anonymous
  • anonymous
Find all critical points of R(t)=3t+77-15ln(t^2-3)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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amistre64
  • amistre64
R' = 3 -(30t)/(t^2-3) i think
amistre64
  • amistre64
3t^2 -30t -9 ------------- t^2- 3
amistre64
  • amistre64
3(t^2 -10 -3) would be the top if I did it right....

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amistre64
  • amistre64
-10t....
anonymous
  • anonymous
I Got R'=3-15/(t^2-3)(2t)
amistre64
  • amistre64
yep, as long as thats just screen clutter and not a glaring error :)
amistre64
  • amistre64
add the parts like fractions, 3(t^2 -3) -30t ------------ t^2 -3
anonymous
  • anonymous
Sorry how did you get 3(t^2-3) on top?
amistre64
  • amistre64
the top becomes a quadratic.... a fancy name for "highest power is a 2"
amistre64
  • amistre64
ok... you pretty much have the form: 30 3 - ---- now how do we add fractions togther? b
anonymous
  • anonymous
You have to have a common denominator?
anonymous
  • anonymous
But if R'(t)=3- 15 (2t) ----- (t^2 -3)
anonymous
  • anonymous
That would be 3 - 30t ------- (t^2 - 3)
amistre64
  • amistre64
30t 3 - ------- is what we got right? (t^2 -3)
anonymous
  • anonymous
That's what I got
amistre64
  • amistre64
then to add the "fractions" mulitply the left part by: 3 t^2-3 -- x ----- right? 1 t^2-3
anonymous
  • anonymous
Oh! Okay
amistre64
  • amistre64
:) deriving...no problem.... fractions? ehhhh :)
amistre64
  • amistre64
3t^2 -30t -9 ----------- t^2 -3
amistre64
  • amistre64
quadratic formula for the top to find the critical points...
amistre64
  • amistre64
5 +- [sqrt(900 - (4)(3)(-9))]/6
anonymous
  • anonymous
Where is the 5 from?
amistre64
  • amistre64
5 +- [sqrt(1008)]/6
amistre64
  • amistre64
30/6 -b/2a
amistre64
  • amistre64
you familiar with the quadratic formula?
anonymous
  • anonymous
yes
amistre64
  • amistre64
good heres what I get: t = 5 +- 2sqrt(63)/3
anonymous
  • anonymous
Did you do quadratic formula of 3t^2-30t-9?
amistre64
  • amistre64
yep..... as long as I kept track of myself
amistre64
  • amistre64
30 sqrt(900-(4)(3)(-9)) --- +- -------------- 6 6
anonymous
  • anonymous
So 30 +- sqrt(900+108) ---------------------- 6
amistre64
  • amistre64
we can simplify that by factoring out a 3 first.... sneaky little bugger it is :)
amistre64
  • amistre64
3(t^2 -10t -3) now quad the left side
amistre64
  • amistre64
10 +- sqrt(100 -(4)(-3)(1)) ----------------------- 2
amistre64
  • amistre64
10 +- sqrt(112) --- -------- same result, just easier to see 2 2
amistre64
  • amistre64
112 = 16*7 4sqrt(7)
anonymous
  • anonymous
So 5 +- 2sqrt(7)
amistre64
  • amistre64
thats better :)
amistre64
  • amistre64
you know how to test for min and max?
anonymous
  • anonymous
So those are the critical points? The neg and pos value from that?
anonymous
  • anonymous
No
amistre64
  • amistre64
yep... those are the critical points. to see how they are behaving, you take a second derivative....derive the derivative.
anonymous
  • anonymous
I don't think I have to do that.
amistre64
  • amistre64
you sure? its fun :) if you have to determine if they are MIN MAX or inflections, then we would, but if you just gotta find criticals, thats it.... dont forget to include the end points of the graph as well, they might be important
anonymous
  • anonymous
Wow, thanks! Can you help me with one more?
amistre64
  • amistre64
i dunno....... i feel a bout of stupidy brewing amongst me brain cells :) ........sure why not :)
anonymous
  • anonymous
Find all critical points of P(t)=-cos(4t)+39-2t
anonymous
  • anonymous
Holy cow sorry my computer spazzed out
amistre64
  • amistre64
thats just the cosine function moved about.... should be rather simple to determine
anonymous
  • anonymous
I have found P'(t) if that helps
amistre64
  • amistre64
P' = 4sin(4t) - 2 right?
anonymous
  • anonymous
Yes that's what I got
amistre64
  • amistre64
good, then we are either both genuises or idiots :) Ill go with genuisesss
anonymous
  • anonymous
(=
amistre64
  • amistre64
0 = 4sins(4t) -2 solve for sin(4t)
anonymous
  • anonymous
sin(4t)=1/2
amistre64
  • amistre64
what does 4t have to equal for sin to be 1/2?
amistre64
  • amistre64
there are 2 angles, that will satisfy this
anonymous
  • anonymous
I have no idea.
amistre64
  • amistre64
what? you gotta have SOME idea.... its just a circle :)
amistre64
  • amistre64
sin(30) = 1/2 and sin(150) = 1/2
anonymous
  • anonymous
How do I figure the two angles that will satisfy that?
amistre64
  • amistre64
you can either remember back to trig that a 30-60-90 triangle has sides of 1-2-sqrt(3) and interpolate from that, OR use the sin inverse function on a calculator
anonymous
  • anonymous
so sin^-1(1/2)=4t
amistre64
  • amistre64
yep
anonymous
  • anonymous
what from that tells me that 30 and 150 are what I'm looking for?
anonymous
  • anonymous
Sorry I want to understand
amistre64
  • amistre64
sin^-1 will only give you one angle tho...30. You have to remember to cross over the yaxis to get to the other one...150
amistre64
  • amistre64
4t = 30 and 4t = 150 solve for t :)
amistre64
  • amistre64
might be better to use radians instead of degrees, half dozen of one or 6 of the other
anonymous
  • anonymous
Oh! Okay thanks. Moving on
amistre64
  • amistre64
4t = pi/6 4t = 5pi/6
anonymous
  • anonymous
whoa where did that come from?
amistre64
  • amistre64
what!!! you see a bee??
anonymous
  • anonymous
No I mean the 4t=pi/6 and 4t=5pi/6
anonymous
  • anonymous
Wait is that 30 degrees and 150 degrees in radians?
amistre64
  • amistre64
:) there are a few ways to express angles. you can use degrees: 30 and 150 are degrees. OR you can use "radians" which measures the angle using the radius itself as a measuring stick. a circle is 360 degrees around OR 2pi radians pi/6 = 30 degrees 5pi/6 = 150 degrees
anonymous
  • anonymous
Okay
amistre64
  • amistre64
To get a number for "t" we should use radians :)
anonymous
  • anonymous
Okay
amistre64
  • amistre64
t = pi/24 t = 5pi/24 is what I come up with
amistre64
  • amistre64
30/4 or 150/4 depending on what your answers are looking for
anonymous
  • anonymous
so t=.1309 + pin/2
amistre64
  • amistre64
t = pi/24 = .1309 or t = 5pi/24 = .6545
amistre64
  • amistre64
for degrees... t = 7.5 degrees or t = 37.5 degrees
anonymous
  • anonymous
Okay. I have a question.
amistre64
  • amistre64
go ahead and ask :)
anonymous
  • anonymous
usually before solving for t my professor asks "which quadrants is sin negative in" why did we not have to look for that in this problem?
amistre64
  • amistre64
sin(angle) = 1/2 sin is only positive in Q1 and Q2 .... no need to look any where else, unless you can think of a reason to..
anonymous
  • anonymous
Okay. That makes sense. Thank you sooo much!
amistre64
  • amistre64
when sin = -1/2, it puts a different value into the equation that is not a critical point
amistre64
  • amistre64
you are soooo welcome :)
amistre64
  • amistre64
if you have any more questions, just start a new posting :)

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