Esercizi tipo Analisi matematica I - Germano/Martinelli

Esercizi

1. Limiti di funzioni (Punti di accumulazione)

2. \(\lim_{{P}\to{P_0}} \phi(P) = l\)

   \(\forall \epsilon > 0 \exists \delta > 0\) sse \(\Rightarrow | \phi(P) - l | < \epsilon\)

   \(\Rightarrow d(P, P_0) < \delta \Rightarrow P \in E \Rightarrow \phi(P) \in \left[ \phi(P) < \overline{K} \right]\)

   lim \((x,y) \to (0,0) \dfrac{x^4}{x^2+y^2} \neq 0\)

   \(\lim_{{g\to0}} \Rightarrow 0\);con \(\cos \theta \Rightarrow \beta \cos \theta \Rightarrow\) (x_0,y_0)

3. f(x,y) = \(\dfrac{x^4}{x^2+y^2} \neq 0\)
4. \(x_0= 0; y_0= 0\)
5. \(x \to 4 \neq 0\)
6. \(\lim_{{x,y}\to{(0,0)}} \dfrac{x^4}{x^2+y^2} = 0\)
7. \(\forall \epsilon > 0 \Rightarrow \exists \delta > 0\) sse \(\Rightarrow \dfrac{x^4}{x^2+y^2} < \epsilon\)
8. PF \(x_0=0, y_0=0\)
9. Dim. fate una maggioranza a \(\Rightarrow S.B.C.\)
10. A \(\Rightarrow\)
• \(\dfrac{x^4}{x^2+y^2}\)
• BRD per il prossimo esempio \({\sqrt{x^2+y^2}} \leq \epsilon\)

1. x^4 \(\to \dfrac{x^4}{x^2+y^2} \leq\)
2. \(\sqrt{x^2+y^2} \leq \epsilon \Rightarrow \delta_{\epsilon} \dfrac{= \sqrt{\epsilon}}\)
3. lim \(\Rightarrow\ x=y\), \(\dfrac{x^4}{x^2+y^2} \to 0\)

3) f(x,y)=\frac{x^2}{x^2+4y^2}

1. x^2+4y^2≠0

2. \lim_{(x,y)\to(0,0)}\frac{x^2}{x^2+4y^2}?

\frac{x^2}{x^2+4y^2}?

4. A ≤ B < ε

A = \frac{x^2}{x^2+4y^2} < \frac{x^2+4y^2}{x^2+4y^2} =1 < ε

\lim_{(x,y)\to∞} \frac{x^2}{x^2+4y^2}?

A = \frac{x}{x^2+4y^2} < \sqrt{x^2+4y^2} < B

\lim_{(x,y)\to∞} \frac{x}{x^2+4y^2} =0

5) f(x,y) = \frac{xy^2}{x^2+4y^4}

1. x^2+4y^4≠0

2. \lim_{(x,y)\to(0,0)}\frac{xy^2}{x^2+4y^4}?

3. ∀ε > 0 ∃δ: \sqrt{x^2+4y^4}∞

1/(log(sup>k))

(4/e^x -1)

1/(e^(x+4))

1/(log(8(sup>(k+1))))

lim

log(sup>k/(e^(x+4)k)

+∞

u e^x/(e^x+4)

{ 2u/(e^x-4)

lim

log e^x - log(4 + e^x))

log log(sup>k)

{ e e^x/(e^x -4)

2u/

u -e^x log(k/sup>k)))

DIVIAs

CONVERGE

1.

{2u/{u > 0

u/e^x -4

u > 0

4

1. u > e^x/sup>3

u e^x/=4

u -e^{u x}

3.

{u2 u > 4

u e > e^x/=4

u

2.

u < e^x

u e^x/=4

u e < e^x + 4

u < 0

8.

4 u e < e^x/sup>4

u e^{x< 4 +}

u 4 < 2u

2u/{u > 0

u > 0

4

q(u) < 0.1 >=4

u e^x/sup>3

u < e^x/sup>4

u -24 + e^x e^x-4

CONVERGE

1)

y' - 2y = x²

y' - 2y = x² eq. diff. d 1° grado non omogenea

• a(x) = -2, b(x) = x²
• ∫ a(x)dx = ∫ -2dx = -2∫ dx = -2x
• e∫a(x)dx = e^-2x

y_h = ce^-2x

y_p = x f(x)

• f'(x) + 2f(x) = 1
• ∫xe^-2xdx = x · -e^-2x
• ∫e^-2xdx + c
• -e^-2xx - e^-2xe^-2x + e^-2x + c
• - x · e^-2x
• ∫x · e^-2x dx = x · e^-2x -∫ e^-2x dx - x · e^-2x + 2 · e^-2x
• ∫ e^-2x dx = - x · e^-2x +∫e^-2xdx
• x · e^-2x x · e^-2x
• -∫x · e^-2x + x · e^-2x
• ∫ e^-2xdx = - x · e^-2x +∫ e^-2x dx
• 2x · e^-2x
• -e^-2x · x · e^-2x · e^-2x + e^-2x -

y = -^x²/⁄₂ - x/2 + 1/2 + c/e^-2x

U(x) = x²/2 + x/2 + 1 + c → U(x) = -1/2(x² + x + A) + C

3)

y' = 1/x + 1/x sin x eq. diff. d 1° grado non omogenea

• y' = 1/x + 4/x + b(x) = 2, b(x) = x · k e^kx = 1
• ∫x

Ve = ∫sin x dx + c - cos x + c

u^ie = -x cos x + c → u^ie = -x cos x + C

Ve = 4/x - 1/x + sin x/x

y = 4/x - 1/x + x sin x

y(π/2) = -π/2 cos π + π/2 x/2

Ve = cos/