Problem $4[22$ marks$].$ Supposeanobjectwithweight$100$kgis dragged along a horizontal

plane by a force acting along a rope attached to the object. If the rope makes an angle $\backslash$theta

with the plane, then the magnitude of the force F is

F $=$

$20$

$0\mathrm{.}2$ sin$\backslash$theta $+$ cos$\backslash$theta

$2$

STAT$805/$ENGE$800$

Assignment $2$

Due Date: $29/05/2024$

a$.[10$ marks$]$ Evaluate F$$ at $\backslash$theta $=1$

$3$

$\backslash$pi by using two different approximations: centered

differences schema of order $4$ and Lanczos derivative method, each with three

different step sizes h$.$ Subsequently, calculate truncation errors Rt for both methods

across different step sizes.

b$.[7$ marks$]$ Assume that the truncation error Rt depend on h$,$ which is$,$

Rt $$Chp

Use the results in part $($a$)$ to calculate both C and p$.$ Also find the largest possible

step sizes for both numerical methods that can be used if that $|$Rt$|<=108.$

c$.[5$ marks$]$ Compare the two methods which uses the results in part b and provide

commentary on their calculation speed and accuracy. $($Hint: use MATLAB function

$$timeit$$ to measure the execution time$)$