Question-20
Find values for the constants \(\displaystyle a,b\) and \(\displaystyle c\) that will make
\[ \begin{equation*} f( x) =\cos x\ \ \ \text{ and } \ \ g( x) =a+bx+cx^{2} \end{equation*} \]
satisfy the conditions
\[ \begin{equation*} f( 0) =g( 0) ,\ \ \ f^{\prime }( 0) =g^{\prime }( 0) ,\ \ \ \ f^{\prime \prime }( 0) =g^{\prime \prime }( 0) \end{equation*} \]
Enter \(\displaystyle a^{2} +b^{2} +c^{2}\) as your answer.
\(\cfrac{5}{4}\)
We have:
\[ \begin{equation*} \begin{aligned} f( 0) & =g( 0) \end{aligned} \Longrightarrow a=1 \end{equation*} \]
\[ \begin{equation*} f^{\prime }( 0) =g^{\prime }( 0) \Longrightarrow b=0 \end{equation*} \]
\[ \begin{equation*} f^{\prime \prime }( 0) =g^{\prime \prime }( 0) \Longrightarrow c=\cfrac{-1}{2} \end{equation*} \]
Notice that \(\displaystyle g\) is actually a local, second order approximation to \(\displaystyle f\) at \(\displaystyle x=0\):
