this is a very confusing question. i think i nkow what you’re trying to ask, but you worded it very poorly.

i think you’re talking about Euler’s Identity, the formula for which is e^(i*pi)-1=0 where w is the angle. you can derive it by substituting the value i*Pi into the infinite series expansion of cosine & sine. geometrically this gives you the upper half of a radius 1 circle in the complex plane. pi can be substituted for any angle theta to give you the portion of a circle completed by rotating the radius theta-many radians about the origin.

google “euler’s identity” it will do a better job of explainign this than i do in a text post. i could explain this better if i had a convenient way to type formulas into the reply.

Well, this is a rather general question, but some advice for you is:

1. PRACTICE, PRACTICE, PRACTICE! You can read every book in the world about mathematics, and you’ll be no better at math unless you sit down with some pencil and paper and actualy try some problems. The best way to learn math is by doing math. So practice it until your eyes shrivel up and you literally can’t stare at the paper!
2. Don’t skip over proofs. Proofs explain why important and essential facts are true. If you skim over the proofs and don’t make an honest effort to understand them, you’ll never understand anything.
3. Try to prove things for yourself. It’s one thing to read the proof, but if you can construct it on your own, that means you truly know the knowledge.
4. Don’t be afraid to ask for help. Even Dr. Math gets stumped sometimes. And when that happens and you’ve tried everything you can think of, sometimes it will help to go to a person more knowledgable than yourself and ask for hints.

Once again, the single most important thing you can do to improve your skill at mathematics is to practice it and actually do it.

Now that I am done for the semester I am bored and deprived of math

I feel like a heroin junkie going through withdrawls

Yes I realize I could pick up a book, but I enjoy helping you guys as well!

Ahh, sorry I just received this message. sadly I am not so knowledgable in statistics. However if any follower thinks they know the answer, by all means, I invite you to submit me your response. I will, of course, give credit where credit is due.

Glad I could be of service =]

I hope it helped you on your exam!

since a few weeks ago

but yeah you should have expected it to happen eventually lol

Feel free to pepper me with question! Just understand I have my own exams I need to study for, so I will try to balance helping you with my own studying haha. But for now:

1. You’re 90% of the way there. What you said is all true, but since you’re evaluating a function over an interval [a,b], you must also consider the value of the function at those endpoint, in addition to the value at critical points.
   
   Also, and this is EXTREMELY (no pun intended) important, YOU CANNOT EVALUATE EXTREMUM OF A REAL VALUED FUNCTION OVER AN OPEN INTERVAL!!! I don’t know if they’d throw you that sort of curve ball, but basically, you can’t do this because no matter how close you get to the endpoint of an open interval, a functions behavior towards the endpoint becomes monotone and thus you have no limit point to serve as a definitive end point. Once again that’s not the main point, but keep that in mind. I took the AP test myself and I know they like to throw you trick questions, so be aware.
2. Depending on how fine you want to make the partition (IE the width of the rectangle), all you need to do is partition the interval you’re approximating the integral over into n-many rectangles of a given width, then multiply the width by the value of the function. If it’s a right Riemann Sum, use the rightmost value. Left Riemann Sum = Leftmost value.
   
   In your example, if you want to approximate that function using Left RS’s, partition the interval into [0,1];[1,2];[2,3]. Since you’re using Left RS’s, evaluate the function at the left endpoint of each interval IE f(0)*1 + f(1)*1 + f(2)*1. use right endpoints for a right sum.
3. I posted an entry on the Trapezoid Rule, I direct you to that entry for more information.
4. Optimization problems, for your purposes, are nothing more than a fancy way of saying “Find absolute extrema”, so always remember that.
5. Net change can be a lot of things, can you give me a little more context? Net change of a function over an interval? Net rate of change over an interval? What exactly do you mean?

Hope that helps :)

My version of a 10 minute introduction to topology

combattant-de-la-liberte:

A Maryland business owner has successfully discredited the camera equipment used by traffic enforcers – ironically by using their own technology against them to prove errors in the system.

Will Foreman, the owner of Eastover Auto Supply in Oxon Hill, Maryland, has managed to prove reasonable doubt five times before three separate judges, by bringing photos snapped on the highway of his company’s vehicles into court and proving that there is no way they were travelling over the speed limit.

But how did he do it? 

Foreman took a close look at the photos snapped on Maryland’s Indian Head Highway by Optotraffic. The company’s devices first use sensors to detect vehicles traveling at least 12 miles over the imposed speed limit, and then snap two time-stamped image of the vehicle 50 feet down the road, at 0.363 second intervals. 

The allegedly speeding motorist is then sent the pictures and a $40 ticket.

After superimposing the two photographs into one image - using the vehicle’s length as a frame of reference - Foreman was able to calculate the vehicle’s speed, given the distance and the elapsed time of the shots, and was able to prove that the vehicles were not in fact speeding. 

‘I’ve never seen this before…You’ve produced an elegant defence and I’m sufficiently doubtful,’ Judge Mark T. O’Brien said in court before throwing the tickets out. 

Foreman says that he is waiting to prove the system’s technology wrong on at least 40 more tickets that drivers at his company have received. 

‘This whole thing…I can’t wrap my head around it. They’re stealing from the public. It’s highway robbery,’ Foreman said.

‘It p***** me off. I’m trying to run a business…they’re raping the public,’ he added, calling the ticketing of the public for driving the speed limit ‘appalling’.

Dear Mathies:

If you ever wondered why you’re learning trigonometry in high school, and how you can ever practically apply it, well here you go: you can apply it to fighting traffic tickets in court!

-Dr Math