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 ≈

- anonymous

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- chestercat

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- inkyvoyd

set 2sin(x^2)=0

- inkyvoyd

Wait, over an interval -_-

- inkyvoyd

Ignore the interval, and just solve that trig equation.

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## More answers

- anonymous

i know that the answer for part "a" is

- anonymous

square(pi)

- inkyvoyd

square or sqrt?

- anonymous

i dont know how to do part b

- anonymous

sqrt

- inkyvoyd

Well, your interval s defined only up to 3. Is that supposed to be that way, or is the three suposed to be pi?

- anonymous

up to 3

- anonymous

a)\[2*\sin(2*x)=0\rightarrow x=k*pi/2\]
In this interval x=pi/2

- inkyvoyd

Do you understand a though?

- anonymous

well i got x= |dw:1334814197335:dw|

- inkyvoyd

|dw:1334814169184:dw|

- anonymous

and is the right answer

- inkyvoyd

|dw:1334814236680:dw|

- inkyvoyd

Naturally, sqrt pi would be the largest value up to 3, because it squares out to pi.

- anonymous

the equation is f(x) = 2sin(x^2)

- anonymous

sorry about that

- anonymous

\[F(x)=-\cos(2*x)+K'\]
\[F(1)=-\cos(2*1)+K'=9\]

- inkyvoyd

I know - That is sin x

- inkyvoyd

but, sin x^2 is similar to the graph in that it has its highest values at sqrtpi

- anonymous

sqrt(pi) is the right answer for part a is marking it right

- anonymous

the part that i dont get how to do is part b

- inkyvoyd

Yes. Can you tell me what calc course you are enrolled in?

- anonymous

calculus I

- inkyvoyd

Alright, give me a second.

- anonymous

2sin(x^2) = 0
x^2 = pi
x = sqrt(pi)

- anonymous

thats how i got my answer

- inkyvoyd

Have you learned how to approximate integrals?

- anonymous

kinda

- anonymous

thats what we are learning

- inkyvoyd

What's it called?

- anonymous

antiderivatives

- inkyvoyd

No, what kind of approximations have you learned?

- inkyvoyd

I just figured out how I can solve the problem.

- inkyvoyd

Here's the graph, btw

- anonymous

i dont know

- inkyvoyd

http://www.wolframalpha.com/input/?i=integrate+2+sin%28x%5E2%29

- anonymous

the tittle of this chapter is antiderivatives

- inkyvoyd

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

- anonymous

so how do i do part b?

- inkyvoyd

I'm thinking ;)

- inkyvoyd

Have you learned trapozoidal approximation?

- anonymous

no

- anonymous

I also have a question for part b... help us figure this out plz

- inkyvoyd

http://www.wolframalpha.com/input/?i=+2+sin%28x%5E2%29

- inkyvoyd

Alright. that is the graph of f(x)=2 sin(x^2) right?

- inkyvoyd

Notice how the waves become wider and wider close to 0.

- inkyvoyd

after that, they become thinner and thinner.

- inkyvoyd

Now, give me a second to explain something.

- inkyvoyd

Let me get htis plot into mathematica, and show you what I'm talking about.

- anonymous

k

- inkyvoyd

##### 1 Attachment

- inkyvoyd

Ok. The area of a>b>c>d Tell me if you understand.

- inkyvoyd

@Ldaniel
?

- anonymous

yes i do

- inkyvoyd

So, because of this, we know that from 0 to 3, the area enclosed in a is the greatest

- anonymous

yes

- inkyvoyd

So we are essentially looking for the area there.

- inkyvoyd

@FoolForMath , I give up/

- anonymous

;/

- inkyvoyd

I can give you the answer though, me being a cheater cheater.

- anonymous

what's the answer?

- anonymous

i wish i could do this math. -_-

- inkyvoyd

Give me a second.

- inkyvoyd

I'm going to numerically evaluate the integral with wolfram Mathematica

- inkyvoyd

1.78966

- inkyvoyd

@lgbasallote , here you go.

- anonymous

the answer is wrong

- inkyvoyd

WHAT?

- inkyvoyd

Try the square root of pi then.

- anonymous

as the answer?

- inkyvoyd

Yes.

- anonymous

wrong

- inkyvoyd

btw, try 1.790 if you haven't already.

- anonymous

i did its wrong

- inkyvoyd

Ahem.

- anonymous

;(

- lgbasallote

lol why call me =)) i only know algebra :P i dont like these stuffs

- anonymous

:((

- inkyvoyd

Wrong, call >.<

- inkyvoyd

http://www.wolframalpha.com/input/?i=integral%5B2+sin%28x%5E2%29%2C%7Bx%2C0%2Csqrt%28pi%29%7D%5D

- inkyvoyd

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

Got any more info about what you are learning right now?

- inkyvoyd

If I used the methods they gave, I might just get the answer they are looking for.

- anonymous

i got the answer is y = 10.16912...

- inkyvoyd

That's th correct answer?

- anonymous

|dw:1334818371099:dw|

- anonymous

yes

- dumbcow

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

Well, then I'm completely useless.

- anonymous

10.170 is the answer

- inkyvoyd

Nice job.

- anonymous

thanks anyways

- anonymous

how old are you?

- inkyvoyd

15

- inkyvoyd

I'm still not sure how I missed that problem so badly. :/

- anonymous

15? really?

- dumbcow

http://www.wolframalpha.com/input/?i=integrate+2sin%28x^2%29+dx+from+1+to+sqrt%28pi%29
add 9 to get answer

- anonymous

yes dumbcow

- dumbcow

sorry for being redundant, i know you already got the answer
just putting it out there for everyones benefit

- apoorvk

I believe graph would go like this. (the pitch decreasing with increase in x)
|dw:1334819574505:dw|

- inkyvoyd

btw, yes, I'm trying to jump the gun and finish calculus and some math afterwards before I enter college.

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