Ldaniel
Consider the function f(x) = 2sin(x2) on the interval 0 ≤ x ≤ 3.
(a) Find the exact value in the given interval where an antiderivative, F, reaches its maximum.
x =
(b) If F(1) = 9, estimate the maximum value attained by F. (Round your answer to three decimal places.)
y ≈
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inkyvoyd
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set 2sin(x^2)=0
inkyvoyd
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Wait, over an interval -_-
inkyvoyd
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Ignore the interval, and just solve that trig equation.
Ldaniel
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i know that the answer for part "a" is
Ldaniel
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square(pi)
inkyvoyd
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square or sqrt?
Ldaniel
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i dont know how to do part b
Ldaniel
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sqrt
inkyvoyd
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Well, your interval s defined only up to 3. Is that supposed to be that way, or is the three suposed to be pi?
Ldaniel
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up to 3
RaphaelFilgueiras
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a)\[2*\sin(2*x)=0\rightarrow x=k*pi/2\]
In this interval x=pi/2
inkyvoyd
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Do you understand a though?
Ldaniel
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well i got x= |dw:1334814197335:dw|
inkyvoyd
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|dw:1334814169184:dw|
Ldaniel
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and is the right answer
inkyvoyd
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|dw:1334814236680:dw|
inkyvoyd
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Naturally, sqrt pi would be the largest value up to 3, because it squares out to pi.
Ldaniel
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the equation is f(x) = 2sin(x^2)
Ldaniel
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sorry about that
RaphaelFilgueiras
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\[F(x)=-\cos(2*x)+K'\]
\[F(1)=-\cos(2*1)+K'=9\]
inkyvoyd
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I know - That is sin x
inkyvoyd
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but, sin x^2 is similar to the graph in that it has its highest values at sqrtpi
Ldaniel
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sqrt(pi) is the right answer for part a is marking it right
Ldaniel
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the part that i dont get how to do is part b
inkyvoyd
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Yes. Can you tell me what calc course you are enrolled in?
Ldaniel
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calculus I
inkyvoyd
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Alright, give me a second.
Ldaniel
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2sin(x^2) = 0
x^2 = pi
x = sqrt(pi)
Ldaniel
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thats how i got my answer
inkyvoyd
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Have you learned how to approximate integrals?
Ldaniel
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kinda
Ldaniel
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thats what we are learning
inkyvoyd
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What's it called?
Ldaniel
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antiderivatives
inkyvoyd
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No, what kind of approximations have you learned?
inkyvoyd
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I just figured out how I can solve the problem.
inkyvoyd
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Here's the graph, btw
Ldaniel
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i dont know
Ldaniel
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the tittle of this chapter is antiderivatives
inkyvoyd
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Note how the function uses something called the fresnal S, which I don't know either. But mor eimportantly, notice how it gets to its peak at about 1,7
Ldaniel
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so how do i do part b?
inkyvoyd
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I'm thinking ;)
inkyvoyd
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Have you learned trapozoidal approximation?
Ldaniel
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no
noginogi
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I also have a question for part b... help us figure this out plz
inkyvoyd
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Alright. that is the graph of f(x)=2 sin(x^2) right?
inkyvoyd
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Notice how the waves become wider and wider close to 0.
inkyvoyd
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after that, they become thinner and thinner.
inkyvoyd
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Now, give me a second to explain something.
inkyvoyd
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Let me get htis plot into mathematica, and show you what I'm talking about.
Ldaniel
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k
inkyvoyd
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inkyvoyd
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Ok. The area of a>b>c>d Tell me if you understand.
inkyvoyd
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@Ldaniel
?
Ldaniel
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yes i do
inkyvoyd
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So, because of this, we know that from 0 to 3, the area enclosed in a is the greatest
Ldaniel
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yes
inkyvoyd
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So we are essentially looking for the area there.
inkyvoyd
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@FoolForMath , I give up/
Ldaniel
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;/
inkyvoyd
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I can give you the answer though, me being a cheater cheater.
noginogi
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what's the answer?
mr.luna
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i wish i could do this math. -_-
inkyvoyd
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Give me a second.
inkyvoyd
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I'm going to numerically evaluate the integral with wolfram Mathematica
inkyvoyd
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1.78966
inkyvoyd
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@lgbasallote , here you go.
Ldaniel
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the answer is wrong
inkyvoyd
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WHAT?
inkyvoyd
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Try the square root of pi then.
Ldaniel
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as the answer?
inkyvoyd
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Yes.
Ldaniel
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wrong
inkyvoyd
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btw, try 1.790 if you haven't already.
Ldaniel
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i did its wrong
inkyvoyd
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Ahem.
Ldaniel
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;(
lgbasallote
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lol why call me =)) i only know algebra :P i dont like these stuffs
Ldaniel
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:((
inkyvoyd
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Wrong, call >.<
inkyvoyd
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So, you are supposed to use an estimation method that isn't accurate, nor do I know. Or, the problem is set up funy.
inkyvoyd
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Got any more info about what you are learning right now?
inkyvoyd
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If I used the methods they gave, I might just get the answer they are looking for.
Ldaniel
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i got the answer is y = 10.16912...
inkyvoyd
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That's th correct answer?
Ldaniel
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|dw:1334818371099:dw|
Ldaniel
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yes
dumbcow
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so\[\int\limits_{1}^{\sqrt{\pi}}f(x) = F(\sqrt{\pi}) - F(1)\]
all you need to do is find approximation for definite integral
inkyvoyd
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Well, then I'm completely useless.
Ldaniel
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10.170 is the answer
inkyvoyd
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Nice job.
Ldaniel
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thanks anyways
Ldaniel
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how old are you?
inkyvoyd
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15
inkyvoyd
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I'm still not sure how I missed that problem so badly. :/
Ldaniel
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15? really?
Ldaniel
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yes dumbcow
dumbcow
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sorry for being redundant, i know you already got the answer
just putting it out there for everyones benefit
apoorvk
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I believe graph would go like this. (the pitch decreasing with increase in x)
|dw:1334819574505:dw|
inkyvoyd
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btw, yes, I'm trying to jump the gun and finish calculus and some math afterwards before I enter college.