Comme un jeu

Find the Maclaurin polynomial of degree 4, $P_4(x)$, for $f(x)=\cos(x)$.

Use this polynomial to approximate the value of $\cos(0.5)$.

The Lagrange form of the remainder term, $R_n(x)$, gives the exact error of a Maclaurin approximation of degree $n$, and is defined as: $$ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1} $$ for some value $c$ between $0$ and $x$. Write down the Lagrange form of the remainder $R_4(x)$ for the approximation in part (b).

By finding the maximum possible absolute value of $R_4(0.5)$, determine the upper bound for the error in your approximation of $\cos(0.5)$.