The last two weeks, we studied derivation, with topics like

• mean value theorem
• intermediate value theorem
• Taylor expansion

Exercises:

• Read Ross p257, Example 3 about smooth interpolation between $0$ for $x \leq 0$ and $e^{-1/x}$ for $x>0$. Construct a smooth function $f: \R \to \R$ such that $f(x)=0$ for $x\leq 0$ and $f(x)=1$ for $x\geq 1$, and $f(x) \in [0,1]$ when $x \in (0,1)$.
• Rudin Ch 5, Ex 4 (hint: apply Rolle mean value theorem to the primitive)
• Rudin Ch 5, Ex 8 (ignore the part about vector valued function. Hint, use mean value theorem to replace the difference quotient by a differential)
• Rudin Ch 5, Ex 18 (alternative form for Taylor theorem)
• Rudin Ch 5, Ex 22