Applicationes derivatarvm: introdvctio

Antequam vnitatem incipias, pvnctvm vnde proficisceris metire. Brevis est sui ipsivs probatio: non aestimatvr, et eivs exitvs in solo tvo navigatro servatvr.

1. Qvaestio 1Qvo ex his casibvs theorema Weierstrassii praestat fvnctionem valorem svvm maximvm in copia indicata attingere?

   $f(x) = \abs{x^2 - 1}$ in $\icc{0}{2}$

   $f(x) = \arctan x$ in $\mathbb{R}$

   $f(x) = x^2$ in $\ico{0}{1}$

   Fvnctio qvae $x$ in $\ico{0}{1}$ et $0$ in $1$ valet, in intervallo $\icc{0}{1}$

2. Qvaestio 2Fvnctio $f$ in $\mathbb{R}$ derivabilis est, cvm $f(0) = 1$ et $\abs{f'(x)} \leq 2$ pro omni $x$. Qvid de $f(3)$ praestari potest?

   $-5 \leq f(3) \leq 7$

   $f(3) = 7$

   $\abs{f(3)} \leq 6$

   Nihil, qvia derivata in nvllo pvncto nota est.

3. Qvaestio 3Compvta $\lim_{x \to 0} \frac{x^2 \sin\frac{1}{x}}{\sin x}$.

   $0$

   Non exstat, qvia qvotiens derivatarvm limitem non habet.

   Non exstat, qvia $\sin\frac{1}{x}$ limitem non habet cvm $x \to 0$.

   $1$

4. Qvaestio 4In $\mathbb{R}$ estne stricte crescens fvnctio $f(x) = x + \cos x$?

   Est: derivata eivs, $1 - \sin x$, negativa non est neqve in omnibvs pvnctis vllivs intervalli aperti evanescit.

   Non est: derivata eivs in infinitis pvnctis evanescit.

   Crescens est, sed non stricte, qvia derivata eivs non in omni pvncto positiva est.

   Monotona non est, qvia cosinvs ascendit et descendit.

5. Qvaestio 5Fvnctio $f$ in $\icc{0}{4}$ continva est et in omni pvncto ex $\ioo{0}{4}$ derivabilis praeter $1$, et derivata eivs in solo $3$ evanescit. Si $f(0) = 2$, $f(1) = -1$, $f(2) = 3$, $f(3) = 5$ et $f(4) = 4$, qvi svnt valores eivs maximvs et minimvs absolvti in $\icc{0}{4}$?

   Maximvm est $5$, minimvm $-1$.

   Maximvm est $5$, minimvm $2$.

   Maximvm est $4$, minimvm $2$.

   Sine formvla $f$ sciri non potest.

6. Qvaestio 6Qvae svnt pvncta inflexionis fvnctionis $f(x) = x^5 - 5x^4$?

   Solvm $3$.

   $0$ et $3$, qvia in eis derivata secvnda evanescit.

   $0$ et $4$, qvia in eis derivata evanescit.

   Nvllvm, qvia derivata tertia in $0$ evanescit.

7. Qvaestio 7Qvae est circvmferentia oscvlans ad delineationem fvnctionis $f(x) = \frac{1}{x}$ in pvncto $(1, 1)$?

   Ea cvivs centrvm est $(2, 2)$ et radivs $\sqrt{2}$.

   Ea cvivs centrvm est $(0, 0)$ et radivs $\sqrt{2}$.

   Ea cvivs centrvm est $\left(\frac{3}{2}, \frac{3}{2}\right)$ et radivs $\frac{\sqrt{2}}{2}$.

   Ea cvivs centrvm est $\left(1, \frac{3}{2}\right)$ et radivs $\frac{1}{2}$.

8. Qvaestio 8Qvae pvncta parabolae $y = x^2$ pvncto $(0, 2)$ proxima svnt?

   $\left(-\sqrt{\frac{3}{2}}, \frac{3}{2}\right)$ et $\left(\sqrt{\frac{3}{2}}, \frac{3}{2}\right)$

   Vertex $(0, 0)$.

   Solvm $\left(\sqrt{\frac{3}{2}}, \frac{3}{2}\right)$.

   Nvllvm: cvm parabola circvmscripta non sit, distantia fortasse minimvm non attingit.

9. Qvaestio 9Fvnctio $f(x) = e^x - 3$ in $\mathbb{R}$ crescens et convexa est, et radix eivs est $\ln 3$, prope $1{,}0986$. Qvid facivnt iterata Newtoni et Raphsoni a pvncto initiali $x_0 = 0$?

   Primvm in altervm radicis latvs salit, ad $x_1 = 2$, et inde ad $\ln 3$ decrescvnt neqve eam praeterevnt.

   Ad $\ln 3$ crescvnt neqve eam praeterevnt, qvia a sinistra eivs incipivnt.

   Non convergvnt, qvia $f(x_0) < 0$ hypothesi theorematis non satisfacit.

   Alternis in vtroqve radicis latere iacent, semper propivs.

10. Qvaestio 10Ex vnitatibvs prioribvsFvnctio $f$ in $\icc{0}{1}$ continva est, cvm $f(0) = 1$ et $f(1) = -1$. Qvid praestari potest?

    Eam in aliqvo pvncto ex $\ioo{0}{1}$ evanescere.

    Eam in vno prorsvs pvncto ex $\ioo{0}{1}$ evanescere.

    Eam in $\icc{0}{1}$ decrescentem esse.

    Eam in $\frac{1}{2}$ evanescere.

11. Qvaestio 11Ex vnitatibvs prioribvsFvnctio $f$ in intervallo aperto qvod $a$ continet definita est, et $f'(a) > 0$. Qvid praestari potest?

    Exstare aliqvod $\delta > 0$ tale vt $f(x) < f(a)$ si $a - \delta < x < a$, et $f(x) > f(a)$ si $a < x < a + \delta$.

    $f$ in aliqvo intervallo aperto qvod $a$ continet crescentem esse.

    Valere $f(x) > f(a)$ pro omni $x > a$ intervalli.

    Valere $f'(x) > 0$ pro omni $x$ prope $a$.

12. Qvaestio 12Ex vnitatibvs prioribvsCompvta $f''(1)$ pro $f(x) = x^2 \ln x$.

    $3$

    $1$

    $2$

    $-2$

13. Qvaestio 13Ex vnitatibvs prioribvsAeqvatio $x^2 + xy + y^2 = 3$ prope pvnctvm $(1, 1)$ fvnctionem derivabilem $y(x)$ definit, cvm $y(1) = 1$. Qvantvm valet $y'(1)$?

    $-1$

    $-\frac{3}{2}$

    $-5$

    $1$

14. Qvaestio 14Ex vnitatibvs prioribvsCompvta $\lim_{x \to 0} \frac{\ln\left(1 + x^2\right)}{1 - \cos x}$.

    $2$

    $1$

    $\frac{1}{2}$

    Non exstat, qvia qvotiens formae $\frac{0}{0}$ est.