anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
this should say for what value of x on the interval... Sorry!
amistre64
  • amistre64
deriving trig...aint had to do this yet :)
anonymous
  • 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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amistre64
  • amistre64
Dx(-7.2 sin(.3x)) = -7.2(.3) cos(.3x)
amistre64
  • amistre64
lol .... I was getting there :)
anonymous
  • anonymous
sorry! lol
amistre64
  • amistre64
lets multiply by 100 to get rid of some of those decimals
amistre64
  • amistre64
-54 sin(.3x) - 216(cos.3x) = 0
amistre64
  • amistre64
can we factor out anything from 54 and 216? I know 2 comes out...
anonymous
  • anonymous
you can take out a 9 i think
amistre64
  • amistre64
54/216 = 27/108..try 3 now 9/36 try 3 again 3/12 and 3 again 1/4
amistre64
  • amistre64
looks like 54 factors out completely... 216/54 = anything useful? 4 yay!!
amistre64
  • amistre64
-54(sin(.3x) + 4cos(.3x)) this look right to you so far?
anonymous
  • anonymous
yeah I'm following I think. lol, that's what I have when I do the same work on my calculator. :)
amistre64
  • amistre64
so, lets see what sin(.3x) + cos(.3x) gets us to equal 0
amistre64
  • amistre64
they gotta be opposites of each other, so lets throw one to the other side
amistre64
  • amistre64
and dont forget that 4 that I left out lol
anonymous
  • anonymous
yeah i was going to ask why we weren't using it, lol but nevermind
amistre64
  • amistre64
4cos(.3x) = - sin(.3x) now picture these graphs for a minute
amistre64
  • amistre64
the cos is just stretched 4 tall, so the 4 doesnt matter the sin is flipped over upside down
amistre64
  • amistre64
.3 = the frequency of the graphs
amistre64
  • amistre64
these cross points at 2 places every time they go thru one frequency...does that make sense?
amistre64
  • amistre64
once in Q2 and another in Q4
amistre64
  • amistre64
the only angle that has sin and cos equal is 45 degrees
anonymous
  • anonymous
right.. I put them in my graphing calculator. lol so that means I have more than one answer right?
amistre64
  • amistre64
yes... more than one answer, and it will have something to do with 45 degrees
amistre64
  • amistre64
-9.65 and 11.28 is our interval
amistre64
  • amistre64
.3(sqrt(2)/2) = ?
anonymous
  • anonymous
.2121320344
amistre64
  • amistre64
I might be wrong about the 45 tho :)
amistre64
  • amistre64
derive it again to see if we can get some useful information...
anonymous
  • anonymous
derive the original f(x) or take the second derivative?
amistre64
  • amistre64
second derivative will give us a better picture....
anonymous
  • anonymous
64.8 sin(0.3 x)-16.2 cos(0.3 x)
amistre64
  • amistre64
I got: y'' = .648 sin(.3x) - .162 cos(.3x)
anonymous
  • anonymous
oh, yeah I just had it multiplied by 100 still
amistre64
  • amistre64
it might be helpful, but im still "stuck" :)
anonymous
  • 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
  • amistre64
lets go back to the original.. sin(.3x) + 4cos(.3x)
anonymous
  • 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
  • amistre64
there can be a square root involved, or a phase shift....
amistre64
  • amistre64
sin = sqrt(1-cos^2)
amistre64
  • amistre64
sin^2 = 1-cos^2
anonymous
  • 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
  • amistre64
oww...my eyes.....
anonymous
  • 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
  • amistre64
4cos(.3x) = -sin(.3x) square both sides to get 16cos^2(.3x) = sin^2(.3x) 16cos^2(.3x) = 1 - cos^2(.3x)
amistre64
  • amistre64
then: 17cos^2(.3x) -1 = 0 right?
anonymous
  • anonymous
i believe you're correct.
amistre64
  • amistre64
cos^2(.3x) = 1/17
amistre64
  • amistre64
cos(.3x) = sqrt(17)/17 then ....if that helps :)
anonymous
  • anonymous
okay so x ends up being like 4.419392212
amistre64
  • amistre64
cos-1(sqrt(17)/17) is?
anonymous
  • anonymous
cos-1(sqrt(17)/17) is 1.325817664 i think
amistre64
  • amistre64
25.32 is what I get for that x :) does that make any sense?
amistre64
  • amistre64
25.32 radians of course
anonymous
  • anonymous
ohh.. so why am I getting a different number, is my calculator just in the wrong mode?
amistre64
  • amistre64
maybe; and I used 1/sqrt(17) to be simple
amistre64
  • amistre64
lets plug that in to see what we get: sin(.3(25.32)) + 4cos(.3(25.32)) :=: 0 ??
anonymous
  • anonymous
I'm not getting zero. :\
amistre64
  • amistre64
0.1321 + 3.9648 = .... ack!!! lol.
amistre64
  • amistre64
need someone smarter than me I guess :)
anonymous
  • 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
  • amistre64
how many points it take off if it is wrong?
anonymous
  • anonymous
I'm not sure.. there are only 4 sections and this one is technically two parts.
amistre64
  • amistre64
let me review our stuff for a few minutes and see what I can deduce...
anonymous
  • 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
  • anonymous
I don't know how they're getting tangent though.
anonymous
  • anonymous
What if we divided both sides by cos(0.3x) and then had tan(0.3x)=4.. does that work?
amistre64
  • 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
  • amistre64
tan(a) = 4 tan-1(4) = what?
amistre64
  • amistre64
i was getting to it.... :)
amistre64
  • amistre64
75.96 degrees is a divide that by.3
anonymous
  • anonymous
so we have 253.212=x ?
anonymous
  • anonymous
that sounds large. lol
amistre64
  • amistre64
remember that tan-1 spits out an angle between -90 and 90
amistre64
  • amistre64
(75.96 + 180) /.3 ?? maybe
amistre64
  • amistre64
(75.96 - 90)/.3?
anonymous
  • anonymous
that one gives us -46.8
amistre64
  • amistre64
i see that..... maybe our interval itself gives us a max or min at the end points?
anonymous
  • anonymous
if you put those in.. for -9.65, we get 0.0119783705 and for 11.28, we get -0.190832377
amistre64
  • amistre64
.3x gives us one cycle every 20pi/3 does that helps us out?
anonymous
  • 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
  • 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
  • amistre64
6 and 2/3 pi
anonymous
  • anonymous
x = 20/3 (pi n+tan-1(1/4 (1+sqrt(17))))
amistre64
  • amistre64
What is the value of our interval in terms of pi?
amistre64
  • amistre64
x(pi) = -9.65 x = pi/-9.65
amistre64
  • amistre64
-.325 pi to .279 pi right?
amistre64
  • 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
  • anonymous
okay.. I'm just lost at this point. lol I really don't even know how to get any further in this.
amistre64
  • amistre64
im at a loss for the moment as well :) its all those decimals that are messing me up
anonymous
  • 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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