Melody

Use Desmos graphing calc to graph it

I have started it for you

https://www.desmos.com/calculator/ta2h7syry8

THEN give us the link and we will help you with the next bit.

I started with 0.01 and the penalty doubled every day. What is the equation?

The penalty on day one is   0.01

The penalty on day two is   0.01 * 2

The penalty on day three is   0.01 * 2*2*2 = 0.01*2^2 = 0.01*2^(3-1)

etc

The penalty on day  "d "is   0.01 * 2^(d-1)

P = 0.01 * 2^(d-1)

Deleted

That is a good answer last guest :))

YES try constructing it.  Start wit the longest side and then try to get a triangle using the two short lengths.

See what happens.     

What is the derivative of

\(f(x)=x^3*(5x+1)^{-(2/3)}\\ \mbox{use product rule and composite function rule.}\\ u=x^3\\ u'=3x^2\\ v=(5x+1)^{-2/3}\\ v'=(-2/3)(5x+1)^{(-5/3)}\times 5\\ f'(x)=uv'+vu'\)

You should be able to finish it  :))

          lnx,x>=1

           /

f(x)={

           \

              e^(x-1)+βx-β-1,x<1

f has a derivative at x=1

find b

If there is a derivative at x=1 then the curve must be continuous and smooth at x=1

\(f(1) = ln(1)=0\\ f(x) = lnx \qquad for \;\;x\ge1\\ f ' (x) = \frac{1}{x} \qquad for \;\;x\ge1\\ f ' (1) = \frac{1}{1}=1 \)

Now I am going to ignore the fact that the following function is not defined for x=1

but it needs to have the same f(1) and f ' (1) as    f(x)=ln x does

\(f(x)=e^{(x-1)}+\beta x-\beta-1\\ f(x)=\frac{e^x}{e}+\beta x-\beta-1\\ f'(x)=e^x+\beta\\ NOW\\ f(1)=0\\ f(1)=e^{(1-1)}+\beta* 1-\beta-1\\ f(1)=1+\beta-\beta-1\\ f(1)=0 \qquad good \\also\\ f'(1)=1\\ f'(1)=e^1+\beta\\ 1=e+\beta\\ \beta=1-e \)

I do not think I have displayed this very well but  beta = 1-e

Thanks Guest.

\(\frac{x}{-4}=\frac{-4}{3}\\ \mbox{multiply both sides by -4}\\ \frac{x}{-4}\times -4=\frac{-4}{3}\times -4\\ x=\frac{-4*-4}{3}\\ x=\frac{16}{3}\\ x=5\frac{1}{3}\\\)

You could also just use the quadratic equation :))

Oh dear,

I am so sorry guest.

I have stuffed up your answer.  I am very sorry.    

I must have clicked 'edit' in yours instead of in mine by mistake.  I did not even know that I could do that!

I went back to try and fix it but that didn't help.  I don't think I tried to edit it 4 times, not even by mistake.  I do not know what has happened.  I do not know why it has changed it like that either.

I am SOOOOO  SORRY    

Simplify x(40-2x)(40-2x) over -4xsquared + 1600

If what I have done is what you meant then this would have been the best way to present it.

Simplify         [ x(40-2x)(40-2x) ] / [ -4x^2 + 1600 ]               

I have used difference of 2 squares to factoise the bottom, that is     \(a^2-b^2=(a-b)(a+b)\)

\(\frac{x(40-2x)(40-2x)}{-4x^2+1600}\\ =\frac{x(40-2x)(40-2x)}{1600-4x^2}\\ =\frac{x(40-2x)(40-2x)}{(40)^2-(2x)^2}\\ =\frac{x(40-2x)(40-2x)}{(40-2x)(40+2x)}\\ =\frac{x(40-2x)}{(40+2x)}\\ =\frac{x\times 2(20-x)}{2(20+x)}\\ =\frac{x(20-x)}{20+x}\\ or\\ =\frac{20x-x^2}{20+x}\\\)

Thanks guest, I have only just seen you answer, it was not there when I started :))

NOTE:  The 2 answers are the same.