14.9: Mazoezi ya Mapitio ya Sura ya 14

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Edwin “Jed” Herman & Gilbert Strang
OpenStax

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Kwa mazoezi yafuatayo, onyesha kama taarifa hiyo ni ya kweli au ya uongo. Thibitisha jibu lako kwa ushahidi au mfano wa kukabiliana.

1. Uwanja wa$f(x,y)=x^3\arcsin(y)$ ni$ \big\{ (x,y) \, | \, x \in \mathbb R\text{ and }−\pi≤y≤\pi \big\}.$

2. Ikiwa kazi$f(x,y)$ inaendelea kila mahali, basi$f_{xy}(x,y) =f_{yx}(x,y).$

Jibu: Kweli, kwa theorem ya Clairaut

3. Makadirio ya mstari kwa kazi ya$f(x,y)=5x^2+x\tan y$ wakati$(2,π)$ huo hutolewa na$L(x,y)=22+21(x−2)+(y−π).$

4. $(34,916)$ni hatua muhimu ya$g(x,y)=4x^3−2x^2y+y^2−2.$

Jibu: Uongo

Kwa mazoezi yafuatayo, mchoro kazi katika grafu moja na, kwa pili, mchoro safu kadhaa za ngazi.

5. $f(x,y)=e^{−\left(x^2+2y^2\right)}$

6. $f(x,y)=x+4y^2$

Jibu: $Mpango wa Mpangilio kwa kazi z = x + 4y ^ 2$

Kwa mazoezi yafuatayo, tathmini mipaka ifuatayo, ikiwa iko. Ikiwa haipo, thibitisha.

7. $\displaystyle \lim_{(x,y)→(1,1)}\frac{4xy}{x−2y^2}$

8. $\displaystyle \lim_{(x,y)→(0,0)}\frac{4xy}{x−2y^2}$

Jibu: Haipo.

Kwa mazoezi yafuatayo, pata muda mkubwa wa kuendelea kwa kazi.

9. $f(x,y)=x^3\arcsin y$

10. $g(x,y)=\ln(4−x^2−y^2)$

Jibu: Kuendelea katika pointi zote juu ya$xy$ -plane, isipokuwa ambapo$x^2 + y^2 > 4.$

Kwa mazoezi yafuatayo, pata derivatives zote za kwanza za sehemu.

11. $f(x,y)=x^2−y^2$

12. $u(x,y)=x^4−3xy+1,$pamoja$x=2t$ na$y=t^3$

Jibu: $\dfrac{∂u}{∂x}=4x^3−3y,$

$ \dfrac{∂u}{∂y}=−3x,$

$\dfrac{dx}{dt} = 2$na$\dfrac{dy}{dt} = 3t^2$

\ (\ kuanza {align*}\ dfrac {du} {dt} &=\ dfrac {u} {x}\ cdot\ dfrac {dx} {dt} +\ dfrac {u} {y}\ cdot\ dfrac {dt}\ [4pt]
&= 8x ^ 3 -6y -9xt ^ 2\\ [4pt]
&= 8\ kubwa (2t\ kubwa) ^3 - 6 (t ^ 3) - 9 (2t) t ^ 2\\ [4pt]
&= 64t ^ 3 - 6t ^ 3 - 18t ^ 3\\ [4pt]
&= 40t ^ 3\\ mwisho {align*}\)

Kwa mazoezi yafuatayo, pata derivatives yote ya pili ya sehemu.

13. $g(t,x)=3t^2−\sin(x+t)$

14. $h(x,y,z)=\dfrac{x^3e^{2y}}{z}$

Jibu: $h_{xx}(x,y,z) = \dfrac{6xe^{2y}}{z},$
$h_{xy}(x,y,z) = \dfrac{6x^2e^{2y}}{z},$
$h_{xz}(x,y,z) = −\dfrac{3x^2e^{2y}}{z^2},$
$h_{yx}(x,y,z) = \dfrac{6x^2e^{2y}}{z},$
$h_{yy}(x,y,z) = \dfrac{4x^3e^{2y}}{z},$
$h_{yz}(x,y,z) = −\dfrac{2x^3e^{2y}}{z^2},$
$h_{zx}(x,y,z) = −\dfrac{3x^2e^{2y}}{z^2},$
$h_{zy}(x,y,z) = −\dfrac{2x^3e^{2y}}{z^2},$
$h_{zz}(x,y,z) = \dfrac{2x^3e^{2y}}{z^3}$

Kwa mazoezi yafuatayo, pata usawa wa ndege ya tangent kwenye uso maalum katika hatua iliyotolewa.

15. $z=x^3−2y^2+y−1$katika hatua$(1,1,−1)$

16. $z=e^x+\dfrac{2}{y}$katika hatua$(0,1,3)$

Jibu: $z = x - 2y + 5$

17. Takriban$f(x,y)=e^{x^2}+\sqrt{y}$ katika$(0.1,9.1).$ Andika makadirio yako linear kazi$L(x,y).$ Jinsi sahihi ni makadirio ya jibu halisi, mviringo kwa tarakimu nne?

18. Kupata tofauti$dz$ ya$h(x,y)=4x^2+2xy−3y$ na takriban$Δz$ katika hatua$(1,−2).$ Hebu$Δx=0.1$ na$Δy=0.01.$

Jibu: $dz=4\,dx−dy, \; dz(0.1,0.01)=0.39, \; Δz = 0.432$

19. Kupata derivative directional ya$f(x,y)=x^2+6xy−y^2$ katika mwelekeo$\vecs v=\mathbf{\hat i}+4\,\mathbf{\hat j}.$

20. Kupata maximal directional derivative ukubwa na mwelekeo kwa ajili ya kazi$f(x,y)=x^3+2xy−\cos(πy)$ katika hatua$(3,0).$

Jibu: $3\sqrt{85}\langle 27, 6\rangle$

Kwa mazoezi yafuatayo, pata gradient.

21. $c(x,t)=e(t−x)^2+3\cos t$

22. $f(x,y)=\dfrac{\sqrt{x}+y^2}{xy}$

Jibu: $\vecs \nabla f(x, y) = -\dfrac{\sqrt{x}+2y^2}{2x^2y}\,\mathbf{\hat i} + \left( \dfrac{1}{x} + \dfrac{1}{\sqrt{x}y^2} \right) \,\mathbf{\hat j}$

Kwa zoezi zifuatazo, tafuta na uainishe pointi muhimu.

23. $z=x^3−xy+y^2−1$

Kwa mazoezi yafuatayo, tumia multipliers ya Lagrange ili kupata maadili ya juu na ya chini ya kazi na vikwazo vilivyopewa.

24. $f(x,y)=x^2y,$chini ya kikwazo:$x^2+y^2=4$

Jibu: kiwango cha juu:$\dfrac{16}{3\sqrt{3}},$ kiwango cha chini:$-\dfrac{16}{3\sqrt{3}},$

25. $f(x,y)=x^2−y^2,$chini ya kikwazo:$x+6y=4$

26. Mchezaji ni kujenga koni ya mviringo sahihi nje ya block ya alumini. Mashine inatoa hitilafu ya$5\%$ urefu na$2\%$ katika radius. Pata kosa la juu kwa kiasi cha koni ikiwa mashine hujenga koni ya urefu wa$6$ cm na cm ya radius$2$.

Jibu: $2.3228$cm ³

27. Compactor ya takataka iko katika sura ya cuboid. Fikiria compactor ya takataka imejaa kioevu kisichoweza kuingizwa. Urefu na upana hupungua kwa viwango vya$2$ ft/sec na$3$ ft/sec, kwa mtiririko huo. Pata kiwango ambacho kiwango cha kioevu kinaongezeka wakati urefu ni$14$ ft, upana ni$10$ ft, na urefu ni$4$ ft.