3.6E: Mazoezi ya Sehemu ya 3.6

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Katika mazoezi ya 1 - 6, iliyotolewa$y=f(u)$ na$u=g(x)$,$\dfrac{dy}{dx}$ tafuta kwa kutumia maelezo ya Leibniz kwa utawala wa mnyororo:$\dfrac{dy}{dx}=\dfrac{dy}{du}\dfrac{du}{dx}.$

1)$y=3u−6,\quad u=2x^2$

2)$y=6u^3,\quad u=7x−4$

Jibu: $\dfrac{dy}{dx} = 18u^2⋅7=18(7x−4)^2⋅7= 126(7x−4)^2$

3)$y=\sin u,\quad u=5x−1$

4)$y=\cos u,\quad u=-\frac{x}{8}$

Jibu: $\dfrac{dy}{dx} = −\sin u⋅\left(-\frac{1}{8}\right)=\frac{1}{8}\sin(-\frac{x}{8})$

5)$y=\tan u,\quad u=9x+2$

6)$y=\sqrt{4u+3},\quad u=x^2−6x$

Jibu: $\dfrac{dy}{dx} = \dfrac{8x−24}{2\sqrt{4u+3}}=\dfrac{4x−12}{\sqrt{4x^2−24x+3}}$

Kwa kila moja ya mazoezi yafuatayo,

a. kuharibu kila kazi kwa fomu$y=f(u)$ na$u=g(x)$, na

b. kupata$\dfrac{dy}{dx}$ kama kazi ya$x$.

7)$y=(3x−2)^6$

8)$y=(3x^2+1)^3$

Jibu: a.$f(u)=u^3,\quad u=3x^2+1$;

b.$\dfrac{dy}{dx} = 18x(3x^2+1)^2$

9)$y=\sin^5(x)$

10)$y=\left(\dfrac{x}{7}+\dfrac{7}{x}\right)^7$

Jibu: a.$f(u)=u^7,\quad u=\dfrac{x}{7}+\dfrac{7}{x}$;

b.$\dfrac{dy}{dx} = 7\left(\dfrac{x}{7}+\dfrac{7}{x}\right)^6⋅\left(\dfrac{1}{7}−\dfrac{7}{x^2}\right)$

11)$y=\tan(\sec x)$

12)$y=\csc(πx+1)$

Jibu: a.$f(u)=\csc u,\quad u=πx+1$;

b.$\dfrac{dy}{dx} = −π\csc(πx+1)⋅\cot(πx+1)$

13)$y=\cot^2x$

14)$y=−6\sin^{−3}x$

Jibu: a.$f(u)=−6u^{−3},\quad u=\sin x$;

b.$\dfrac{dy}{dx} = 18\sin^{−4}x⋅\cos x$

Katika mazoezi 15 - 24, tafuta$\dfrac{dy}{dx}$ kwa kila kazi.

15)$y=(3x^2+3x−1)^4$

16)$y=(5−2x)^{−2}$

Jibu: $\dfrac{dy}{dx}=\dfrac{4}{(5−2x)^3}$

17)$y=\cos^3(πx)$

18)$y=(2x^3−x^2+6x+1)^3$

Jibu: $\dfrac{dy}{dx}=6(2x^3−x^2+6x+1)^2⋅(3x^2−x+3)$

19)$y=\dfrac{1}{\sin^2(x)}$

20)$y=\big(\tan x+\sin x\big)^{−3}$

Jibu: $\dfrac{dy}{dx}=−3\big(\tan x+\sin x\big)^{−4}⋅(\sec^2x+\cos x)$

21)$y=x^2\cos^4x$

22)$y=\sin(\cos 7x)$

Jibu: $\dfrac{dy}{dx}=−7\cos(\cos 7x)⋅\sin 7x$

23)$y=\sqrt{6+\sec πx^2}$

24)$y=\cot^3(4x+1)$

Jibu: $\dfrac{dy}{dx}=−12\cot^2(4x+1)⋅\csc^2(4x+1)$

25) Hebu$y=\big[f(x)\big]^3$ na tuseme kwamba$f′(1)=4$ na$\frac{dy}{dx}=10$ kwa$x=1$. Kupata$f(1)$.

26) Hebu$y=\big(f(x)+5x^2\big)^4$ na tuseme kwamba$f(−1)=−4$ na$\frac{dy}{dx}=3$ wakati$x=−1$. Kupata$f′(−1)$

Jibu: $f′(−1)=10\frac{3}{4}$

27) Hebu$y=(f(u)+3x)^2$ na$u=x^3−2x$. Ikiwa$f(4)=6$ na$\frac{dy}{dx}=18$ wakati$x=2$, tafuta$f′(4)$.

28) [T] Kupata equation ya mstari tangent kwa$y=−\sin(\frac{x}{2})$ katika asili. Tumia calculator kwa grafu kazi na mstari wa tangent pamoja.

Jibu: $y=-\frac{1}{2}x$

29) [T] Kupata equation ya mstari tangent kwa$y=\left(3x+\frac{1}{x}\right)^2$ katika hatua$(1,16)$. Tumia calculator kwa grafu kazi na mstari wa tangent pamoja.

30) Pata$x$ -kuratibu ambayo mstari wa tangent$y=\left(x−\frac{6}{x}\right)^8$ ni usawa.

Jibu: $x=±\sqrt{6}$

31) [T] Kupata equation ya mstari kwamba ni ya kawaida kwa$g(θ)=\sin^2(πθ)$ katika hatua$\left(\frac{1}{4},\frac{1}{2}\right)$. Tumia calculator kwa grafu kazi na mstari wa kawaida pamoja.

Kwa mazoezi 32 - 39, tumia habari katika meza ifuatayo ili kupata$h′(a)$ thamani iliyotolewa$a$.

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	-2	3	0
2	4	4	1	-1
3	3	1-3	2	3

32)$h(x)=f\big(g(x)\big);\quad a=0$

Jibu: $h'(0) = 10$

33)$h(x)=g\big(f(x)\big);\quad a=0$

34)$h(x)=\big(x^4+g(x)\big)^{−2};\quad a=1$

Jibu: $h'(1) = −\frac{1}{8}$

35)$h(x)=\left(\dfrac{f(x)}{g(x)}\right)^2;\quad a=3$

36)$h(x)=f\big(x+f(x)\big);\quad a=1$

Jibu: $h'(1) = −4$

37)$h(x)=\big(1+g(x)\big)^3;\quad a=2$

38)$h(x)=g\big(2+f(x^2)\big);\quad a=1$

Jibu: $h'(1) = −12$

39)$h(x)=f\big(g(\sin x)\big);\quad a=0$

40) [T] Kazi ya nafasi ya treni ya mizigo hutolewa na$s(t)=100(t+1)^{−2}$, na$s$ katika mita na$t$ kwa sekunde. Wakati$t=6$ s, kupata treni ya

a. kasi na

b. kuongeza kasi.

c. Kuzingatia matokeo yako katika sehemu a. na b., ni treni kasi au kupunguza kasi?

Jibu: a.$v(6) = −\frac{200}{343}$ m/s,

b.$a(6) = \frac{600}{2401}\;\text{m/s}^2,$

c. treni ni kupunguza kasi tangu kasi na kuongeza kasi na ishara kinyume.

41) [T] Misa kunyongwa kutoka spring wima ni katika mwendo rahisi harmonic kama iliyotolewa na zifuatazo nafasi kazi, ambapo$t$ ni kipimo katika sekunde na$s$ ni katika inchi:

\[s(t)=−3\cos\left(πt+\frac{π}{4}\right).\nonumber \]

a Kuamua nafasi ya spring saa$t=1.5$ s.

b Kupata kasi ya spring katika$t=1.5$ s.

42) [T] jumla ya gharama ya kuzalisha$x$ masanduku ya Thin Mint Girl Scout cookies ni$C$ dola, ambapo$C=0.0001x^3−0.02x^2+3x+300.$ Katika$t$ wiki uzalishaji inakadiriwa kuwa$x=1600+100t$ masanduku.

Kupata gharama ndogo$C′(x).$

b. nukuu Matumizi Leibniz kwa utawala mnyororo$\dfrac{dC}{dt}=\dfrac{dC}{dx}⋅\dfrac{dx}{dt}$,, kupata kiwango kwa heshima na muda$t$ kwamba gharama ni kubadilisha.

c Tumia matokeo yako katika sehemu b. kuamua jinsi gharama za haraka zinaongezeka wakati wa$t=2$ wiki. Jumuisha vitengo na jibu.

Jibu: a.$C′(x)=0.0003x^2−0.04x+3$

b.$\dfrac{dC}{dt}=100⋅(0.0003x^2−0.04x+3) = 100⋅(0.0003(1600+100t)^2−0.04(1600+100t)+3) = 300t^2 +9200t +70700$

c. takriban $90,300 kwa wiki

43) [T] Fomu ya eneo la mduara$r$ ni$A=πr^2$ wapi eneo la mduara. Tuseme mduara unapanua, maana yake ni kwamba eneo$A$ na radius$r$ (kwa inchi) zinapanua.

Tuseme$r=2−\dfrac{100}{(t+7)^2}$$t$ wapi wakati katika sekunde. Tumia utawala wa mnyororo$\dfrac{dA}{dt}=\dfrac{dA}{dr}⋅\dfrac{dr}{dt}$ ili kupata kiwango ambacho eneo hilo linapanua.

Tumia matokeo yako katika sehemu ya a. kupata kiwango ambacho eneo hilo linapanua saa$t=4$ s.

44) [T] Fomu ya kiasi cha tufe ni$S=\frac{4}{3}πr^3$, ambapo$r$ (kwa miguu) ni radius ya tufe. Tuseme snowball ya spherical inayeyuka jua.

Tuseme$r=\dfrac{1}{(t+1)^2}−\dfrac{1}{12}$$t$ wapi muda katika dakika. Tumia utawala wa mnyororo$\dfrac{dS}{dt}=\dfrac{dS}{dr}⋅\dfrac{dr}{dt}$ ili kupata kiwango ambacho snowball inayeyuka.

b Tumia matokeo yako katika sehemu ya a. kupata kiwango ambacho kiasi kinabadilika saa$t=1$ min.

Jibu: a.$\dfrac{dS}{dt}=−\dfrac{8πr^2}{(t+1)^3} = −\dfrac{8π\left( \dfrac{1}{(t+1)^2}−\dfrac{1}{12} \right)^2}{(t+1)^3}$

b. kiasi ni kupungua kwa kiwango$−\frac{π}{36}\; \text{ft}^3$ cha/min

45) [T] joto kila siku katika digrii Fahrenheit ya Phoenix katika majira ya joto inaweza kuwa inatokana na kazi$T(x)=94−10\cos\left[\frac{π}{12}(x−2)\right]$, ambapo$x$ ni masaa baada ya usiku wa manane. Pata kiwango ambacho joto linabadilika saa 4 p.m.

46) [T] Kina (kwa miguu) cha maji kwenye kizimbani kinabadilika na kupanda na kuanguka kwa mawimbi. Ya kina kinaelekezwa na kazi$D(t)=5\sin\left(\frac{π}{6}t−\frac{7π}{6}\right)+8$, wapi idadi$t$ ya masaa baada ya usiku wa manane. Pata kiwango ambacho kina kinabadilika saa 6 a.m.

Jibu: $~2.3$ft/hr

\(x\)	\(f(x)\)	\(f'(x)\)	\(g(x)\)	\(g'(x)\)
0	2	5	0	2
1	1	-2	3	0
2	4	4	1	-1
3	3	1-3	2	3