I am working through a book in mortgages which has the following question:

Determine the forward rates from the following spot curve. Then, determine the par-yield curve from the spot rates and the forward par-yield curve from the forward spot rates.

Finding the forward rates from the given spot rates is straightforward. Essentially, investing in a 2 year bond now earns 9% compounded. So, the one year rate one year from now (x) can be found so that 1.09 * 1.09 = 1.1 * (1+x)

The current par-yield curve is also easily determined. For e.g., if the current par yield curve for year 2 is 8 % it means that discounting 8 by 1.1 (from the spot yield curve above for year 1) and discounting 108 by 1.09 * 1.09 (from the spot yield curve above for year 2) should yield 100/par.

So far, so good. I am completely unclear about what the "forward par-yield curve" even is. The answers provided are as follows at the back of the book. I have been able to obtain the forward zero rates, but the "forward par rates" are completely unclear how to obtain. Any help is appreciated.

Forward zero rates:

Forward par rates (?? how is the table below obtained??):

For things like this how did they determine the radius and centre? For c I sort of get it cause it's an easier example but for e.g d how do they determine this?

How do you do these questions? Every method I try gives me a different result...

In this question is it correct that you do f(x)-r(x)? When I do that I get c=-4 and d=-12, is that correct?

third derivative and still dont know where its going

I need to prove that: x - y  ≤ | x^2 \* ln(x) - y^2 * ln(y) |  ≤ 3 * e * | x - y | and x,y ∈ [1,e]

Let f ( t ) = t^2 * ln(t). Using Lagrange theorem.

This function is defined and continuous on [0,+∞) and thus on the interval [1,e]. This function is differentiable on [1,e] .

From the Lagrange theorem it follows that there is a point c ∈ (1,e) such that f'(c) * ( y -x) = f ( y ) - f ( x ), this means that the derivative at that point is 2 * c * ln(c) + c.

This means that it is
2 * c * ln(c) + c = ( y^2 * ln(y) - x^2 * ln(x) ) / ( y - x )

We want to have 2 * c * ln(c) + c bounded.
Since c ∈ (1,e) , then 1 < c< e, so then we have c ≥ 1 and ln(c) ≥ 0 , this means that
2 * c * ln(c) + c ≥ 2 * 1 * 0 + 1 = 1

Then c ≤ e and ln(c) ≤ 1 , that is 2 * c * ln(c) + c ≤ 2 * e * 1 + e = 3 * e.

Now we have 1 ≤ 2 * c * ln(c) + c ≤ 3 * e that is
( y - x ) ≤ y^2 * ln(y) - x^2 * ln(x) ≤ 3 * e ( y - x ).

When we take the absolute value we can replace that is
| x - y | ≤ | x^2 * ln(x) - y^2 * ln(y) | ≤ 3 * e * | x - y | , but i need to get

x - y ≤ | x^2 * ln(x) - y^2 * ln(y) |.

Could it be mistake in exercise setup that my professor forgot about absolute value?

Thank you

The answer to part ii is this. How do they know that point D is on the y-axis?

I can’t seem to figure out how to do this. I need to be able to identify which color is which molecule/atom but I have no idea where to start. This crystalroster is Mg3Si4O10(OH)2 (I know OH is supposed to count as one ball) any help is much appreciated!

As title says, i'm a bit confused on how to even start this problem? I tried using an AI app (Gauth) to explain it and try to get to the answers, and i would then fact check with the answers provided, however the answers it said were correct, the website I submit the homework through said it was incorrect.

For each answer, how would you specifically solve for it? Is there a step i took the complete wrong path for?

Question/Work images provided.

Hi! So for the last month of my first term in uni, our physics professor handed out 31 questions to help study for next exam until the end of next month and I'm hard-stuck on the last one and would love some help (I emailed him but since its Christmas break he hasn't answered and both of my friends from uni gave up on the last question after many failed attempts), the question is originally in Swedish but ill do my best to translate it as accurately as possible. The question is in the field of nuclear physics but as of now we have not gone into molar masses and avogadros constant so this should be doable without it.

Question:

In a body of water there is 6500 the amount of hydrogen-atoms as deuterium-atoms. Assume you can utilize all the deuterium atoms in the 1m3 body of water to produce energy according to this reaction equation:

How much energy can you extract/obtain?

Now in my book I have the masses for neutrons, deuterium and helium nucleus, the water I need to calculate by myself and I get it to 18,01u per molecule but not sure how correct it is or if its even needed but I did it anyways. After that I went into getting the mass defect and I got it to 0,00349u, using the massdefect I could calculate how much energy I get per reaction (I got a hint from a friend to do so but I still don't really understand it?) but after getting the energy for the mass defect I'm kind of lost, I tried using the mass of one water molecule to get the number of molecules in water and the total number of particles of deterium knowing it has a ratio of 1/6500 but I still go wildly wrong answers form 0,02 GJ to 5 TJ to 20TJ

I checked what the answer is in a answer sheet the professor posted but it did not include a method only that the final answer is 3 TJ.

So yeah I would love to get some help here :)

TYVM

Edit: Forgot picture

Edit: Ok nvm I finally solved it, I think.. Got it to 2,6 TJ which I assume rounded up to 3 TJ from the significant figure of "1" in "1 cubic meter of water"

my answer: 3.2-3 / 0.2, using this:

I have tried solving this but i have no idea how to continue from this step. The only real answer is x1= 0. Does anyone have any idea how to solve this without guessing the answer?

I need to understand how to calculate the first component of this vector.

Hello. I need to write a research paper about Khalid ibn al Walid and I'm using Chicago for the citations. I found a primary source by a historian name al-Baladhuri, who wrote a book called Kitab Futuh al Buldan, which was translated to English. The link to the book is here: https://archive.org/details/in.ernet.dli.2015.175260

How am supposed to cite this in Chicago style, since it is a primary source found in a book? Any advice is useful. Thank you.

I have a formula for 3. I have no formula for 4(a) or 4(b). I'm helping a kid prepare for this and I don't have the foggiest how to handle the triangle stuff quickly or formulaically (as opposed to just counting and trying to remember)

I didn't put a specific grade because this is open to kids of all grades up to some limit so I don't know what would be appropriate