for what value of on the interval (-9.65, 11.28) will f(x) take on a minimum if f(x)=−7.2sin(0.3x)+1.8cos(0.3x). What is f(x) at this x value? Same question for the maximum on the same interval.

- anonymous

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- anonymous

this should say for what value of x on the interval... Sorry!

- amistre64

deriving trig...aint had to do this yet :)

- anonymous

well the derivative is -0.54sin(0.3x)-2.16(cos(0.3x) if you need it. I know I need to set it to zero to get the critical values, but I don't know how to solve for x lol

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## More answers

- amistre64

Dx(-7.2 sin(.3x)) = -7.2(.3) cos(.3x)

- amistre64

lol .... I was getting there :)

- anonymous

sorry! lol

- amistre64

lets multiply by 100 to get rid of some of those decimals

- amistre64

-54 sin(.3x) - 216(cos.3x) = 0

- amistre64

can we factor out anything from 54 and 216? I know 2 comes out...

- anonymous

you can take out a 9 i think

- amistre64

54/216 = 27/108..try 3 now 9/36 try 3 again 3/12 and 3 again 1/4

- amistre64

looks like 54 factors out completely...
216/54 = anything useful? 4 yay!!

- amistre64

-54(sin(.3x) + 4cos(.3x)) this look right to you so far?

- anonymous

yeah I'm following I think. lol, that's what I have when I do the same work on my calculator. :)

- amistre64

so, lets see what sin(.3x) + cos(.3x) gets us to equal 0

- amistre64

they gotta be opposites of each other, so lets throw one to the other side

- amistre64

and dont forget that 4 that I left out lol

- anonymous

yeah i was going to ask why we weren't using it, lol but nevermind

- amistre64

4cos(.3x) = - sin(.3x)
now picture these graphs for a minute

- amistre64

the cos is just stretched 4 tall, so the 4 doesnt matter
the sin is flipped over upside down

- amistre64

.3 = the frequency of the graphs

- amistre64

these cross points at 2 places every time they go thru one frequency...does that make sense?

- amistre64

once in Q2 and another in Q4

- amistre64

the only angle that has sin and cos equal is 45 degrees

- anonymous

right.. I put them in my graphing calculator. lol so that means I have more than one answer right?

- amistre64

yes... more than one answer, and it will have something to do with 45 degrees

- amistre64

-9.65 and 11.28 is our interval

- amistre64

.3(sqrt(2)/2) = ?

- anonymous

.2121320344

- amistre64

I might be wrong about the 45 tho :)

- amistre64

derive it again to see if we can get some useful information...

- anonymous

derive the original f(x) or take the second derivative?

- amistre64

second derivative will give us a better picture....

- anonymous

64.8 sin(0.3 x)-16.2 cos(0.3 x)

- amistre64

I got:
y'' = .648 sin(.3x) - .162 cos(.3x)

- anonymous

oh, yeah I just had it multiplied by 100 still

- amistre64

it might be helpful, but im still "stuck" :)

- anonymous

When I asked a friend, she told me I needed to use the quadratic. However, she had her math major friend do the problem for her, and can't tell me anything farther than that lol

- amistre64

lets go back to the original..
sin(.3x) + 4cos(.3x)

- anonymous

Is there a way for us to write them both in the same terms, like a way for us to make cos to sin?

- amistre64

there can be a square root involved, or a phase shift....

- amistre64

sin = sqrt(1-cos^2)

- amistre64

sin^2 = 1-cos^2

- anonymous

If it helps, when I tried using wolfram alpha, it gives me the answer.. x = 20/3 (pi n+tan^(-1)(1/4 (1+sqrt(17))))

- amistre64

oww...my eyes.....

- anonymous

But I don't actually even know what that means, lol so I that wasn't very helpful either. It was kind of a a last resort as it was, because my assignment is due at midnight. :\ lol

- amistre64

4cos(.3x) = -sin(.3x)
square both sides to get
16cos^2(.3x) = sin^2(.3x)
16cos^2(.3x) = 1 - cos^2(.3x)

- amistre64

then:
17cos^2(.3x) -1 = 0 right?

- anonymous

i believe you're correct.

- amistre64

cos^2(.3x) = 1/17

- amistre64

cos(.3x) = sqrt(17)/17 then ....if that helps :)

- anonymous

okay so x ends up being like 4.419392212

- amistre64

cos-1(sqrt(17)/17) is?

- anonymous

cos-1(sqrt(17)/17) is 1.325817664 i think

- amistre64

25.32 is what I get for that x :) does that make any sense?

- amistre64

25.32 radians of course

- anonymous

ohh.. so why am I getting a different number, is my calculator just in the wrong mode?

- amistre64

maybe; and I used 1/sqrt(17) to be simple

- amistre64

lets plug that in to see what we get:
sin(.3(25.32)) + 4cos(.3(25.32)) :=: 0 ??

- anonymous

I'm not getting zero. :\

- amistre64

0.1321 + 3.9648 = .... ack!!!
lol.

- amistre64

need someone smarter than me I guess :)

- anonymous

oh no. lol what do I do now? Do you have any ideas of where else I can get help b y midnight?

- amistre64

how many points it take off if it is wrong?

- anonymous

I'm not sure.. there are only 4 sections and this one is technically two parts.

- amistre64

let me review our stuff for a few minutes and see what I can deduce...

- anonymous

alright. If it helps, on the thread for the problem, it says to take the derivative equal to zero, and that you should get tan=#, then take the inverse.

- anonymous

I don't know how they're getting tangent though.

- anonymous

What if we divided both sides by cos(0.3x) and then had tan(0.3x)=4.. does that work?

- amistre64

-4x -y=0...-y=4x....-y/x = 4
-4x +y = =..... y=4x......y/x=4
this is the tan they get at....
-sin(.3x)
------- = 4
cos(.3x)
-sin(x) = sin(-x) so thats the same

- amistre64

tan(a) = 4
tan-1(4) = what?

- amistre64

i was getting to it.... :)

- amistre64

75.96 degrees is a divide that by.3

- anonymous

so we have 253.212=x ?

- anonymous

that sounds large. lol

- amistre64

remember that tan-1 spits out an angle between -90 and 90

- amistre64

(75.96 + 180) /.3 ?? maybe

- amistre64

(75.96 - 90)/.3?

- anonymous

that one gives us -46.8

- amistre64

i see that..... maybe our interval itself gives us a max or min at the end points?

- anonymous

if you put those in.. for -9.65, we get 0.0119783705 and for 11.28, we get -0.190832377

- amistre64

.3x gives us one cycle every 20pi/3 does that helps us out?

- anonymous

well in the wolfram alpha, it said something about 20/3 pi * n, but n is supposed to mean any integer, so I didn't really understand it.

- amistre64

n = the angle measurement
a normal cycle is 2pi
but we disturb that with .3.... so it increases the frequency of the cycle to 20pi/3

- amistre64

6 and 2/3 pi

- anonymous

x = 20/3 (pi n+tan-1(1/4 (1+sqrt(17))))

- amistre64

What is the value of our interval in terms of pi?

- amistre64

x(pi) = -9.65
x = pi/-9.65

- amistre64

-.325 pi to .279 pi right?

- amistre64

that aint gonna make a difference, becasue our interval is telling us the value of "x" which is the angle involved.... not the period....

- anonymous

okay.. I'm just lost at this point. lol I really don't even know how to get any further in this.

- amistre64

im at a loss for the moment as well :) its all those decimals that are messing me up

- anonymous

I've got it!!!! lol I graphed it and just followed the graph and put in the numbers :P thanks though hahah

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