5

3u = t,2 € R.t > Ou(z. 0) = 2 ~ € R...

Question

3u = t,2 € R.t > Ou(z. 0) = 2 ~ € R

3u = t,2 € R.t > Ou(z. 0) = 2 ~ € R



Answers

$$\frac{d r}{d \theta}=\frac{r^{2}+2 r \theta}{\theta^{2}}$$

Let us look under the left hand side of the equation first, which is art are one hard to R. Three. You know, this can be written us our history to my s. Uh one is dealt upon S -E. This is delta of on s minus B in the delta upon s minus C, Which is nothing. But tell that to the power of 4th divided by S. S minus E as minus me and s minus C. We know that the area of the triangle has given us under route has s minus E. Yes, may not be S minus E. This is the Iran's formula. This becomes delta powerful data square which is nothing but data square, which is the area of the triangle square. This is seen as the right hand side of the equation, hence.

In this problem let us first look. And the left hand side of the equation Which is on one or two R three. This becomes delta upon s minus C. Into delta upon s minus B. Into delta upon s minus C. This becomes data cube upon as minus C. As minus B. S minus C. Multiplying and dividing my S. This becomes as delta cube divided by sister the square. This is delta times. Yes. Now let us look at the right hand side of the equation. Use the fact that caught off A by two, there's nothing but under root. S S minus A upon yes minus B into s minus C. Similarly carter B by two. Yes as s minus B upon S minus E. S minus C. And caught off see back to as as S minus C upon s minus A. And to s minus B. So from here the right hand side of the equation becomes this is our cube. And do my application. Oh Court Square. A court squared B by 20 courts by sea by too well give us mrs S s minus A upon s minus B as minus C. Into us as minus B upon s minus E into s minus C into us S minus C. Born s minus E into the s minus B. As fantasy. As fine as being s minus C. Each get cancelled we are left with rQ into s cube divided by s minus E. S minus B as minus C. Multiplying this by S this bias are into ss delta. So this is delta cube into a divided by this is delta square, which comes as delta into us. And we see that Alleges is equals two our ages hence.

Eso. Here's another one that I would read write. The problem is, the cube root of something is the same thing is taking it to the 1/3 power. Um, and you can double check that with a calculator, like if If you know, the cube root of eight, uh is equal to two with a calculator. Maybe you can try is eight to the 1/3 power, also equal to two. And I'm telling you that's true. If you didn't know that the reason why I do that for my students is I asked them to look for parentheses because that's your clear indicator of an inter function and an hour function. So what you do when you're asked to take the derivative S o D. Q D R uh, you do the outer function first. That's 1/3 to the negative 2/3 power, because one my 1/3 minus one is 2/3. So you leave that inter function alone, and then you take the derivative of it and you multiply by that function. So the drift of to ours to the derivative of R squared is to our and then the only thing to do. Now is simplify that, Um, not really much to do to do with two minus two are So that's in your numerator. Clearly, three is in the denominator. 1/3. Now this negative exponents sticks it back in the denominator. It's the cube root of that two ar minus R squared, squared. In case you didn't know this, um, the numerator of your rational explanations becomes the exponents, and then the denominator of irrational exponents is the index, the cube root. So this is how you get this answer?

Okay, so the function g of T equals to zero 40 less than two and t plus one 40 greater legal in two. So which is the permit to say Jerry of T equals two T plus one tons U of T minus two, because the function vanishes that, uh, before tables two. And it's equal to P plus one after tables, too. So that's the definition of the equipment's definition of J. T. So now we fly. Let's transform here and again we're gonna ply the shipping technique. So this sequel students eat too connected to us times you of two plus one plus two. So now this is heed to the US and left us of L a times l of U of t plus three. And the answer will be eaten into s tons one nowhere. I swear. It was three over ass. That's it.


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