please delete

1) Has it dawned on you that the cosine function is an EVEN function?

2) It's possible that you have it correct, but I can think of wrong ways to get where you got. Show intermediate steps for an actual determination.

3) Have you noticed that x*e^(x^4) is symmetric about the origin?

Sometimes, they are not as hard as they seem if you first ponder the nature of what you are doing, rather than just ploughing into it.
 
You've successfully done the grunt work, Lisa, now observe the general results you've reached. Your examples each have f(u) being odd or even -- sin(-u) = -sin(u), etc. Read Tk's post again if still unsure.


The other, less deliberate result concerns \(\displaystyle \L\mbox{ \frac{d}{dx} \int^{h(x)}_{g(x)} f(u) du}\), where f is cont. & g, h, diff.
 
If each if g and h is a differentiable function and f is integrable, then
\(\displaystyle \L
\frac{d}{{dx}}\left( {\int\limits_{g(x)}^{h(x)} {f(u)du} } \right) = f(h(x))h'(x) - f(g(x))g'(x)\).

Therefore, one does not have to find any antiderivative to do your problems.

What Tkh wanted you to see is \(\displaystyle \L
\cos (x) + \cos ( - x) = 2\cos (x)\)
 
lisasayzhi said:
...our professor...his whole grading philosophy is that it's more important how you got to the answer, than the answer itself....
Click to expand...
But that shouldn't preclude teaching the class useful and clever conceptual shortcuts.

When the tutors pointed out the even-versus-odd considerations, I would suspect that they were working under the assumption that of course your professor had taught you this helpful concept. They weren't meaning to criticize; they were trying to point out something very helpful, presuming that your professor had bothered to mention it.

But he hadn't. :cry:

Eliz.
 
I think not. Even or Odd is an Algebra concept. You should have it engrained into your soul by the time you get to calculus.

The point of the Even function for #1 is that f(x) = f(-x). If your limits are on opposite sides of the Origin, what does that mean for the value of your integral? As long as "something" is in the Domain and the function is well-behaved,

f(something) - f(-something) = ???

I could have said the same thing as #3. The sine function is symmetric about the Origin. This is VERY USEFUL if your integration limits are the same distance from the Origin.
 
"get snapped at" - No one is doing that. You are being encouraged to think. That is OK. It needn't be a bad experience.

"None of the other people on here do that" - Pretty much most of the time we all try to be as helpful as we can. Sometimes there are style problems. It happens. We should get past them and learn something anyway.

"make me feel like an idiot" - Here's where we have a problem. If you feel like an idiot, you may be choosing to do that on your own. I can't make you feel that way. It's your choice. Simplest solution? Stop choosing it.

"some people don't even try to do it themselves" - I have found you cooperative and I appreciate your efforts.

"everytime I post a question, you make me feel stupid." - There it is again. You have your personal agency. You do not have to feel any such way. Perhaps my language reminds you of someone who hated you when you were 5 years old and you remember only the words. Let's get past that. I have no malice. I have no ill will.

"but you don't learn about integrals" - It is a common deficiency in the education of many simply to forget what one knows already. There is a reason why undergraduate mathematics is taught in a relatively rigid sequence and courses list prerequisites. Keep it ALL in your head and it will be to your advantage. A few years ago, I had to go look up the definition of an Odd Function. I had forgotten. I remember now and it is to my advantage.

"How am I supposed to know to do that when we haven't been taught it yet?" - This is a GREAT question. Many have asked it. I have to wonder, whatever it is, who learned it first? SOMEONE learned it without being TAUGHT it. Whoever that was probably wasn't a whole lot smarter than you, if at all. They just THOUGHT about it longer or maybe they had a better background in related topics. I think just about everyone would be startled to learn what they could learn if they decided not to be so dependent on what they are TAUGHT. Your education is about LEARNING. Teaching is only one means to that end.

"you don't have to make me feel bad about it..." - Third time's the charm? Stop choosing to feel bad. Take responsibility for your own feelings and let's move on to learn some mathematics. Personally, I like honest math students who really are trying to learn, rather than just get by or pass the exam.
 
It is often the student who struggles with "style" who, if survival can be accomplished, discovers much more than imagined was learned in the surviving. Nevertheless, I really don't look at names before I respond to an initial posting. It's difficult to agree to such an arrangement. We'll just have to see how it goes.
 
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