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good grief I get rusty so fast


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Right now we are back to doing something that requires derivatives.  And I already freaking forgot the details.  Not a huge deal.  After some quick review I'm just about caught up.  I just wonder how to keep stuff sharp so I can keep going with this.  Are there some good workbooks out there with calc problems? 

 

I wish ALEKS had anything higher than pre calc because that was so helpful!

 

 

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Udemy.com is having a black Friday sale right now.  I purchased a calculus refresher class for my husband, who teaches math to our daughters.  Perhaps one of the classes will work for you..

 

Also, Annenberg Learner has some mathematics workshops/courses for teachers.

Edited by llama
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Are you having enough sleep? Sleep would affect retention and I am an insomniac.

 

Schuam series is good if you need more work examples as library has those.

The Humongous Book of Calculus Problems is another popular book at my local libraries. It says it’s for Calc 1 and Calc 2 though. Are you in Calc 3 now?

https://www.amazon.com/Humongous-Book-Calculus-Problems/dp/1592575129

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Are you having enough sleep? Sleep would affect retention and I am an insomniac.

 

Schuam series is good if you need more work examples as library has those.

The Humongous Book of Calculus Problems is another popular book at my local libraries. It says it’s for Calc 1 and Calc 2 though. Are you in Calc 3 now?

https://www.amazon.com/Humongous-Book-Calculus-Problems/dp/1592575129

 

I sleep plenty.

I have the Humongous book.  I have 3 of the Schaum books, but not for calc.  I guess I could get those. 

 

I'm in 2.  If I end up taking 3, that won't be until the fall.  All that time to wait.  Ugh.  They only offer it in the fall. 

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If my kids (or I) haven't done enough example problems (and it takes A LOT for most of us) to really cement something into our heads, we have to do this short review each time we come back around to a topic. (We aren't AoPS types where a couple of examples & a handful of problems is enough.) 

 

Also, the more practice we get, the more certain things make sense or appear to "look right" in our heads. (There are some things that I could mindlessly practice over & over again and still not intuit enough to repeat them without a model. Solving a Rubix cube, for example. I have to try to understand why I'm doing what I'm doing while I'm doing it over & over again instead of just following an algorithm.)

 

This thought came to me when you commented on your recent math question thread about a process looking "pretty convoluted". I've worked with trig functions enough on my own and then with dd#1 last year that I could "see" there would be a simplification of your first answer, but would have had to think through it a minute to get the right answer. 

 

Just a thought.

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If my kids (or I) haven't done enough example problems (and it takes A LOT for most of us) to really cement something into our heads, we have to do this short review each time we come back around to a

This thought came to me when you commented on your recent math question thread about a process looking "pretty convoluted". I've worked with trig functions enough on my own and then with dd#1 last year that I could "see" there would be a simplification of your first answer, but would have had to think through it a minute to get the right answer. 

 

Just a thought.

 

I had all the identities in front of me, but for some reason I didn't see it.

 

Sometimes I work on stuff so much I start getting confused.  LOL  I think that is what happened with that one.

 

But then with some of them, you could keep switching stuff all around so then I start to wonder why one answer is better than another (if it is at all).  I could still find the slope with my answer so it wasn't exactly wrong either. 

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Khan Academy has a fair number of calculus problems now. 

 

So, when I said go back to basics on the other thread, I meant turn the entire thing into stuff involving sin and cos only, and then simplify. Of course, sometimes you can't... but anyway... I think it tends to help.

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Btw, as far as retention is concerned... repeated practice, spread out over longer and longer periods of time (so, at first you're going to have to practice something every day, but then you can practice it every other day, once a week, once a month, once every few months, w/e etc... of course, it varies per person and per topic as to how quickly you can spread out your practice, and yes, it's frustrating). (and that's about all I retained from cognitive science, lol)

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Khan Academy has a fair number of calculus problems now. 

 

So, when I said go back to basics on the other thread, I meant turn the entire thing into stuff involving sin and cos only, and then simplify. Of course, sometimes you can't... but anyway... I think it tends to help.

 

I've found a lot of their videos really helpful.

 

In terms of simplifying..with a calculator it feels like it hardly matters.  KWIM? 

 

But yeah this is one area that I didn't seem to get enough practice with when I went through it.  I did a bunch of them today and that helped.  It's no longer my least favorite thing at least.  LOL

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In terms of simplifying..with a calculator it feels like it hardly matters.  KWIM? 

 

But once you have the argument as a variable and not as a number, the calculator won't help. And if you need your result in a subsequent calculation, simplifying really helps.

 

Btw, the way to simplify trig expressions like the one in your question is usually pretty standard: convert everything that's not a sine or cosine to sines and cosines first and then see what cancels.

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In terms of simplifying..with a calculator it feels like it hardly matters.  KWIM? 

 

 

With a calculator (computer) it probably hardly matters at all whether you learn trig or calculus at all... I'm pretty sure that there are programs that will find you the answers to equations you put in. Just like with arithmetic. But, you learn it anyway, because the computer thus far doesn't come up with useful questions and equations that need to be solved, and you need a certain facility with the math in order to come up with that stuff and to be aware when the answer the computer gives you makes no sense. 

 

Wrt Khan I meant the exercises, not the videos, since you were asking for more practice. I don't think they've got all of calc 1 and 2 yet though.

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With a calculator (computer) it probably hardly matters at all whether you learn trig or calculus at all..

 

That anyway, but nobody can seem to answer my question about why simplify it to THAT particular point.  Maybe if I've been doing this for years and years that would be apparent, but I haven't and it isn't.

 

Maybe it is a stupid question, but better to ask and risk sounding stupid than to just go on not knowing.

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That anyway, but nobody can seem to answer my question about why simplify it to THAT particular point.  Maybe if I've been doing this for years and years that would be apparent, but I haven't and it isn't.

 

 

I think that's mostly a style or personal preference issue. Many mathematicians like things to look 'elegant'. 2csc(t) is certainly a lot less clunky-looking than 2sec(t)/tan(t). That said, I probably would've left it at 2/sin(t). Not that I'm a mathematician or anything... but I do think that both 2csc(t) and 2/sin(t) are improvements over the original. 

Edited by luuknam
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If you can visualize the graph of csc(t) (because you've seen it many times or w/e), then 2csc(t) would be the better choice, because you'd easily be able to visualize what 2csc(t) would be. 

 

I don't remember what that graph looks like anyway, so I'd have to draw it based on my knowledge of the graph of the sine, at which point it doesn't matter if someone wrote 2csc(t) or 2/sin(t) (that said, presumably anyone who's memorized the graph of csc(t) can easily mentally convert 2/sin(t) into 2csc(t)).

Edited by luuknam
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That anyway, but nobody can seem to answer my question about why simplify it to THAT particular point.  Maybe if I've been doing this for years and years that would be apparent, but I haven't and it isn't.

 

Maybe it is a stupid question, but better to ask and risk sounding stupid than to just go on not knowing.

 

Writing it as a single function or term is simpler than writing it as a ratio of two functions or terms.

 

There are often several equivalent expressions;  I don't see the csc t as simpler than a 1/sin t (and I would have left it as 1/sin t), but it is definitely simplified compared to sec t over tan t. Just like you would not give a final answer as 6/12, but simplify it to 1/2

Edited by regentrude
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That anyway, but nobody can seem to answer my question about why simplify it to THAT particular point. Maybe if I've been doing this for years and years that would be apparent, but I haven't and it isn't.

My exams back in the 80s/90s did ask exam candidates to put their answer in the most simplified form for free response questions. Since those are manually graded, you at most lose a point or two per question if further simplification can be done.

 

My kids while doing their AoPS online class homework would get points taken off (as in not get full points) for their written homework if their answer can be simplified further.

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Thinking about graphing does help with understanding what is the simpler way of expressing it.  But in that particular instance, I wasn't asked to graph anything.  So in that case it didn't matter.  Doing problem after problem and seeing the solutions vary in terms of how far they get twisted and turned makes it more confusing than helpful.

 

I rely on math to not be vague.  :laugh:

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If it helps, I don't think anybody's ever counted it wrong when I've left stuff as 2/sin(t) instead of 2csc(t). Whereas points do sometimes get subtracted for leaving it as a convoluted jumble. 

 

I am 100% certain she would not count it wrong. 

 

I just want to understand it inside and out.

 

why why why...it's like I'm 3

 

:laugh:

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Honestly the reason I'd type it as 2 csc t is because then I don't have to typeset a fraction. If handwriting it I'd probably leave it as 2/sin t. I find it more useful to leave it in terms of sin/cos/tan because then if I need to solve it, evaluate it, or take another derivative it's easier for me (I never did memorize the derivatives of anything but sin/cos until I ended up teaching calculus). But this is something that varies by teacher -- similar to rationalizing denominators. 

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Honestly the reason I'd type it as 2 csc t is because then I don't have to typeset a fraction. If handwriting it I'd probably leave it as 2/sin t. I find it more useful to leave it in terms of sin/cos/tan because then if I need to solve it, evaluate it, or take another derivative it's easier for me (I never did memorize the derivatives of anything but sin/cos until I ended up teaching calculus). But this is something that varies by teacher -- similar to rationalizing denominators. 

 

Yeah I'm beginning to think mostly only teachers have these things memorized!

 

She allows us to have a cheat sheet.  So I put the trig stuff on it!

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Yeah I'm beginning to think mostly only teachers have these things memorized!

 

She allows us to have a cheat sheet.  So I put the trig stuff on it!

 

It's one of those things that you either need to have memorized or be able to derive really fast if you're in calculus, and I'm horrible at memorizing. 

 

For example, if I needed the derivative of sec x, every time, I'd write sec x = (cos x)^{-1}, the derivative of which is -1(cos x)^{-2}(-sin x) = (sin x)/cos^2 x = tan x sec x. Except I'd do it on the back of the test so that if I needed it later in the test I could look back there and find it. 

 

But if we'd had a cheat sheet it would totally have been on there. 

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