Naloge brez postopka reševanja

Irina Cristea; Hashem Bordbar; Alessandro Linzi

23 Naloge brez postopka reševanja

Naloga 1: Naj bosta dani matriki $A=\left(\begin{array}{ccccc}3&&0&&1\\0&&1&&2\\-1&&1&&0\end{array}\right)$ in $B=\left(\begin{array}{ccccc}1&&0&&1\\0&&-1&&1\\1&&2&&0\end{array}\right)$ .

Rešite matrično enačbo $2\cdot X-A^2\cdot X=I_3+3\cdot B^T\cdot X$ .

$R: X=(2\cdot I_3-A^2-3\cdot B^T)^{-1}=\left(\begin{array}{ccccc}-\frac{3}{32}&&\frac{5}{64}&&\frac{1}{16}\\\\-\frac{1}{160}&&-\frac{9}{320}&&-\frac{21}{80}\\\\-\frac{1}{40}&&-\frac{9}{80}&&-\frac{1}{20}\end{array}\right)$

Naloga 2: Izračunajte vrednost determinante matrike $A$ .

$A=\begin{pmatrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{pmatrix}$ $R: det(A)=xy+xz+yz$
$A=\begin{pmatrix} -3 & 1 & 1 & 0\\ 1 & 0 & 1 & 2\\ 1 & 1 & 1 &-2\\ 0 & 0 & -1& 4 \end{pmatrix}$ $R: det(A)=26$
$A=\begin{pmatrix} i & 1 & -i \\ 0 & 1+i & 1 \\ -1 & 0 & -2-i \end{pmatrix}$ $R: det(A)=3-2i$

Naloga 3: Rešite sistem linearnih enačb.

$\begin{equation*} \begin{cases} x+y+z-w-u = -6 \\ -x+2y-z-w+u = 11\\ 2x -y+2z+3u= -2 \end{cases} \end{equation*}$

$R: x=-3+w-z-\frac{5+2w}{3}, y=\frac{5+2w}{3}, u=3, z,w\in\mathbb{R}$ .

Naloga 4: Rešite sistem linearnih enačb.

$\begin{equation*} \begin{cases} 3x-y+2z= 7 \\ -5x+y-2z = 0\\ 10x -2y+4z= 6 \end{cases} \end{equation*}$

$R$ : Sistem ni rešljiv.

Naloga 5: Rešite sistem linearnih enačb.

$\begin{align*} \begin{cases} 2x - y + z = 0 \\ x + 2y - 2z=0 \\ 3x + y - z = 0 \end{cases} \end{align*}$

$R: x=0, y=t, z=t, t\in\mathbb{R}$ .

Naloga 6: Rešite sistem linearnih enačb.

$\begin{equation*} \begin{cases} x_1-x_2+2x_3-x_4+x_5= 0 \\ 4x_1-3x_2-x_3+x_4-2x_5 = 0\\ 3x_1 -2x_2-3x_3+2x_4-3x_5= 0\\ 6x_1-5x_2+3x_3-x_4=0 \end{cases} \end{equation*}$

$R: x_1=7t-4s+5u, x_2=9t-5s+6u, x_3=t, x_4=s, x_5=u, t,s,u\in\mathbb{R}$ .

Naloga 7: Poiščite lastne vrednosti in lastne vektorje matrike $A$ ter algebraične in geometrične večkratnosti njenih lastnih vrednosti.

$A=\left(\begin{array}{ccccc}8&&-2&&2\\-2&&5&&4\\2&&4&&5\end{array}\right)$
$R: \lambda_1=0, \lambda_2=\lambda_3=9, v_1=\begin{pmatrix} -1\\ -2\\ 2\end{pmatrix}$ , $v_2=\begin{pmatrix} -2\\ 1\\ 0\end{pmatrix}, v_3=\begin{pmatrix} 2\\ 0\\ 1\end{pmatrix}$
$A=\left(\begin{array}{ccccc}2&&0&&0\\1&&2&&1\\-1&&0&&1\end{array}\right)$
$R: \lambda_1=1, \lambda_2=\lambda_3=2, v_1=\begin{pmatrix} 0\\ -1\\ 1\end{pmatrix}$ , $v_2=\begin{pmatrix} 0\\ 1\\ 0\end{pmatrix}, v_3=\begin{pmatrix} -1\\ 0\\ 1\end{pmatrix}$
$A=\left(\begin{array}{ccccc}2&&1&&0\\1&&2&&1\\0&&1&&2\end{array}\right)$
$R: \lambda_1=2, \lambda_2=2+\sqrt{2}, \lambda_3=2-\sqrt{2}$ , $v_1=\begin{pmatrix} 1\\ 0\\ -1\end{pmatrix}$ $v_2=\begin{pmatrix} 1\\ \sqrt{2} \\ 1\end{pmatrix}, v_3=\begin{pmatrix} 1\\ -\sqrt{2}\\ 1\end{pmatrix}$
$A=\left(\begin{array}{ccccccc}2&&1&&0&&0\\0&&2&&0&&0\\0&&0&&1&&-1\\ 2&&0&&1&&1\end{array}\right)$
$R: \lambda_1=\lambda_2=2, \lambda_3, \lambda_4\not\in\mathbb{R}$ , $v_1=\begin{pmatrix} 1\\ 0\\ -1\\ 1\end{pmatrix}$
$A=\left(\begin{array}{ccccccc}1&&0&&2&&-1\\0&&1&&4&&-2\\2&&-1&&0&&1\\ 2&&-1&&-1&&2\end{array}\right)$
$R: \lambda_1=1, a(\lambda_1)=4$ , $v_1=\begin{pmatrix} 2\\ 4\\ 0\\ 0\end{pmatrix}, v_2=\begin{pmatrix} -1\\ 0\\ 2\\ 4\end{pmatrix}$ , $g(\lambda_1)=2$

Naloga 8: Zapišite enačbo ravnine $\pi$ , ki vsebuje točko $A(3,1,5)$ in je vzporedna premicama $p$ in $q$ .

$p:\begin{cases} x=t\\ y=2+t\\ z=-3+t,\end{cases} t\in\mathbb{R}, \ \ \mbox{in}\ \ q:\begin{cases} x-y+1=0\\ y+z-2=0.\end{cases}$

$R$ : Enačba ravnine $\pi: x-y=2$ .

Naloga 9: Poiščite enačbo premice $q$ , ki leži v ravnini $\sigma: x-4y+2z=3$ in ki pod pravim kotom seka premico $p$ , podano s presekom ravnin $\pi_1: x-2y-4z=-3$ ter $\pi_2: x+y-3z=-1$ .

$R$ : $\vec{s}_q{=}\vec{s}_p\times \vec{n}_{\sigma}{=}(-10,17,39); p\cap \sigma{=}A(15,-1,5); q: \begin{cases} x{=}15-10t\\ y{=}-1+17t\\ z{=}5+39t, t\in\mathbb{R}\end{cases}$

Naloga 10: Izračunajte naslednje limite:

$\displaystyle\lim_{x\to -\infty} \displaystyle\frac{\sqrt{3x^2+5}+x}{x+7}$ $R: -\sqrt{3}+1$
$\displaystyle\lim_{x\to +\infty} \left(\displaystyle\frac{3x^2+5x+1}{3x^2-7x+3}\right)^{2x+5}$ $R: e^8$
$\displaystyle\lim_{x\to 2} \displaystyle\frac{x^3-5x^2+8x-4}{x^3-3x^2+4}$ $R: \displaystyle\frac{1}{3}$
$\displaystyle\lim_{x\to 1} \displaystyle\frac{2x-\sqrt{x^2-x+4}}{x^2-1}$ $R: \displaystyle\frac{7}{8}$
$\displaystyle\lim_{x\to -\infty} \left(\displaystyle\frac{3x+2}{3x-7}\right)^{-5x-8}$ $R: e^{-15}$
$\displaystyle\lim_{x\to -2^+} \displaystyle\frac{x^2-x-6}{2x^2+8x+8}$ $R: -\infty$
$\displaystyle\lim_{x\to +\infty} (\sqrt{2x^2+3x+1}-\sqrt{2x^2-5x-2})$ $R: 2\sqrt{2}$
$\displaystyle\lim_{x\to -\infty} \displaystyle\frac{\ln (1+e^{2x})}{\ln (1+e^{5x})}$ $R: +\infty$
$\displaystyle\lim_{x\to 0} \displaystyle\frac{e^x-\pi ^x}{\sin (2x)}$ $R: \displaystyle\frac{1-\ln \pi}{2}$
$\displaystyle\lim_{x\to +\infty} (\sqrt{x^2+4x+5}-x-2)$ $R: 0$
$\displaystyle\lim_{x \rightarrow -\infty}\left(\sqrt{1+4x^2}-\sqrt{3+x^2}\right)$ $R: +\infty$
$\displaystyle\lim_{x \to 0} \displaystyle\frac{\sin (\sin x)}{x}$ $R: 1$
$\displaystyle\lim_{x \to +\infty} (e^x+x)^{\frac{1}{x}}$ $R: e$
$\displaystyle\lim_{x \to 1} \displaystyle\frac{x^{\frac{2}{3}}-1}{1-\sqrt{x}}$ $R: -\displaystyle\frac{4}{3}$
$\displaystyle\lim_{x \to \infty} x^{\frac{\ln{2}}{1+\ln{x}}}$ $R: 2$

Naloga 11: Poiščite konstanti $a$ in $b$ tako, da velja

$\displaystyle\lim_{x\to +\infty} \left(\displaystyle\frac{x^2+2}{x+3}-ax-b\right)=0.$

$R: a=1, b=3$

Naloga 12: Poiščite konstanti $a$ in $b$ tako, da velja

$\displaystyle\lim_{x\to 1} \displaystyle\frac{\tan (ax-a)+b-2}{x-1}=3.$

$R: a=3, b=2$

Naloga 13: Določite, če obstajajo, asimptote naslednjih funkcij:

$f(x)=\sqrt{\displaystyle\frac{x+2}{x+1}}$ $R: y=1; x=1$
$f(x)=\displaystyle\frac{x^3+2x+1}{x^2-3}$ $R: x=\pm \sqrt{3}; y=x$
$f(x)=\displaystyle\frac{ x-5}{\sqrt{x^2-9}}$ $R: y=\pm 1; x=\pm 3$
$f(x)=(2x-1)\cdot e^{-x+3}$ $R: y=0$
$f(x)=3x+e^{-x}$ $R: y=3x$

Naloga 14: Naj bo funkcija $f:D\longrightarrow\mathbb{R}$ podana s predpisom $f(x)=\displaystyle\frac{x^2+2x+3}{x^2-6x+a}$ , kjer je $a\in\mathbb{R}$ ter $D$ domena funkcije $f$ .

Določite vrednost parametra $a$ tako, da je premica $x=2$ navpična asimptota dane funkcije.
$R: a=8$
Določite domeno funkcije $f$ ter vse njene asimptote.
$R: D=\mathbb{R}\setminus\{2,4\}$ $x=2, x=4, y=1$

Naloga 15: Naj bo funkcija $f:\mathbb{R}\setminus \{\displaystyle\frac{m}{2}\}\longrightarrow \mathbb{R}$ podana s predpisom $f(x)=\displaystyle\frac{x^2+mx+6}{2x-m}$ . Določite vrednost parametra $m$ tako, da je premica $y=\displaystyle\frac{1}{2}x+\displaystyle\frac{3}{2}$ poševna asimptota dane funkcije.
$R: m=2$

Naloga 16: Določite vrednost parametra $m\in\mathbb{R}$ tako, da je funkcija $f:\mathbb{R}\longrightarrow\mathbb{R}$ , podana s predpisom $f(x)=\begin{cases} \sqrt{x^3+3e^x}, x\leq 0 \\ 5x-m, x>0 \end{cases},$ zvezna pri točki $x=0$ .
$R: m=-\sqrt{3}$

Naloga 17: Določite vrednost realnega parametra $a$ tako, da je funkcija $f:\mathbb{R}\longrightarrow\mathbb{R}$ , podana s predpisom $f(x)=\begin{cases} a, \ \ \mbox{če je} \ \ x\leq 0 \\ \displaystyle\frac{\sqrt{x}-1}{\sqrt[3]{x}-1},\ \ \mbox{če je} \ \ x>0 \end{cases},$ zvezna na $\mathbb{R}$ .
$R: a=\displaystyle\frac{3}{2}$

Naloga 18: Izračunajte odvod funkcije $f$ , podane s predpisom.

$f(x) = \ln (x^2\cdot e^{3x})$ $R: f'(x) = \displaystyle\frac{2}{x} + 3$
$f(x) = \sqrt{\sin \sqrt{x}}$ $R: f'(x) = \displaystyle\frac{\cos(\sqrt{x})}{4\sqrt{x\cdot \sin(\sqrt{x})}}$
$f(x) {= }\cos (e^{\sqrt{\tan (3x)}})$ $R: f'(x) {=} -\sin(e^{\sqrt{\tan (3x)}}) \cdot e^{\sqrt{\tan (3x)}} \cdot \frac{\frac{3}{\cos^2(3x)}}{2\sqrt{\tan (3x)}}$
$f(x) = \sqrt{ x \cdot\ln(x^4)}$ $R: f'(x) = \displaystyle\frac{2\cdot (\ln(x) + 1)}{\sqrt{4x \cdot\ln(x)}}$
$f(x){=} \cos(\sin(\cos(e^x)))$ $R: f'(x) {=} \sin(\sin(\cos(e^x))) \cdot \cos(\cos(e^x)){ \cdot \sin(e^x)\cdot e^x}$
$f(x) = x^2\cdot \sin^2(2x^2)$ $R: f'(x) = 2x\cdot \sin^2(2x^2) + {8x^3 \cdot \sin(2x^2)\cdot \cos(2x^2)}$
$f(x) = \displaystyle\frac{1}{\sqrt[3] {x+\sqrt{x}}}$ $R: f'(x) = -\displaystyle\frac{1 + \displaystyle\frac{1}{2\sqrt{x}}}{3\cdot (x+\sqrt{x})^{\frac{4}{3}}}$

Naloga 19: Po l’Hôpitalovem pravilu izračunajte naslednje limite:

$\displaystyle\lim_{x\to +\infty} x^n\cdot e^{-x}$ $R: 0$
$\displaystyle\lim_{x\to 0} \displaystyle\frac{x- (x+1)\cdot \ln (1+x)}{x^2\cdot (x+1)}$ $R: -\displaystyle\frac{1}{2}$
$\displaystyle\lim_{x\to 1} \displaystyle\frac{e^{2\cdot (x-1)}-x}{(2x-1)\cdot (x-1)}$ $R: 1$
$\displaystyle\lim_{x\to 0} \displaystyle\frac{\ln (\cos x+x^2)}{x^2+5}$ $R: \displaystyle\frac{1}{2}$
$\displaystyle\lim_{x\to 0} \displaystyle\frac{e^{3x}-e^{\sin x}}{\sin (3x)}$ $R: \displaystyle\frac{2}{3}$
$\displaystyle\lim_{x\to 0} \displaystyle\frac{\cos x-e^{x}+x}{x\cdot \sin x}$ $R: -1$ .
$\displaystyle\lim_{x \to 0} \displaystyle\frac{\sin x}{2x^2 - x}$ $R: -1$
$\displaystyle\lim_{x \to 1^+} \ln{x}\cdot \tan{(\displaystyle\frac{\pi x }{2})}$ . $R: -\displaystyle\frac{2}{\pi}$
$\displaystyle\lim_{x\to 0} \displaystyle\frac{\sin x}{\ln (1+x+x^2)}$ $R: 1$
$\displaystyle\lim_{x\to -\infty} x^2\cdot e^x$ $R: 0$