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Point) Find y as a function of t if576y" + 336y' + 49y = 0,y(0) = 4 y (0) = 5. y...

Question

Point) Find y as a function of t if576y" + 336y' + 49y = 0,y(0) = 4 y (0) = 5. y

point) Find y as a function of t if 576y" + 336y' + 49y = 0, y(0) = 4 y (0) = 5. y



Answers

You are given the parametric equations of a curve and a value for the parameter $t$. Find the coordinates of the point on the curve corresponding to the given value of $t$. $$x=2-4 t, y=3-5 t, t=0$$

Section 3.6 Problem number sixties. I'm dealing with derivatives here that involved the chain rule. So what I see years, this is something raised to a power. So the power rules, the first thing is gonna be negative. Five and something raised the negative six. I'm gonna have a lot of negative exponents and met to make sure I don't make mistakes with the algebra. This one, I can easily transform this and just rewrite it as five t plus 2/3 T minus four raised to the fifth power. And now I'm done with all my negative exponents so that maybe an easier thing to integrate. So to integrate that so d y d t start with the power rule. So that is going to be five times all of that expression five t plus 2/3 T minus four raised to the fourth Power times the derivative of that expression. Well, that's in the form of, you know, f over G, which tells me quotient rule. So to take the derivative of what's inside that parentheses with the derivative of the numerator, which is five times the denominator minus the derivative of the denominator times the numerator hold that over the denominator squared And now the calculus is done were into algebraic simplification here. So this is going to be five times five t plus two to the fourth over three T minus four, toothy six power And then what I'm left with here is 15 T minus 20 and then minus 15 T, then minus six So you could see that the tea terms canceled here and this is minus 26. So it's five times minus 26. So this gives me a final answer of minus 1 30 then five t plus two to the fourth over three. T minus four to the sixth power and that is my final answer. So you would have gotten to the same answer if you kept this negative exponents issues. You had had a couple more algebraic steps. So just one of those things that integrating with positive exponents a little bit simpler because you reduce the number you're reducing both ways. But there's more cancellation these to happen with these negative exponents. So anyway, just and people tend to make more with meticulous takes with the negative number, So just my choice There

In this question given the thanks in cultures coroner, T and a Y in co June June deeper as far. And now for the first thing question congee Endara on the exam I d d you e coach you know one armatures Karuna a day. And for the second one we confined it. Do you wanna arm a TT? Then we get equal to that too. And then from here Giuliana, we confined it do Why are pretty thanks exactly Accord Unity 100 et give anybody Thanks on what did they they were getting? Quit that you are the one I would choose Gardiner Day and then you go to the phones clarity. And now because the take what you want Never forget culture and at the ICO to one and then again in coaching afar. And now want to find a ding? Why the square? Why already X exam? It isn't in question to deal with the Ex Im the d white Bram. Then we get equal to that. The wife remember DT They running by the banks of a deity. So from here And we do that, dearie with respect unity in a coach in the far armatures. Coroner t the ex Im ready to get equal to one off Joe Square under that they So we see we can consider this on my own. Today we get equal to the far So both answer here. Could you fall?

Section three at six Problems 66 wise Acquittal four. Sign of the square root of one plus the square root of T. This is just testing. Are you solid with chain role? So d y d t well got a constant in front that sticks around for the differentiation. So the outermost function is the sine function I see there, What's the derivative of the sine function? It is the co sign. So it's the co sign of this quantity. Now it balls down to now finding what is the derivative of what you see there. So that is the square root of a quantity. So this is going to be the driven or there is 1/2 Um and then you're gonna have one plus square to t to the minus 1/2. So you could just write that right here. So that took care of this radical. And then you have to take now the derivative of this quantity that's inside the radical. So that is going to be 1/2 square root of t. So what you do, you start with the outermost. This is the sign of a quantity. Okay. With that quantity turned out to be a radical. Take the different differential of that radical. There was a radical inside of that. Take the differential of that radical as well. So that's where it all goes down to the calculus. And now we just see like, Is there any simplification that can happen? It looks like you've got a four and a two and a two. That's gonna all cancel out nicely. So it looks like we're going to be left with one over the square root of tea and then the square root of one plus the square root of T times, the co sign of square root of one plus square root of tea. And so that's our final answer. So again, you could just guess it doesn't make it different, but is the co side of one plus rooty over root tea and then one plus root tea? So that's our final answer

Section 3.6 Number 69 were working with problems that requires to know the chain rule for differentiation. So here to take the derivative of y with respect to t besides the chain role, I'm gonna have to use the product rule. This looks like some function of after times a function of G. So that's going to be F crime. So that's three times G. So two T squared minus five to the fourth plus G Prime F said the derivative of G is going to be four times two t squared on its five to the third power times the derivative of what you see inside the parentheses, which is just four t So that's G prime times F, which is three t. And so if we factor out, what do we have in common here? It looks like there is in common three and then two t squared minus five to the third power. So when you factor this out, um, and then see what we are left with. So factoring out three to t squared minus five to the third power, that leaves me to t squared minus five and then it looks like plus four times for tea. And so this is three to t squared, minus five cubed and this looks to be, what, 16 18 t squared minus five. So 18 t squared, minus five. And so that's our final answer. Three times to T Square, but it's five cubed times 18 t squared, minus five.


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