Derivatae: 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 1Fvnctio $f(x) = \frac{1}{2x + 1}$ in $0$ definita est. Ex definitione derivatae, qvae est recta tangens ad delineationem eivs in pvncto $(0, 1)$?

   $y = 1 - 2x$

   $y = 1 + \frac{x}{2}$

   $y = -2x$

   $y = 1 + 2x$

2. Qvaestio 2Compvta derivatam fvnctionis $f(x) = \frac{x^2 + 5}{x - 1}$ in $3$.

   $-\frac{1}{2}$

   $6$

   $\frac{1}{2}$

   $-1$

3. Qvaestio 3Compvta derivatam fvnctionis $f(x) = e^x \cos x$ in $\frac{\pi}{4}$.

   $0$

   $\sqrt{2} \, e^{\frac{\pi}{4}}$

   $-\frac{\sqrt{2}}{2} e^{\frac{\pi}{4}}$

   $\frac{\sqrt{2}}{2} e^{\frac{\pi}{4}}$

4. Qvaestio 4Compvta derivatam fvnctionis $f(x) = \cos^2(2x)$ in $\frac{\pi}{8}$.

   $-2$

   $-1$

   $2$

   $\sqrt{2}$

5. Qvaestio 5In qvo horvm intervallorvm theorema Bolzani praestat aeqvationem $x^3 + 2x = 5$ solvtionem habere?

   $\icc{1}{2}$

   $\icc{0}{1}$

   $\icc{2}{3}$

   In nvllo, qvia theorema tantvm aeqvationes formae $F(x) = 0$ tractat.

6. Qvaestio 6Fvnctio $f(x) = x^5 + x$ continva et stricte crescens est in $\mathbb{R}$. Qvantvm valet $\left(f^{-1}\right)'(2)$?

   $\frac{1}{6}$

   $\frac{1}{81}$

   $6$

   $81$

7. Qvaestio 7Compvta derivatam fvnctionis $F(x) = \left(x^2 + 1\right)^x$ in $1$.

   $2 + 2\ln 2$

   $2$

   $2\ln 2$

   $1 + \ln 2$

8. Qvaestio 8Compvta derivatam ordinis $5$ fvnctionis $f(x) = x e^{2x}$ in $0$.

   $80$

   $16$

   $0$

   $160$

9. Qvaestio 9Rectangvlvm aream svam $6$ centimetrorvm qvadratorvm servat dvm basis eivs ratione $3$ centimetrorvm in secvndo crescit. Qvam celeriter altitvdo eivs mvtatvr eo instanti qvo basis $2$ centimetra metitvr?

   Decrescit ratione $\frac{9}{2}$ centimetrorvm in secvndo.

   Non mvtatvr: altitvdo valet $3$ et derivata eivs est $0$.

   Crescit ratione $\frac{9}{2}$ centimetrorvm in secvndo.

   Decrescit ratione $\frac{3}{2}$ centimetrorvm in secvndo.

10. Qvaestio 10Ex vnitatibvs prioribvsCompvta $\lim_{h \to 0} \frac{\cos h - 1}{h}$.

    $0$

    $1$

    $-\frac{1}{2}$

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

11. Qvaestio 11Ex vnitatibvs prioribvsCompvta $\lim_{h \to 0} \frac{e^{-2h} - 1}{h}$.

    $-2$

    $1$

    $2$

    $0$

12. Qvaestio 12Ex vnitatibvs prioribvsQvae harvm fvnctionvm est $\operatorname{o}(h)$ cvm $h \to 0$?

    $h \sin h$

    $\sin h$

    $\sqrt{\abs{h}}$

    $2h + h^2$

13. Qvaestio 13Ex vnitatibvs prioribvsSi $f(x) = 2x^3 + 3$, qvantvm valet $f^{-1}(19)$?

    $2$

    $8$

    $\frac{1}{13721}$

    $13721$

14. Qvaestio 14Ex vnitatibvs prioribvsQvantvm valet $\cos\left(\arcsin\left(-\frac{3}{5}\right)\right)$?

    $\frac{4}{5}$

    $-\frac{4}{5}$

    $\frac{16}{25}$

    $-\frac{3}{5}$