Differentiation and trigonometric integrals

Good afternoon everybody! I recently started taking calculus 2 at university and the class picks up like we never missed a day between calc 1 and calc 2. We jumped right in and I'm rather rusty. I use khan academy often but was hoping somebody could help me review/teach me with differentiation and trig integrals.

Anyways, here are my problems that I'm working on. Yes, they're rather simple but I've not done calculus in 4 or so months:

Differentiate:
1. y= 4x^3 -2x

Spoiler for Answer:

12x^2 -2

2. y= sqrt(2x-4)

Spoiler for Answer:

(2x+4)^-.5

3. y= x^3 tan(x)

Spoiler for Answer:

(3x^2)(tan x)+(x^3)(sec^2 x)

4. y= sin(x^2)

Spoiler for Answer:

2x cos(x^2)

5. y= (x+4)/(x-2)

Spoiler for Answer:

(x+4)((-x+2)^2)+(x-2)^-1

Integrate:
6. ∫ (cos x- sin x)dx

Spoiler for Answer:

sin x+ cos x+ C

7. ∫ sqrt(x)dx (Definite- solved at 4 and 1)

Spoiler for Answer:

2x^1.5 (Waiting to calculate this one :P)

8. ∫ cos(x^2)xdx

Spoiler for Answer:

1-sin^2 x + C

9. ∫ (sin^2 x cos x)dx

Spoiler for Answer:

sin^3 x/3 + C

10. ∫ (tan x sec^2 x)dx

Spoiler for Answer:

sec^2 x + C

Those are the questions I'm currently working on. I'd just like somebody to discuss something related with me or check my work. Thanks in advance! :)

Here is a perfect resource for the stuff you'll be doing. If you want to be awesome, just spend two or three days working through the lecture course.

And here is a great tool for checking your work.

Quote:

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Originally Posted by Xei

Here is a perfect resource for the stuff you'll be doing. If you want to be awesome, just spend two or three days working through the lecture course.

And here is a great tool for checking your work.

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Yes, indeed! I've heard of Wolfram | Alpha but I've never used it. I'm unsure of how I'll like the MIT lectures but I give them a whirl. I really like what Salman Khan does with his lectures. I get bored very quickly so shorter lectures are often preferred. Thanks Xei!

No problem. In my experience MIT are great at realising that the best way to teach something is reduce it to very simple and intuitive components which any sane person could understand; nothing is very tricky. And the lectures are terse; the guy doesn't say anything which you shouldn't know. If you're used to university workloads, the learning to work ratio here is a bargain.

Wolfram is very easy to use, if you want the answers for the questions above you can just type in plain English, 'what is the integral of sqrt(x) with respect to x from 1 to 4', and it'll work out what you mean and then give you a clear answer.

If you're very stuck with the unanswered ones just give me a shout.

Sure thing! I'm pretty confident with my ability to differentiate but the integrals are still a bit foreign to me. Perhaps I'm over thinking these things.

It seems that problem #10 is = sec^2 x + C? Is this true?

Spoiler for Problem #10:

10. ∫(tan x sec^2 x) dx
Let u=sec^2 x
du/dx= tan x
∫(u)(du/dx)(dx)
= ∫(u)(du)
= sec^2 x + C

If it is then I've definitely been making my work harder. XD

Type it into Wolfram, it shows derivations. But there's a slipup here, where you say:

d/dx sec^2(x) = tanx

"discuss something related" isn't really useful.

What are your questions? Could you prove rules for differentiation and integration? That's more up my alley than solving routine problems and will help you understand the stuff forever. Frankly I don't even remember most of the rules (kinda of a lie) but I could be dropped off on a desert island and write a book on them (not a lie). This is because I can figure out how to prove them.

Have you worked on epsilon-delta proofs of limits? Some books toss it in at the elementary level and some don't but it's worth understanding how this stuff actually works rather than memorizing a bunch of rules by rote.

As long as you're working with nice functions, I think it's almost always okay to just use intuition. Perhaps understanding what a limit actually means is good to know, but at his level I wouldn't go much further and start niggling around with epsilons and deltas to prove the chain rule and stuff. I mean, Newton did fine without them.

Quote:

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Originally Posted by PhilosopherStoned

"discuss something related" isn't really useful.

What are your questions? Could you prove rules for differentiation and integration? That's more up my alley than solving routine problems and will help you understand the stuff forever. Frankly I don't even remember most of the rules (kinda of a lie) but I could be dropped off on a desert island and write a book on them (not a lie). This is because I can figure out how to prove them.

Have you worked on epsilon-delta proofs of limits? Some books toss it in at the elementary level and some don't but it's worth understanding how this stuff actually works rather than memorizing a bunch of rules by rote.

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I have yet to work on epsilon-delta proofs but I know they've been brought up or mentioned in my Calculus I class. At my university, they don't expect you to do proofs for almost anything until Calculus III or higher. Personally, without much proof or thoughts, it always just made sense that differentiation gives you the rate of change, and integration gives you the equation from the rate of change.

The thing about proof is it's subjective how far you need to go. You definitely do need to prove the stuff you look at; you don't need to remember the exact details of a proof, but you do need to have some decent intuitive grounding for where something comes from. Epsilon-delta is a 'high level' kind of proof where you break stuff down a long way. At your level it's not that beneficial to go that far, you can probably stop when you get some kind of obvious limit.

Characterising integration simply as 'antidifferentiation' as you just did kind of demonstrates what I'm saying. Integration really isn't that... when you think of an integral you should be thinking of a continuous sum. If you don't think of it like this, you'll really struggle with intuitively understanding physical applications of integrals, and more complex integration stuff. It just happens that antidifferentiation turns out to be a good method for working out the integral, it's not what the integral actually means.

Check out MIT's vid on integration. Seriously... it'll really help.

I don't advocate that epsilon-delta proofs be used for actual limits but proving things like Lim(f + g) = Lim F + Lim G, and hence that D(f + g) = Df + Dg is pretty essential to knowing what you're doing in my opinion.

Unless one is just interested in solving problems that mathematica can do better anyways of course.

KhanAcademy is awesome, and so is PatrickJMT. I have been meaning to start a thread on those two guys' videos. I highly recommend them. Whenever I want to learn about a mathematical concept that is new to me or that I am rusty on, I usually watch a video by one and then a video by the other. That usually makes me get it.