Problem $1$:

Use zero$-$ through second$-$order Taylor series expansions $($three different expansions, as done in class on $2/6/24)$ to predict $f(4\mathrm{.}2)$ for $f(x)=x^4+2x^3-8x^2+3x+11,$ using a base point of $x_0=4\mathrm{.}0.$

For each approximation, list the following:

a$.$ Approximate value of $f(4\mathrm{.}2)$

b$.$ On what order of the step size $h$ is the truncation error?

c$.$ Compute the true percent relative error, ${}_t,$ for each approximation.

For each estimate of $f(4\mathrm{.}2),$ provide the true percent relative error, ${}_t.$ State which approximation is the most accurate. Upload a pdf of your handwritten work and$/$or any Python code or console calculations.