3.5E: Mazoezi ya Sehemu ya 3.5

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Katika mazoezi ya 1 - 10,$\dfrac{dy}{dx}$ tafuta kazi zilizopewa.

1)$y=x^2−\sec x+1$

Jibu: $\dfrac{dy}{dx}=2x−\sec x\tan x$

2)$y=3\csc x+\dfrac{5}{x}$

3)$y=x^2\cot x$

Jibu: $\dfrac{dy}{dx}=2x\cot x−x^2\csc^2 x$

4)$y=x−x^3\sin x$

5)$y=\dfrac{\sec x}{x}$

Jibu: $\dfrac{dy}{dx}=\dfrac{x\sec x\tan x−\sec x}{x^2}$

6)$y=\sin x\tan x$

7)$y=(x+\cos x)(1−\sin x)$

Jibu: $\dfrac{dy}{dx}=(1−\sin x)(1−\sin x)−\cos x(x+\cos x)$

8)$y=\dfrac{\tan x}{1−\sec x}$

9)$y=\dfrac{1−\cot x}{1+\cot x}$

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

10)$y=(\cos x)(1+\csc x)$

Katika mazoezi 11 - 16, pata usawa wa mstari wa tangent kwa kila kazi zilizopewa kwenye maadili yaliyoonyeshwa ya$x$. Kisha kutumia calculator kwa grafu wote kazi na mstari tangent kuhakikisha equation kwa mstari tangent ni sahihi.

11) [T]$f(x)=−\sin x,\quad x=0$

Jibu

$y=−x$

$Grafu inaonyesha dhambi hasi (x) na mstari wa moja kwa moja T (x) na mteremko -1 na y kukatiza 0.$

12) [T]$f(x)=\csc x,\quad x=\frac{π}{2}$

13) [T]$f(x)=1+\cos x,\quad x=\frac{3π}{2}$

Jibu

$y=x+\frac{2−3π}{2}$

$Grafu inaonyesha kazi ya cosine imebadilishwa moja na ina mstari wa moja kwa moja T (x) na mteremko 1 na y kupinga (2 - 3π) /2.$

14) [T]$f(x)=\sec x,\quad x=\frac{π}{4}$

15) [T]$f(x)=x^2−\tan x, \quad x=0$

Jibu

$y=−x$

$Grafu inaonyesha kazi kama kuanzia saa (-1, 3), kupungua hadi asili, kuendelea kupungua polepole hadi karibu (1, -0.5), ambapo inapungua haraka sana.$

16) [T]$f(x)=5\cot x, \quad x=\frac{π}{4}$

Katika mazoezi 17 - 22,$\dfrac{d^2y}{dx^2}$ tafuta kazi zilizopewa.

17)$y=x\sin x−\cos x$

Jibu: $\dfrac{d^2y}{dx^2} = 3\cos x−x\sin x$

18)$y=\sin x\cos x$

19)$y=x−\frac{1}{2}\sin x$

Jibu: $\dfrac{d^2y}{dx^2} = \frac{1}{2}\sin x$

20)$y=\dfrac{1}{x}+\tan x$

21)$y=2\csc x$

Jibu: $\dfrac{d^2y}{dx^2} = 2\csc(x)\left(\csc^2(x)+\cot^2(x)\right) $

22)$y=\sec^2 x$

23) Pata$x$ maadili yote kwenye grafu ya$f(x)=−3\sin x\cos x$ ambapo mstari wa tangent ni usawa.

Jibu: $x = \dfrac{(2n+1)π}{4}$,$n$ wapi integer

24) Pata$x$ maadili yote kwenye grafu ya$f(x)=x−2\cos x$$0<x<2π$ ambapo mstari wa tangent una mteremko 2.

25) Hebu$f(x)=\cot x.$ Tambua pointi kwenye grafu ya$f$ kwa$0<x<2π$ wapi mstari wa tangent (s) ni (ni) sambamba na mstari$y=−2x$.

Jibu: $\left(\frac{π}{4},1\right),\quad \left(\frac{3π}{4},−1\right),\quad\left(\frac{5π}{4},1\right),\quad \left(\frac{7π}{4},−1\right)$

26) [T] wingi juu ya spring bounces juu na chini katika rahisi harmonic mwendo, inatokana na kazi$s(t)=−6\cos t$ ambapo s ni kipimo katika inchi na$t$ ni kipimo katika sekunde. Kupata kiwango ambacho spring ni oscillating katika$t=5$ s.

27) Hebu nafasi ya pendulum swinging katika mwendo rahisi harmonic kutolewa na$s(t)=a\cos t+b\sin t$. Kupata constants$a$ na$b$ vile kwamba wakati kasi ni 3 cm/s,$s=0$ na$t=0$.

Jibu: $a=0,\quad b=3$

28) Baada ya diver kuruka bodi ya kupiga mbizi, makali ya bodi oscillates na nafasi iliyotolewa na$s(t)=−5\cos t$ cm kwa$t$ sekunde baada ya kuruka.

a. mchoro kipindi moja ya nafasi ya kazi kwa$t≥0$.

pata kazi ya kasi.

c. mchoro kipindi moja ya kazi kasi kwa$t≥0$.

d Tambua nyakati ambapo kasi iko$0$ juu ya kipindi kimoja.

e Pata kazi ya kuongeza kasi.

f. mchoro kipindi moja ya kazi kuongeza kasi kwa$t≥0$.

29) Idadi ya hamburgers zinazouzwa katika mgahawa wa chakula cha haraka huko Pasadena, California, hutolewa na$y=10+5\sin x$ wapi$y$ idadi ya hamburgers zinazouzwa na$x$ inawakilisha idadi ya masaa baada ya mgahawa kufunguliwa saa 11 asubuhi hadi saa 11 jioni, wakati duka linafungwa. Kupata$y'$ na kuamua vipindi ambapo idadi ya burgers kuuzwa ni kuongezeka.

Jibu: $y′=5\cos(x)$, kuongezeka$\left(0,\frac{π}{2}\right),\;\left(\frac{3π}{2},\frac{5π}{2}\right)$, na$\left(\frac{7π}{2},12\right)$

30) [T] kiasi cha mvua kwa mwezi katika Phoenix, Arizona, inaweza kuwa takriban na$y(t)=0.5+0.3\cos t$, ambapo$t$ ni miezi tangu Januari. Pata$y′$ na utumie calculator kuamua vipindi ambapo kiasi cha mvua kuanguka kinapungua.

Kwa mazoezi 31 - 33, tumia utawala wa quotient ili kupata equations iliyotolewa.

31)$\dfrac{d}{dx}(\cot x)=−\csc^2x$

32)$\dfrac{d}{dx}(\sec x)=\sec x\tan x$

33)$\dfrac{d}{dx}(\csc x)=−\csc x\cot x$

34) Tumia ufafanuzi wa derivative na utambulisho$\cos(x+h)=\cos x\cos h−\sin x\sin h$ ili kuthibitisha hilo$\dfrac{d}{dx}(\cos x)=−\sin x$.

Kwa mazoezi 35 - 39, pata derivative ya juu ya utaratibu wa juu kwa kazi zilizopewa.

35)$\dfrac{d^3y}{dx^3}$ ya$y=3\cos x$

Jibu: $\dfrac{d^3y}{dx^3} = 3\sin x$

36)$\dfrac{d^2y}{dx^2}$ ya$y=3\sin x+x^2\cos x$

37)$\dfrac{d^4y}{dx^4}$ ya$y=5\cos x$

Jibu: $\dfrac{d^4y}{dx^4} = 5\cos x$

38)$\dfrac{d^2y}{dx^2}$ ya$y=\sec x+\cot x$

39)$\dfrac{d^3y}{dx^3}$ ya$y=x^{10}−\sec x$

Jibu: $\dfrac{d^3y}{dx^3} = 720x^7−5\tan(x)\sec^3(x)−\tan^3(x)\sec(x)$