Problem $2.($Module $8)5$ points

Consider the function $g(t)=x^2{e}^{-\frac{x}{2}}.$

Determine true derivative of $g^{\text{'}}(t)$ computed analytically in the point $x_0=0\mathrm{.}6$ using MATLAB

program and symbolic operations.

Determine numerical derivative $g^{\text{'}}(t)$ in the point $x_0=0\mathrm{.}6$ and $h=0\mathrm{.}1$ using self$-$developed

MATLAB programs for various difference formulae:

a$)$ Two$-$point backward difference formula.

b$)$ Two$-$point forward difference formula.

c$)$ Two$-$point central difference formula.

d$)$ Taylor series based difference formula.

Find the percentage relative error in each case.

Note: Matlab program Mod$8\_$Example$1.$m can be used; however, it is work only for Forward

difference, and needs to be $\backslash$bar $($a$)$apted to the other formulas.

Put your answer in this field: Represent the result with $6$ decimal places.

Represent the result with $6$ decimal places.

True numerical value of first derivative in the point $0\mathrm{.}6$ is $d\frac{g}{d}t=$

Forward difference: $g^{\text{'}}(t)=$

Error $(\%)=$

Backward difference: $g^{\text{'}}(t)=$

Error $(\%)=$

Central difference:

$g^{\text{'}}(t)=$

Error$(\%)=$

Taylor based difference: $g^{\text{'}}(t)=$

Error$(\%)=$