I know how to solve problems with one Ka value, it's using the Henderson-Hasselbalch equation.

But I don't know how to solve the following problems.

d) 100 ml Na3PO4 (0,20 mol/l) + 100 ml NaH2PO4 (0,10 mol/l)

Answer: 11,9

e) 100 ml Na3PO4 (0,10 mol/l) + 50 ml NaH2PO4 (0,40 mol/l)

Answer: 7,5

Ka,1 = 7,1 x 10^-3

Ka,2 = 6,3 x 10^-8

Ka,3 = 4,4 x 10^-13

How do I get to the answer? What Ka values are used?

ie is there a formula that int f'(x)f(x) = something?

"In a container of stable temperature and variable volume, there's the chemical equilibrium xA(g) + B(s) -> C(g) + 2D(g). At first, the total P=8atm. When we decrease the volume to V/2, the total P becomes P'=16atm. What is the coefficient x"

answers are: x=1, x=2, x=3 or x=4.

I found that C'=2C.

I don't know how to proceed and im getting discouraged because its the 2nd god damn question lol

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??):

i dont want to have to try both, is there a way just by looking at the question in general how to know which to convert to right off the bat

So we are proving inequalities, i know how to prove them by algorithm but i dont understand what am i doing, in other words i have no idea what it means.

For example, prove that tgx>x for x€(0,pi/2). Then by algorithm we form function f(x)=tgx-x and we want to show that this function is positive on (0,pi/2) Then we find derivative of function f'(x)=1/cos² x - 1 now we look where x belongs that is (0,pi/2) and if this is >0 function is increasing function or <0 decreasing function. 1/cos^2 x - 1 <0 so function is decreasimg and because f(0)=0 we have f(x)<0 on (0,pi/2). And thats the end of proof, i have no idea why are we finding derivative why then is it > or <0, i just know by algorithm.

Or another example. Prove that e^x >=1+x , for x>=0. Algorithm, function f(x)=e^x -1-x, then we want to show that function is positive on [0,+infinity). First derivative e^x -1 >0, so function is increasing , has minimum in x=0, so f(0)=0, we have f(x)>=0 for x€[0,+ininity), e^x >=1+x.

Can you explain why are we forming functions , why showing that is positive, why derivative and is it increasing or decreasing? Im intersted in thinking process, thanks.

there is my attempt at it, at some point i just started writing everything as products of primes to try to see any distinction. the denominator seems easy enough, as its just u to the power of 2n (except for the second and first derivative for some reason) but what’s really troubling is the polynomial on the nominator, any direction for a solution is very appreciated, any idea or anything!!

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?

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?

Can some please help me figure this question out I’ve been stuck on it for sometime and can’t figure out how to solve it