Math Analysis Winter 2016-week 1

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