go. The other of them (the analog device) was telling them that the curve was WRONG. Now the smart thing to do, the safe thing to do, would have been to double check the mirror to be sure that the analog device was malfunctioning. But no, they simply trusted that their new fangled latest and greatest computerized instrument was infallibly correct. And they sent the whole thing up into space only to find that, gee, the mirror isn't the right shape.

I mean, it would have been one thing if they only had one instrument, or if BOTH of them were saying the same thing. But they had two, and one of them was warning them of a problem. Arrrgghh! It's almost as bad as that screw up that happened somewhere because one set of scientists was using inches, and the other cm or meters or something. HA! I guess my first year physics teacher was right after all to insist on the importance of unit analysis.

As far as I'm concerned, faith in digital devices (or analog ones for that matter) should be /ubbthreads/images/graemlins/banned2.gif

So how do I manually figure out the "critical points" I'll put in an equation as soon as I've got my math book here. In Math class maybe we were learning it the hardest way possible, but doing it by hand involved a lot of guessing and checking (Aided by the calculator) as well as some synthetic division and the quadratic formula.

[ QUOTE ]
Saaby said:
...So how do I manually figure out the "critical points"...

[/ QUOTE ]

You take the derivative of the function. The derivative is the slope of the line. When the derivative equals zero you are at an inflection point.
If you don't know how to take the derivative (that's a Calculus thing) then you can use the SOLVE function on my ancient HP15C to find the roots of the equation.

The HP SOLVE function works out the roots by an iteritive (repeating) process of plugging in guesses. The solution of each guess is used as the next set of inputs to the equation until you finally "guess" at a solution that turns out to be correct.

Important terms: Function, minimum, maximum, inflection point, Horner's Method, Newton's Method.

Editor's Note: Saaby---You are obviously a VERY bright guy. js was correct when he used the word "handicap". If you are trying to work with an RPN calculator and the teacher is teaching on a TI then you are working with a significant handicap.
I don't know much about golf, but I guess that Tiger Woods would never win a tournament if he had to take a 20-stroke handicap. Even great talents can be hobbled by the wrong tools.
Get off your high horse and get the TI. Use your extra time to understand the MATH, not the calculator.
If you need $$ to get a TI then we can take up a collection here. Rant Off.

Just a technical point--an inflection point is when a curve switches from being convex to concave or vice versa, and it corresponds to the second derivative (often referred to as concavity) being zero. This may coincide with the first derivative being zero (called a critical point) but it doesn't have to.

To find local extrema you solve for the first derivative being zero and then use the second derivate at those points to tell you what kind of extrema you have found. If the 2nd derivative is negative, it's a local maximum. Positive means a local minimum, and zero means an inflection point. Note that it's possible to have an inflection point (meaning it's not accelerating up or down at that point) without the first derivative being zero. Look at a sine wave to see a simple example of this.

A quadratic equation will always have exactly one critical point and it will always be a local (and global) extreme. It's a simple case that is also easy to visualize.

As for TI vs. HP, the issue isn't as clear as it used to be. HP has long been the hands-down winner for serious engineering and scientific use. TI has been more classroom friendly. HP is much nicer to students now with the G series calculators (which have been around for 10 or more years now) but more people use computers for serious work and the TIs have come along far enough that they seem to be dominating the calculator scene. I think HP has dropped a long of their support now and outsources much of the calculator business. From what I've heard, they aren't built as well as they used to be.

RPN takes some getting used to but it's a powerful system with a lot of advantages. If your classes are really geared to a specific calculator then you'll be at a disadvantage if you don't use that one--not because it's better, but because it is the one the course favors. How much of a disadvantage it is and how much that matters to you is something only you can tell.

Good luck,

Russ

[ QUOTE ]
MicroE said:
When the derivative equals zero you are at an inflection point.


[/ QUOTE ]

Just to point out the obvious: you meant critical point, not inflection point, isn't it ?

Saaby,

Yes, you are correct. Doing it by hand involves factoring the polynomial
and/or using the quadratic formula, taking derivatives, finding their
zeros, and so on. It is slow going at first, but pretty soon you
get fast at it. Synthetic division is a useful tool. Really, factoring
and all the other algebraic manipulations are an art form, a skill.
I personally think that we'll all know when education has sunk to
its lowest point when students begin usuing calculators to solve
geometric proofs. It is very important to understand that it is NOT
about finding the answers. It is about learning math; it is about
THINKING math; it is about reasoning and critical thinking. It's not about
the person who can type stuff into a powerful calculator the fastest.
The good mathematician and scientist is the one who knows HOW to
get the answers, and WHY we get the answers. As mentioned before
the good scientist/mathematician already knows what the answer should
be, more or less. He or she is already expecting something definite.
You will never develope this intuition by becoming dependent on
a calculator. Think about it this way: any idiot with the teacher's
manual has all the answers, but that doesn't make him or her a good
math teacher, does it?

Enrico Fermi used to astonish people with his "back of the envelope"
calculations, that would predict the answer to some problem/situation
to within an order of magnitude or less. There's a story I heard
about someone throwing up a handful of scraps of paper just as
the first A-bomb was tested. He then measured the average deflection
away from the drop point and did some calculations on the back of
an envelope and came up with the approx. power of the bomb.

It always used to simply astonish me that students preparing for
the final examine in first year physics would completely cram a sheet
of paper full, front and back, with formulas and example problems.
I suppose they thought that it was best to search their sheet for
the proper formula and then plug in the numbers and (drum roll please)
THE ANSWER! Really, half a sheet of paper was more than enough.
The scientist is not a walking encyclopedia of formulas!
Understanding what is going on is the indispensable prerequisit
to becoming a competent scientist or mathematician. Period.

And another thing. <RANT>I used to tutor math, physics, and engineering
in college, and many of my students would say "Look, I know you
like math and all, but I just need to pass this one class and then
I'm never going to touch math ever again. All I need is a C or a
D." I'd say "THat's fine. I'm not here to make you into a Math
major, but I'm telling you that BY FAR the easiest way to get
your C or D or even better, is to UNDERSTAND what's going on.
When I take the time to explain something, or dwell on what you
think is only a "minor" point, I'm doing it because it is VITALLY
important." It always amazed me how much time and effort was
wasted by people who were trying to avoid a much smaller expenditure
of time an effort spent in understanding.</RANT>

OK. Ignore me if I'm babling. Getting a TI sounds like a good
idea if your class more or less requires it. But whatever you do
try to put understanding first and foremost, even above grades.

First of all sorry I couldn't write a reply this morning. My ride was picking me up early this morning, so just ot be safe I got up late /ubbthreads/images/graemlins/icon15.gif

I have good news and more good news [for me] that's bad news [for you]

First of all I stumbled upon what I need in order to do the calculatons on the calculator. They're on the funciton menu on the graph screen. There's one that helps you find the roots and one that helps you find the extremes (Minimum and Maximim). That's the stuff that's good for me but bad for you.

It's ok though because it turns out that my teacher decided to teach backwards, so everything we learned to do Monday we learned how to do today without a calculator. The only reason she taught us the 'primitive' way is that it's actually a college course (Concurrent enrollment Math 1050/1060) and we have to take the college's final written by professors that don't use calculators. I still don't know the technical term for what we're doing, but it's where you have something like:

h(x) = 6x^3 + 19x^2 + 2x - 3

and you have to find all the roots, y intercept, minimum and maximum and (dun dun dun DUN!) point of inflection. I could have easily figured all of this out on the calculator but I had this thread in the back of my head and used the calculator as a tool instead of a crutch. That's ok isn't it? For example, I had to list all the possible rational roots but instead of guessing and checking to see which ones actually worked I let the calculator graph it and I checked the graph. Then I did the synthetic division on the roots and blah blah....

MicroE--I know it sounds like I'd be farther off with a Ti but I find that in many cases, including this time around, my peers with their Tis plug and chug while I have to actually figure out the math so I can do it on my less user friendly HP. I'm not ready to climb off my high horse yet, but the saddle is slipping.

To find the point of inflection our teacher has us taking the derivitive twice and then setting that equal to zero. Solving gives you the X but not the Y. To get that my peers blindly use the 'Value' function on their Ti, since I can't find the HP equivalent of the 'Value' function I plug what I get for X back into the original equation and (Surprise surprise) I get the same answer.

So my new question is this: Plugging 13.094 back in for X is easy enough I could do it on paper, but it's loads easier to let the calculator do it. Right now I store my 'x' as a variable in the calculator and then manually key in the whole equation. Is there a shorter quicker way to do this already built into the calculator or do I need to just write a small program? As far as this simple substitution and multiplication is concerned, you have my word that I understand the math--but it's more efficient to let the calculator do it--and efficiency is something I like.

PS--My math experience last year was a less than positive one because the teacher used a calculator for almost nothing, but he was very very good at math and liked to skip steps, assuming that we could fill in the gaps. All I am saying is you've got to find balance. He should have, IIHO:

A) Slowed down and taught at a high school level. He taught over our heads which was a serious problem because his personality was also such that we did not feel comfortable asking for help.

B) Showed us how to check our work on the calculator and encourage us to do so.

Some students would have taken the shortcut and just done all their math on the calculator, but those of us who were really interested in learning how to do it the right way would have.

Oh well....my teacher this year is much better, even if she is a little calculator intensive.

Saaby,

OK. Let's consider y(x) = 6x^3 + 19x^2 + 2x - 3.

First of all we can see that our function is going to go to + infinity as x goes to + infinity, and -infinity as x goes to - infinity, because the cubic term will dominate and it is positive. In graphing terms, this means that I should have an arrow going UP to the far right of my function graph and an arrow going DOWN to the far left.

Also, right away, I know that the function is -3 for x=0, and thus I have one intercept. Also, the function must increase for increasing x because all the coefficients in front of x terms are positive. How could it go more negative for increasing positive x? It can't. So try y(1). You get 6+19+2-3 = 24. So right away you know there is a zero of the function between x=0 and x=1.

OK. So much for what should be obvious at a glance. Next, take derivatives.

y'(x) = 18x^2 + 38x + 2.

Since we are going to be interested in when y'=0, divide both sides of the eq. by 2 to make things simpler.

y'/2 = 9x^2 + 19x + 1.

Set this to zero. Use quardratic formula to find roots. You should get (-19 +/- sqrt(325) ) / 18. Or x=-.054 and -2.057. This confirms our earlier analysis because it shows that there will be no change of whether the function rises or falls for positive x. We knew this already, but it is comforting to see that it is confirmed. It's a check on our work so far.

Take the second derivative:

y''(x) = 36x + 38.

y''(x) = 0 when x = -38 / 36 = -1.05, and y'' is positive to the right of this, and negative to the left. So we already know that the graph will be an upside "U" mated to a "U", or in other words that, starting from the far left of the graph (-x) and going to the right (+x), that it will rise from very negative, peak out at when x is -2.057, fall until x is -.054, rise through y=-3 at x=0, go through y=0 somewhere before x=1, hit y=24 at x=1 and keep on rising as x gets larger. Make sense?

So the question now is "does this bad boy have 1 real root or 3?" or in other words, as it is rising from the left, does it go above y=0 peak positive, and then fall through y=0, or does it never get that high.

Easy to answer that. We know there is a local maximum at x=-2.057, right? So plug in x=-2. No don't plug in the whole blasted "minus two point zero five seven" etc. etc! Just try -2. It's nice and friendly. So y(-2) = -6*8 +19*4 + 4 -3 = -48 + 76 + 1 = whatever, but it's a definitely POSITIVE something whatever (= 29 for those who just have to know). The point was not to get an exact number, because what good is it? We didn't do the value exactly at the maximum, just near it to see what sort of animal we had.

In the same way, plug in x=-1 and -3 to get a feeling for the graph. y(-1) = +8, so it's still coming down at this point. So here again, we know that there is a zero of the function between x=-1 and x=0. For x=-3 we get y(-3) = -162 + 171 - 6 - 3 = -171 + 171 = 0. HOT DIGIDY DOG! Now isn't that nice? Now we know that we can factor out an (X+3) from our function, to leave us with a quadratic function, which can be factored by the quadratic formula, and we expect roots between -1 and 0, and between 0 and 1.

OK. I admit that just typing the equation into the calculator and hiting "graph" is faster. And I also admit that we had to use the calculator in this method to get the roots from the quadratic formula, but the point is that there are so many things that you can know about a polynomial right away just by doing some basic analysis and using some common mathematic sense and intuition.

Hope this all wasn't too pedagogical, or should I have said "pedantic"?

Go out and buy you a TI-89...not much more than a L4, legal on all Standardized tests I know, does algebra, trig, calculus, differntal equations, etc

But atleast know whats going on behind what you do on the TI!

In the words of a math teacher of mine."Put it in you TI and smoke it."

darkwater,

I can assure you that the TI DOES NOT do differential equations! /ubbthreads/images/graemlins/twakfl.gif It may take derivatives of polynomials and certain elementary functions, such as sine, cos, exponentials, etc. but NO WAY can it solve a differential equation, not even a general linear one like:

6d^3y/dx^3 + d^2y/dx^2 - dy/dx = 0

And certainly not a non-linear one like (my favorite /ubbthreads/images/graemlins/smile.gif )

dy/dx = x^2 + y^2.

go ahead. I dare you. Try to even enter that on the TI. It will get a /ubbthreads/images/graemlins/whoopin.gif

No that's great Jim. I think I'm going to print it and keep it at hand. I actually do respect and enjoy Math although I realize it's not coming off that way this time around.

You know when you start a reply, and then you go to Wal-Mart and by the time you get back you forget where you were going with the reply? That's this.

*EDIT*
Can my HP do that impossible equation?

If I get another calculator for standardized tests (The HP 46G is /ubbthreads/images/graemlins/banned2.gif ) than it'll be another HP. LIke I said, once you've got that RPN thing going for you it's hard (Well, I find it hard) to switch. I get on the HP and end up hitting the decimal button (Aprox location of the space button on my HP)