Assignment 2

Using Newton and Langrange methods, find polynoms which interpolate the following datas:
a.

latex x$ 1.2 2.1 3.0 3.6

$y$ 0.7 8.1 27.7 45.1

b.

$x$ 1.0 2.0 3.0 4.0 5.0 6.0

$y$ 14.5 19.5 30.5 53.5 94.5 159.5
A researcher wants to investigate the relation between crops produced ( $y$ ) and the amount of fertilizers given ( $x$ ). He conducted 5 experiments, in which he applied different amounts to each of his experiments. The result of his experiments is as follows:

the amount of fertilizer, $x$ (pound)	1	2	3	4	5
the amount of crops, $y$ (kg)	22	23	25	30	28

a. Find the interpolation polynom of the data given in the table above.
b. Use (a) to find the amount of crops produced if a farmer gave 2.5 pound of fertilizer.

Using the Taylor expansion, derive the following formula
$\displaystyle{f'(x) \approx \frac{-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)}{12h}}$ ,
and find the error of this formula.
Using the Taylor expansion, derive the following formula
$\displaystyle{f''(x) \approx \frac{2f(x)-5f(x+h)+4f(x+2h)-f(x+3h)}{h^2}}$ ,
and find the error of this formula.
Let $f(x)=x^3$ and find approximations for $f'(2)$ ,
a. Use central-difference formula, with $h=0.05$ ,
b. Use central-difference formula in no (3), with $h=0.05$ ,
c. Compare with the true value of $f'(2)$
Untuk setiap persamaan diferensial berikut ini, selesaikanlah dengan metodaTaylor berorde 2, Runge-Kutta orde 2 dan Runge-Kutta orde 4. Gunakan h=0.2 sebanyak dua langkah.
a. $y'=t^2-y$ dengan $y(0)=1$ ,
b. $y'=2ty^2$ dengan $y(0)=1$ .