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

## Analyse 1

11. kursusgang

Onsdag den 8. oktober 2014,  kl. 08:15
Sted: Rum NJV14, 4-117.

Dagens program

08:15-10:15
Repetition og forelæsning: kontinuerte og differentiable reelle funktioner.
10:30--12:00
Exercises: page 260 in [Lay], exercises 3, 4, 5 and 6.
Hint for exercise 3:
c). Note that $f(1)=1$. Let $x_n$ be any sequence which converges to $1$ and $x_n<1$. Then $(f(x_n)-f(1))/(x_n-1)=3$ for all $n$, thus the left derivative exists and equals the value $3$. Now let $x_n$ be any sequence which converges to $1$ and $x_n > 1$. Then $(f(x_n)-f(1))/(x_n-1)=(x_n^2-1)(x_n-1)=x_n+1$ for all $n$, thus the right derivative exists and equals the value $2$. Hence $f$ is not differentiable at $x=1$.

Hint for exercise 4:
e). Let $c>0$ and write $(f(x)-f(c))/(x-c)=-1/(\sqrt{x}+\sqrt{c}) \times 1/\sqrt{x} \times 1/\sqrt{c}$. Conclude that $f'(c)=-1/2 c^{-3/2}$.

Hint for exercise 5:
a). Use the identity $a^3-b^3=(a-b)(a^2+ab+b^2)$
b). Since $(f(x)-f(0))(x-0)=1/x^{2/3}$ for $x\neq 0$, we see that if $x_n=1/n$ then $1/(x_n)^{2/3}=n^{2/3}$, which defines a divergent sequence.

Hint for exercise 6:
a). Show that $f'(x)=2x \sin(1/x) -\cos(1/x)$ if $x\neq 0$.
b). We have $(f(x)-f(0))/(x-0)=x\sin(1/x)$ for all $x\neq 0$. The right hand side converges to $0$ when $x$ goes to zero, hence $f'(0)=0$.
c). If $f'$ were continuous at $0$, then from the formula of $f'(x)$ at $x\neq 0$ we could conclude that $\lim_{x\to 0}\cos(1/x)=0$. But this limit does not exist.

Literatur: Vi skal gå gennem afsnit 1 og 2 fra kapitlen 'Differentiation' (starter på side 251) fra [Lay]. De vigtigste sætninger er sætning 1.7 på side 255 (regneregler med differentiable funktioner), sætning 1.10 på side 257 (kædereglen) og sætning 2.2 på side 263 (Rolles sætning).

Disse sider vedligeholdes af Horia Cornean.