MAT3, Mat-Øk 3, Mat-Tek3 -- Efterår 2014

Analyse 1

Fredag den 14. november 2014,  kl. 08:15
Sted: Rum NJV14, 4-117.

Dagens program

08:15-10:00

Repetition og forelæsning: Vi fortsætter med d.7 kapitel i [Lay]: egenskaber af Riemann integralet og Analysens fundamentalsætning.

10:15--12:00

Exercises: I. A lemma from last time: if $f$ is integrable on $[a,b]$, show that $|f|$ is also integrable and

$$|\int_a^b f(x)dx|\leq \int_a^b |f(x)| dx$$.

Hint:

Assume first that $|f|$ is integrable (we prove it later). We have $f(x)\leq |f(x)|$ and $-f(x)\leq |f(x)|$. Using Theorem 7.2.5 we have $ \int_a^b f(x)dx\leq \int_a^b |f(x)|dx$ and $ -\int_a^b f(x)dx\leq \int_a^b |f(x)|dx$. This is the same as $|\int_a^b f(x)dx|\leq \int_a^b |f(x)| dx$.

Let us now show that $|f|$ is integrable. Let $P$ be a partition of $[a,b]$. If $u$ and $v$ belong to $[x_{j-1},x_j]$, then $f(u)-f(v)\leq M_j(f)-m_j(f)$ and also $f(v)-f(u)\leq M_j(f)-m_j(f)$, thus:

$$|f(u)-f(v)|\leq M_j(f)-m_j(f).$$ From the triangle inequality we get:

$$|f(u)|\leq |f(v)|+|f(u)-f(v)|\leq |f(v)|+M_j(f)-m_j(f),$$ or:

$$|f(u)|\leq |f(v)|+M_j(f)-m_j(f).$$ Taking first the supremum over $u$ og and after that the infimum over $v$, we get:

$$M_j(|f|)\leq m_j(|f|)+M_j(f)-m_j(f).$$ In other words:

$$M_j(|f|)- m_j(|f|)\leq M_j(f)-m_j(f).$$ This implies:

$$U(|f|,P)- L(|f|,P)\leq U(f,P)-L(f,P).$$ The proof is now reduced to an application of Theorem 7.1.9.

II. Read Theorems 7.3.1 and 7.3.5 in [Lay].

III. Read Example 7.3.2, Corollary 7.3.3, Example 7.3.4 and Practice 7.3.7 in [Lay].

Literatur: Afsnit 7.3. De vigtigste sætninger er sætning 3.1 og 3.5.