The figure shows the curve $y = e^x$ and its tangent at the point where $x = s$, for some $s > 0$. The curve bends upwards, so the tangent lies below it. The curve, the tangent and the $y$-axis enclose a region, shaded in the figure. Let $A(s)$ be its area. For larger $s$ the tangent crosses the $y$-axis below the origin, but the region is still bounded by the same three lines and curves. Find $A(s)$ as a formula in $s$, and then check each statement. Which of the following statements are true?

Fig. 1: The curve y = e^x and its tangent at a point x = s to the right of the y-axis, with axes. The tangent lies below the curve and crosses the y-axis below the curve's y-intercept. The region between the curve, the tangent and the y-axis is shaded. The point of contact is marked and the value s is labelled on the x-axis.