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helpppp: find the critical points: s(t)= (t-1)^4 (t+5)^3

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@mathcalculus , what have you tried?
yes, i used the product rule: \[\frac{ d }{ dx } (t-1)^{4}* (t+5)^{3}= (t-1)^{4} * 3(t+5)^{2} + (t+5)^{3} * 4(t-1)^{3}\]
then from there, i'm stuck.

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ok, so far so good. Now, factor out the greatest common factor of both terms. For example, they both have a factor of (t-1)^3, but that's not all they have in common. What else?
Don't they both have a factor of t+5 to a power?
oh yes, but are from opposite sides..
doesn't matter the order. AB+CA=A(B+C)
can you explain to me. i dont really see where we are going with factoring
The purpose in factoring is to simplify the derivative into a product of three factors. One of the factors will be t-1 to a power, a second factor will be t+5 to a power, and then you will be left with some factor containing a t as well.
And then once you have your derivative in factored form, you can set each factor equal to zero to find your critical points.
can you show me
i understand what youre saying but not how sure how to do that.
Ok, lets look at the t-1 term in each expression.
What is the largest exponent they share?
now i know that i write (t-1)^3
but what happens to the 4?
Exactly, so, we may factor out a \[\left( t-1 \right)^{3}\] from both expressions. Hang on, we will come to that.
Let's take a looke at the first expression in its entirety. If all you are removing is the \[\left( t-1 \right)^{3}\] You would be left with \[4\left( t+5 \right)^{3}\]
Do you see that?
no if i was to see the common factor.. i would get (t-1)^3+ (t+5)^2.... then what do i do with the numbers that are left...?
ok,, there is your mistake. It isn't a sum. It's a product.
ok so i dont see how you got 4(t+5)^3
Hang on, I was only talking about factoring out the t-1 from the first expression,, not from the whole thing.
well you showed me that
Hang on my pc is giving me fits.
why 4(t+5)?????
On the outside is what they both have in common, on ther inside is what is left after you have factored out that common factor.
i understand this part... (t-1)^3* (t+5)^2
Rememmber when I was saying if all we factored out was the (t-1)^2 You would have the 4(t+5)^3 left.
but then i dont understand why it is 4(t+5) ... i did this..4(t-1) + 3(t+5)
Well, now we include a (t+5)^2 as well, so what is left from the 4(t+5)^3? Just the 4(t+5).
i understand...
im left with 7t+17
t= -17/7
Perfect. So, now that each expression of the derivative is writtern in factored form, we set each of them equal to zero to find your critical points.
That is one of your critical points. There are two others.
ok so (t-1)^3 = 0
x= 0?
Not quite.
If (t-1)^3=0 then t=1
Now you have one more to do.
1? how?
Tinme for lunch, I will come back shortly to see if you have finished it.
the exponent is in the way.
Well, raise each side to the 1/3 power. That will get rid of your exponent.
kkk but how do we find the y
Just go back in and substitute the critical values.
For t=1 and t=-5, you will get y=0. But for t=-17/7, you won't get y=0.
i did....
(1-1)^4 (1+5)^3
and -17/7 i got a huge number.. doesn;t make sense.. :(
i got -432/823543
Well, you will need to raise some fractions with a denominator of 7 to the 4th and 3rd powers, so yeah, it will get rather ugly!
i see that but it keeps saying the critical points are wrong.
ok, I need to see the original problem. And what keeps saying your critical points are wrong?
Are you using an online program?
hang on.
1 Attachment
i attached it.
Your value for t=-17/7 is incorrect
You should get a large numerator divided by a denominator of 7^7.
so i dont know where i went wrong...
The numerator should be (-24)^4(18)^3
we did this step by step... now t is wrong?? really?
No, t is not wrong, but your evaluation of the function for t=-17/7 is incorrect.
so now how do i solve that?
obviously i know its wrong.
ive tried this problem out 3294810347 times.
now what do i do?
Should have (-17/7-1)^4(-17/7+5)^3
\[-\frac{ 17 }{ 7 }-1=-\frac{ 24 }{ 7 }\] and \[-\frac{ 17 }{ 7 }+5=\frac{ 18 }{ 7 }\]
wow this is insane. ive spent an hour to 2 with this problem.
why?? why did we ignore the exponents???
Now raise each of those terms to the appropriate powers.
sn;t it suppose to be part of it?? the original problem we were given??
i did
we didn't I was just showing you step by step.
thats how i got the hugeeeeeeee number
and so did i. i had it on paper.
what i dont understand is that youre basically saying raise those to the power.. in other words, im redoing this whole crap again.
Well, the numerator should be larger than your denominator, and it wasn't according to your answer.
which AGAIN, is leading m to the wrong huge answer i had before
No, you are finding the y-value that goes with the value you ccame out for t.
so there are 4 crtical points now?
or 3??
youre saying raise hem to the power and multiple right???
do you know rasing them to the power is going to GIVE ME A HUGEEEEEE NUMBER??
do you not see that was what i had in the first place??
There are only the three values of t we came up with, but you want them as ordered pairs, yo
ok good3 values done.
obviously i want them in order pairs.
i did EXACTLY what you just did with the -17/7 and got those numbers.
I see that, but your answer for t=-17/7 showed a smaller numerator than denominator and it shouldd be the other way around.
then show me.
whats the y value for -17/7
How did you do that? Each numerator is larger than each denominator and they both have total of 7th powers.
ok find the answer.
i need to see if it's correct
i cant keep wasting 2-3 hours on one problem.
@calcmat, are you stuck?
\[\left(- \frac{ 24 }{ 7 } \right)^{4}\times \left( \frac{ 18 }{ 7} \right)\]
now 18/7 needs to be raised to the 3rd power.
Should gha have a 3 on the second expression for the power.
and i just proved my point. you are taking em to the wrong direction AGAIN.
What are you talking about.
Hey, don't get mad at me, I am the one spending MY TIME helping YOU.
You can answer the question by taking (-24/7)^4(18/7)^3
I got 2349.5 rounded to 1 decimal place.
(-1+t)^3 (5+t)^2 (17+7 t) your 0's = 1,-5,-17/7
nvm. i''ll solve it
thanks everyone.
Those are the ones I got @dan815
that part is correct.
and @calmat01 don't round. thats wrong.
keep the numbers just the way they are. p.s found the answer. thanks.
Ok, so my answer wasn't completely off?
mathcalculus can you prove the divergence theorem for me?
So you are telling me that you should have left the answer as 1934917632/823543? That's insane @mathcalculus
what are all these big numbers?
lol insane.
your explanation was confusing. but the numbers are correct.
Gee, I spent all that time being patient with you, and all I get is my explanation was confusing?
lol truth to be told...truth be told.
I see. Well, if truth be told, the reason it was confusing was because you forgot how to factor. But I am not bitter. Take care and good luck with the rest of your assignment.
I did factor. I just didn't put it up because I was trying to understand you. And also, you were throwing out numbers without reason. & I'm glad you're not so bitter to take my words sensibly:) Thank you.

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