Help with calculus?

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magician13134
 
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Help with calculus?

Post by magician13134 »

I'm sorry, but I'm just having a panic attack now; I don't know how to do the calculus work, our teacher is no help (she acts like she's still in highschool, and she can't understand why anyone wouldn't understand something about calculus) and we have a test tomorrow. I've got all the notes, and I've been pouring over them, but I just can't retain any of the information!

This is all about the first and second derivative tests... Could someone try to explain those to me in very simple terms? I know how to take derivatives (for the most part)... But I'm just not doing well!

Thanks!

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schill
 
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Re: Help with calculus?

Post by schill »

I've got to admit that I wasn't sure what you were talking about (is "first derivative test" a term my teachers/professors used?).

I did do a quick search and these came up:

http://www.math.hmc.edu/calculus/tutorials/extrema/
http://en.wikipedia.org/wiki/First_derivative_test

Is this what you are talking about? I'm familiar with the concept, but I don't remember the name.

I'm sorry I don't really have time to help, but there should be a lot of tutorials out there.

magician13134
 
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Re: Help with calculus?

Post by magician13134 »

Ok thanks, I was just confused about the very basics...
First and second derivative test find extrema...? But the first also finds increasing and decreasing intervals..? So why bother ever using the second? It just seems like more work all around...

niksun
 
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Re: Help with calculus?

Post by niksun »

Simple answer; given a function f(x):

The first derivative, f'(x), tells you the slope of f(x) at any x (supposing we were to draw a line at that point).

The second derivative, f''(x), tells you if f(x) is increasing or decreasing in slope at any x.

A couple of cool things. If we set the first derivative equal to 0, then we can see where the slope of f(x) is 0 (or horizontal). This can tell us if we have a local minimum or maximum of f(x) (is it the top of a hill or the bottom of a valley). We can use the second derivative to see if it is increasing or decreasing to figure out which one it is. If it's < 0 the we have a local maximum; if > 0 then we have a local minimum.

Supposing I knew nothing about a function f(x) = x^3 - x. I could obtain its first derivative f'(x) = 3x^2 - 1. Set it equal to 0:
3x^2 - 1 = 0
3x^2 = 1
x^2 = 1/3
x = +-sqrt(1/3)
x = +-1/sqrt(3)

So it has local minimums (valleys) or maximums (hills) at x=-1/sqrt(3) and x=1/sqrt(3).

Now the second derivative is f''(x) = 6x. We check at x=-1/sqrt(3):
6*-1/sqrt(3) = -6/sqrt(3) < 0 so we have a hill at x=-1/sqrt(3).

We check at 1/sqrt(3):
6*1/sqrt(3) = 6/sqrt(3) > 0 so we have a valley at x=1/sqrt(3).

I took this example from the Web so you could take a look at it. Hope this helps a bit.

Good luck.

eil
 
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Re: Help with calculus?

Post by eil »

magician13134 wrote:our teacher is no help (she acts like she's still in highschool, and she can't understand why anyone wouldn't understand something about calculus)
Man, I sympathize with you there. I'm no good at math. Not my fault, I try my hardest but just don't have a "knack" for it like I do with most other things. I once had an Algebra teacher who few through the material so fast that by the middle of the class I was completely lost. Oh yes, I asked questions, but it got to the point where I was afraid to ask any more because I was starting to get nasty glares from her whenever I spoke up. Even with Google's help, the homework was just about impossible because I hadn't absorbed anything in class. None of the other students ever spoke. To this day I don't know whether it was because they understood all of the material (why take the class?) or because they understood none of it but didn't want to ask questions and risk getting sharp, unhelpful answers like I always got.

Only class in my life I ever dropped. But don't follow my example, please. :P Check to see if your school has a math tutoring program, check out calc books at the library, and Google your ass off. I'm sure you'll get through it.

magician13134
 
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Re: Help with calculus?

Post by magician13134 »

Thanks for the explanation niksun

And eil, this sounds all too familiar, in fact I was about to go to the counselor today and see about switching into another calc class (she only teaches one). I mean, I do feel kind of bad for the calc teachers though, a LEGEND at our school just retired and the two new teachers (literally right out of college) have some huge shoes to fill... But it just gets a little out of hand when a teacher 'friends' some of her students on Facebook... :?
I really think I may look into getting a tutor, I shouldn't have even waited this long, but I've never had much trouble in math before so I'm just a little nervous about asking for help (so really, it's MY fault)...

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westfw
 
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Re: Help with calculus?

Post by westfw »

Hmm. I didn't remember these "tests" by name either. Here's what I think is happening:

Suppose you have a function f(x) and want to find its minimum and maximum values over some region [X1, X2]. There are two possibilities:
1) The minima/maxima are at one (or both) of the endpoints of the regions [X1,X2] (always check for this - it's a favorite trick question.)
2) The function has "bumps" in it that go above or below the endpoints. At these points, the slope is momentarily zero (the function is parallel to the X axis.) Since the first derivative is the slope, the value of the first derivative function f'(x) is also zero at the x values where these bumps happen.

So we have:
A) "Local minimum and maximum values of a function occur at points where the value of the first derivative is zer0. (VERY IMPORTANT!) ("local" means not including the endpoints of a region or limits of the function.)

Now, a "bump" can be either a minimum or a maximum, depending on whether it faces up or down. You can also have a point where the function is momentarily parallel to the X axis (and the first derivative function has a value of zero) that is not a bump, but just a level part of the curve. (all local min/max values occur where f'(x) = 0, but not all points where f'(x)=0 are local min/maxs of the function.)

One way to figure out whether the point is a min or max is to look at the values of f'(s) "near" the critical value of x. If the point is a maximum, the curve will be going "up" before the point (slope and f'(x) are positive) and "down" after the point (slope and f'(x) are negative.) If the bump is a minimum, the reverse will be true. If it's neither, the slope on both sides will be of the same sign. As far as I can tell, this is the "first derivative test."

B)The first derivative test involves checking the values of the first derivative on either side of a critical point to determine what kind of critical point it is.

(You can pick any convenient x values to do the testing, as long as there aren't any other critical points between the one you pick and the one you're testing.)

This test is actually measuring the slope (or the sign of the slope, anyway) of the derivative function at those values of x. "positive before and negative after" where a function is zero means the slope there is negative, and vis versa. (Do you see this? It took be a bit...) And the slope of the derivative function f'(x) is just its derivative f''(x); the second derivative of the original function. This gives you the "second derivative test":

The second derivative test involves checking the values of the second derivative of a function at the critical points. If the value of f''(x) is positive at the critical point, the point is a local minimum. If it's negative, the point is a local maximum. If f''(x)=0, the point is neither a minimum nor a maximum. (called a "point of inflection"

Does that make sense? (Any other comments from everyone else? Did I get it right?)

My sympathies on the new teachers. The great teachers are the ones who not only understand the subject matter, but also understand all the ways that the subject matter can be misunderstood. Alas, I suspect that this only happens via experience. I didn't do so good when my wife was taking calculus. Now I'm practicing for when my kids will take it... (and oh gods, my sympathies for people who do this the first time in college, where the profs are even less likely to be good teachers, the TAs aren't any more experienced than a beginning teacher, and you're cut off from most of the people that understand YOU. All good reasons for taking as many AP classes in HS as you can!)

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westfw
 
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Re: Help with calculus?

Post by westfw »

PS: the "first derivative test" looks a bit pointless to me, aside from being a step in deriving the "second derivative test".

(OTOH, when I took the AP exam, there was a whole "long" question about derivatives and their associated math that was based on a table of numeric values, that most of the people in our class got wrong because we'd been doing so much symbolic stuff for so long. And it was SO easy. Grr.)

Hazard
 
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Re: Help with calculus?

Post by Hazard »

It seems that your questions were answered, so may I make an offtopic question?
How old are you? What grade are you in?
I get confused a lot by the USAmerican education system.

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westfw
 
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Re: Help with calculus?

Post by westfw »

I get confused a lot by the USAmerican education system.
In the USA, most people start "Kindergarten" at age 5 (approximately), and proceed with 1st through 5th grade (Primary school), 6th through 8th grade (Middle school) and 9th through 12th grade (High school), so you usually end up getting out of high school at approximately age 17 go 18. These are the grades handled by the public school system or alternative private schools. School after 12th grade is "college" or "university" (terms that mean about the same thing in the US) that you have to pay for; one way or another.)
In theory, there is little differentiation that happens in these first 13 years of school (in significant contrast to some European school systems, as I understand it.) You get to 12th grade and you decide whether you'll go on in education, or whether you'll go directly to work. In practice, the high school years DO start to split people into groups, with some getting classes that are more explicitly "college prep", and other classes that are more "vocational" (job-training, including "homemaker.") But you can bounce around quite a bit, and pretty much make your decision at the last minute (perhaps because of the very broad range of "colleges" available.)
Now of course within those grades, you have your students that seem to grasp the material better than others, so typically there will be "honors" or "advanced" classes in some subjects for the students that do well, and "remedial" classes for the students who are behind. Get toward 12th grade, and the people who have been in advanced classes are essentially done with the requirements of grade school. These students can take so-called "AP" ("Advanced Placement") classes, which (in theory) teach the same material as the initial classes they would take in college/university, while still in high school. At the end of the year, these students take a standardized exam, and if they do well they can "skip" the equivalent classes in college. Frequently, these are taught at a somewhat slower pace, so that it's not uncommon for an AP math class to take all year to complete a half-years (one semester) worth of college math. 1st semester college math (in technical/scientific courses of study) is almost always calculus, starting with differentiation. 2nd semester is integral calculus. My AP math class covered both, but the test we took only qualified me to skip 1 semester, leaving me quite comfortable in 2nd semester college math...

So I surmise that the Magician is in 12th grade, approximately 17, and headed off to university next year. (I believe he has said as much in other threads...)

niksun
 
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Re: Help with calculus?

Post by niksun »

Grades 1-8 are considered to be pretty good in the US system (competitive and BANNED). Where the system fails is grades 9-12 (high school). The education of our high schoolers is sorely lacking. I tend to believe that it is primarily due to our "everyone-is-equal" mentality that assures every single student a "trophy." You know, "It's OK Joey. There is no such thing as last place in this competition! Here's your trophy! Great job! Here's your pat on the back!" That, coupled with the no-child-left-behind program is leaving our system way behind others in the world. Personally, I think that school is a bit too easy. We also tend to go completely against the rules of nature and hold the group (class) behind to attend to a few stragglers. That stunts those students who may very well excel and be our next Nobel prize winners. In nature, only one or two "parents" go back to attend to a straggler. The remaining pack continues on and is not kept at the pace of the stragglers. Our high school system holds everyone back so as to help the few who shouldn't be in the class to begin with. My opinion.

magician13134
 
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Re: Help with calculus?

Post by magician13134 »

As for the questions about me, I am a senior and will be heading off to college next year, but I'm 18.

As for the high school system failing, I really don't blame the forced equilibrium of students so much as the lack of caring and effort put into school work. Honestly, I do TOPS 15 minutes of homework a night. I have two "challenging classes", and I have very little motivation at this point. I think it's sooooo wrong that I can come home, plop down on my bed and be on my computer from 2:15 until 11:11, and still get by with a near 4.0 GPA. THAT'S where our system is failing. They make it so easy to slack off...

And as for math, thanks for the help. I studied for THREE HOURS last night (totally contradictory to my above statement), I completed a seven page study guide (very anti-climactic, it was a one page test, albeit front and back) and even went into class with a highlighter to force myself to do exactly what the question asked (that's where I usually lose points) and breezed through the test. Then I got home and checked my grade... C :? I'm SO upset about that. I studied for three hours last night and for two study-halls and lunch today and couldn't get better than a C? I really need to get to school to see what I did wrong, and I won't be able to sleep until I know that! (Now I can only hope that I aced the physics test to help make up for this embarrassment in calculus; it was my favorite subject, DC Circuits!)

It's shaping up to be a baaaad weekend, Michigan hockey lost, basketball lost, football will probably lose to OSU tomorrow (they never let me live that one down here in Ohio) and I won't know what I did wrong on my calc test :(

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westfw
 
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Re: Help with calculus?

Post by westfw »

High schools, even the public ones, seem to vary a great deal. The one my daughter is in has a reputation for being sort of bi-modal; it's supposed to be good for lesser students to catch up, and it's supposed to be good for the top students (something like 12 different AP classes, for example.) But... if you're in the middle; perhaps not so great... (although we're now assured that things have improved in this department.) (It's a sort of melting pot of a school. 30% white, 30% black, 30% Hispanic, 10% other, getting people from both rich and poor neighborhoods...) So far (9th grader), so good, and there has been very little "slacking off" in the homework department.

The European system, which as I understand it generally involves separating the career-bound students from the university-bound students at what USisans would consider a very early age, tends to be viewed with some horror here. We sort of expect students themselves to have a say in what they'll do, and don't expect them to have the maturity or exposure to make that sort of decision till they reach high school. But its certainly easy to see how it would lead to more efficient instruction in the later grades; a lot of the problems in the US system can be traced to people needing to take classes that aren't relevant to what they think they'll be doing...

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schill
 
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Re: Help with calculus?

Post by schill »

westfw wrote:...a lot of the problems in the US system can be traced to people needing to take classes that aren't relevant to what they think they'll be doing...
It works the other way, too.

koolkat
 
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Re: Help with calculus?

Post by koolkat »

The high school I go to gives me 1-2 hours of homework most days.
I have all strait A's except french which is a B+ becuase his test's have questions that arent even what we studied in class...

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