# Chapter 1

## 1.1 Functions

• Kai: Follow carefully example 6 ( page 5-7). On our meeting next thursday explain carefully the proof of $\lim_{x\to \infty} e^{-1/x^2}=0$ using the definition with $$\epsilon$$ and $$\delta$$.

• equation 1.1-9 is an alternative definition of continuity:

$\left|f(x) -f(x_0) \right|< \epsilon \quad \mbox{whenever} \quad |x-x_0|<\delta$

• we need to review the squeeze theorem ( or sandwich theorem)

• we also need to know that if $$f(x)$$ is continuous and $$\{ x_n \}$$ is a sequence with $$x_n \to x_0$$, then $\lim_{n \to \infty} f(x_n) = f(x_0)$

Problems section 1.1

• 1 c), d), f) and g)

• 2 a), b) and e)

• 4 a) and d)

• 15, 16, 19, 25

## 1.11 Derivatives

There are two definitions of a derivative. Be sure to convince yourself that they are equivalent.

• Be sure to understand (be able to fill the details of) the proof of Theorem 1:

Theorem 1: If $$f$$ is differentiable at a point $$x_0$$, then it is continuous at $$x_0$$

• Kai Explain next thursday example 5

Problems 1.11

• 1, 2, 3, 7

• 8, 22 (compare these two problems)

• 23

• 24 this one seems hard, but interesting!

## 1.12 Maxima and Minima

In order to be consistent along the book we need to use the book’s definition of relative maximum and minimum:

$$f$$ has a relative maximum at $$x_0$$ if there is a neighborhood $$(a, b)$$ of $$x_0$$ with $f(x) \leq f(x_0 ) \quad \mbox{for all }\quad x\in (a, b)$

Example: This allows to any point of the function $$f(x) = 5$$ to be a relative maximum ( and relative minimum).

Problems 1.12

• 1, 3, 9, 10, 12, 13, 14, 16