Schedule! and Some Easy Problems
by djmathman, Aug 23, 2012, 4:40 PM
I just found out my schedule for the 2012-2013 school year:
Period 1, AP Calc BC: This should be fun. The teacher is awesome, and I've already got a good grip on the subject so the class shouldn't be too hard. Hopefully I'll be able to ace out of the final.
Period 2, Chamber Strings: Our orchestra teacher is awesome. Our chamber orchestra is going to go to a festival somewhere near Brown University for the first time in a long time! He'll be stricter on us, but it'll be worth it. Even though some of our best players won't be there this year due to AP Physics conflicting.
Period 3, English II Honors: Our teacher is well respected and well liked by people who had him. English isn't my best subject, but it will probably be much better than last year, if only because of the teacher.
Period 4, Biology Honors: Eh, don't know the teacher so well, and Biology is my least favorite of the high school sciences. Then again I haven't actually taken Bio yet so how should I know
Period 5, LUNCH
Period 6, Java/C++: Love the teacher, love the class. Same room as AP Calc. This is going to be awesome.
Period 7, PE/Driver's Ed Q1: I actually seem to enjoy Gym. Maybe because I don't suck at it as much as my nerd image seems to suggest. We vary teachers throughout the year, so we get different experiences each quarter.
Period 8, Spanish III Honors: This teacher I had in 8th Grade, and she moved up to the High School this year. Although she'll be different this year than she was back then, I loved her. Difficulty compared to Spanish II Honors is unknown at this point.
Period 9, US History Honors I (or APUSH): This will be the toughest class that I will have ever taken. I mean, the teacher's nice, but she's super super super hard. Some of my friends who had her joked that getting a 100 on a test in her class is harder than getting a 15 on the AIME. We'll see how it goes.
Anyway, on to the problems.
Period 1, AP Calc BC: This should be fun. The teacher is awesome, and I've already got a good grip on the subject so the class shouldn't be too hard. Hopefully I'll be able to ace out of the final.
Period 2, Chamber Strings: Our orchestra teacher is awesome. Our chamber orchestra is going to go to a festival somewhere near Brown University for the first time in a long time! He'll be stricter on us, but it'll be worth it. Even though some of our best players won't be there this year due to AP Physics conflicting.
Period 3, English II Honors: Our teacher is well respected and well liked by people who had him. English isn't my best subject, but it will probably be much better than last year, if only because of the teacher.
Period 4, Biology Honors: Eh, don't know the teacher so well, and Biology is my least favorite of the high school sciences. Then again I haven't actually taken Bio yet so how should I know
Period 5, LUNCH
Period 6, Java/C++: Love the teacher, love the class. Same room as AP Calc. This is going to be awesome.
Period 7, PE/Driver's Ed Q1: I actually seem to enjoy Gym. Maybe because I don't suck at it as much as my nerd image seems to suggest. We vary teachers throughout the year, so we get different experiences each quarter.
Period 8, Spanish III Honors: This teacher I had in 8th Grade, and she moved up to the High School this year. Although she'll be different this year than she was back then, I loved her. Difficulty compared to Spanish II Honors is unknown at this point.
Period 9, US History Honors I (or APUSH): This will be the toughest class that I will have ever taken. I mean, the teacher's nice, but she's super super super hard. Some of my friends who had her joked that getting a 100 on a test in her class is harder than getting a 15 on the AIME. We'll see how it goes.
Anyway, on to the problems.
Problem wrote:
Compute 
.
.
.Problem wrote:
Show that there do not exist reals 
and 
with 
such that ![\[\int_a^b \sin x \, dx = \dfrac{1}{2} \qquad\text{and}\qquad \int_a^b \sin 2x \, dx = \dfrac{5}{8}.\]](//latex.artofproblemsolving.com/d/9/8/d987d5c8c93fa837b84dfac72d2646b87f81649b.png)
and 
with 
such that ![\[\int_a^b \sin x \, dx = \dfrac{1}{2} \qquad\text{and}\qquad \int_a^b \sin 2x \, dx = \dfrac{5}{8}.\]](http://latex.artofproblemsolving.com/d/9/8/d987d5c8c93fa837b84dfac72d2646b87f81649b.png)
for some 
.
as 
,
.
and ![\[\int 2\sin x \cos x \, dx = \int 2u \, du = u^2 = \sin^2 x + C.\]](http://latex.artofproblemsolving.com/1/4/f/14fc28c9d1ef6905d1e850fd9a2a2ad036c3cd4e.png)
![\[\begin{cases}\cos b - \cos a = -1/2\\\sin^2 b - \sin^2 a = 5/8\end{cases}\]](http://latex.artofproblemsolving.com/7/5/9/759cfc601f8b648e372beef97ff8ee743a161baf.png)
,
,
.
Problem wrote:
Show that the system of equations ![\[\begin{cases}a+b+c=4,\\ab+ac+bc=8\end{cases}\]](//latex.artofproblemsolving.com/3/8/e/38ed7bde94ac830c89eff08a0d7826a69bc622dd.png)
has no real solutions.
![\[\begin{cases}a+b+c=4,\\ab+ac+bc=8\end{cases}\]](http://latex.artofproblemsolving.com/3/8/e/38ed7bde94ac830c89eff08a0d7826a69bc622dd.png)
has no real solutions.
are real. Note that 
Hence 
,
.This post has been edited 2 times. Last edited by djmathman, Apr 6, 2015, 3:08 AM
Reason: align tags
Reason: align tags
April 2025
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