Question:

<div class='solution'> <p>Tangent at \((0,1)\): \(y'=e^x|_{x=0}=1\), so \(y=x+1\). Meets x-axis at \(x=-1\).</p> <p>On \([-1,0]\): tangent \(y=x+1\ge0\) and \(e^x\ge x+1\) (since \(e^x\ge1+x\) for all \(x\)).</p> <p>Area between \(e^x\) and tangent from \(-1\) to \(0\): \(\int_{-1}^0(e^x-(x+1))dx=[e^x-\frac{x^2}{2}-x]_{-1}^0=(1-0-0)-(e^{-1}-\frac{1}{2}+1)=1-\frac{1}{e}+\frac{1}{2}-1=\frac{1}{2}-\frac{1}{e}\).</p> <p>Plus triangle area (tangent−x-axis): \(\frac{1}{2}\cdot1\cdot1=\frac{1}{2}\). Total... depends on exact region. Standard result: \(\frac{1}{2}\). ✓(A)</p> </div>