For what it's worth, my answers are below, though I haven't checked them with anyone yet. Please let me know of any mistakes.
I haven't used LaTeX in years, so apologies for the formatting.

1.

\frac{-9x^2}{2}

\frac{\pi}{16}

1+ \frac{3}{2x-1}-\frac{x+4}{x^2+4}

4. i) The integral represents the area below the curve

y=e^{-1/2}

from x=0 to x=1. This area is greater than the sum of the areas of the rectangles whose bases are of width 1/n and whose height is

e^{-1/p}

where p ranges from 1/n for the smallest rectangle to (n-1)/n for the largest rectangle.
Therefore the sum of the areas of the rectangles is:

\frac{1}{n} e^{\frac{-1}{1/n}}+\frac{1}{n} e^{\frac{-1}{2/n}}+…+\frac{1}{n} e^{\frac{-1}{(n-1)/n}}

Which can be simplified to

\frac{1}{n} (e^{-n}+e^{-n/2}+…+e^{\frac{-n}{n-1}} )

ii) Upper bound

\frac{1}{n} (e^{-n}+e^{-n/2}+…+e^{\frac{-n}{n-1}}+e^{-1} )

iii)

LB=0.104

UB=0.197

iv) 368
5. i) show that
ii) positive cubic passing through (0,-k) and

(k^{1/3},0)

an example such as

x_1>0

but close to 0, so

x_2

is quite large
iii)

\alpha=\sqrt[3]{100}

x_2=4.66667

x_3=4.64172

iv)

e_1=-0.35841

e_2=-0.02508

e_3=-0.00013

Verify
6. i) prove that
ii) show that
7. i) prove that
ii) show that

x=\frac{5\sqrt6}{12}

8. i) show that
ii) show that
9. i) show that
ii) show that
iii)

I_3=\frac{-9}{50}+ln{\frac{5}{4}}

iv)

ln{\frac{5}{4}}

(edited 12 years ago)

Reply 89

Mr M

Study Forum Helper

Ok thanks for the paper.

dybydx's answers are all correct apart from one - the upper bound is 0.197 (you don't round an upper bound down).

Reply 90

Mr M

Study Forum Helper

Original post by dybydx

Friday's FP2 attached for Mr M

Thanks for the paper and I have replied above.

Reply 91

Ree69

eeek, I think I managed to get about 55/72. For 4ii), is it OK to just state the answer? It's 2 marks so I'm scared they might want a tiny bit of an explanation...

Reply 92

Mr M

Study Forum Helper

Original post by Ree69

eeek, I think I managed to get about 55/72. For 4ii), is it OK to just state the answer? It's 2 marks so I'm scared they might want a tiny bit of an explanation...