本题目来源于试卷: Differential Calculus，类别为 IB数学
[问答题]
Let $y=e^{-\frac{x}{2}} \cos \left(\frac{x}{2}\right)$
1. Find an expression for $ \frac{\mathrm{d} y}{\mathrm{~d} x}$ .
2. Show that $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{1}{2} e^{-\frac{x}{2}} \sin \left(\frac{x}{2}\right)$
Consider the function f defined by $f(x)=e^{-\frac{x}{2}} \cos \left(\frac{x}{2}\right),-\pi \leq x \leq \pi$ .
3. Show that the function f has a local maximum value when $ x=-\frac{\pi}{2} $.
4. Find the x -coordinate of the point of inflexion of the graph of y=f(x) .
5. Sketch the graph of y=f(x) , clearly indicating the positions of the local maximum point, the point of inflexion and the intercepts with the axes.
6. Find the area of the region enclosed by the graph of y=f(x) and the x -axis.
The curvature at any point (x, y) on a graph is defined as $\kappa=\frac{\left|\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right|}{\left[1+\left[\frac{\mathrm{d} y}{\mathrm{~d} x}\right]^{2}\right]^{\frac{3}{2}}}$ .
7. Find the value of the curvature of the graph of y=f(x) at the local maximum point.
8. Find the value of $\kappa $ for x=0 and comment on its meaning with respect to the shape of the graph.
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