Taylor remainder estimates for $e^x$ and $\sqrt{1+x}$

How accurately does \(1 + x + x ^ { 2 } / 2\) approximate \(e ^ { x }\) for \(- 1 \leq x \leq 1 ?\) Can you find a polynomial that approximates \(e ^ { x }\) to within 0.01 on this interval?

Use the Lagrangian Remainder Estimate to determine how well the polynomial \(1 + x / 2\) approximates \(\sqrt { 1 + x }\) on \([ 0 , 1 / 10 ]\)