4.10E: Mazoezi ya Sehemu ya 4.10

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Katika mazoezi ya 1 - 20, pata antiderivative\(F(x)\) ya kila kazi\(f(x).\)

1)\(f(x)=\dfrac{1}{x^2}+x\)

2)\(f(x)=e^x−3x^2+\sin x\)

Jibu: \(F(x)=e^x−x^3−\cos x+C\)

3)\(f(x)=e^x+3x−x^2\)

4)\(f(x)=x−1+4\sin(2x)\)

Jibu: \(F(x)=\dfrac{x^2}{2}−x−2\cos(2x)+C\)

5)\(f(x)=5x^4+4x^5\)

6)\(f(x)=x+12x^2\)

Jibu: \(F(x)=\frac{1}{2}x^2+4x^3+C\)

7)\(f(x)=\dfrac{1}{\sqrt{x}}\)

8)\(f(x)=\left(\sqrt{x}\right)^3\)

Jibu: \(F(x)=\frac{2}{5}\left(\sqrt{x}\right)^5+C\)

9)\(f(x)=x^{1/3}+\big(2x\big)^{1/3}\)

10)\(f(x)=\dfrac{x^{1/3}}{x^{2/3}}\)

Jibu: \(F(x)=\frac{3}{2}x^{2/3}+C\)

11)\(f(x)=2\sin(x)+\sin(2x)\)

12)\(f(x)=\sec^2 x +1\)

Jibu: \(F(x)=x+\tan x+C\)

13)\(f(x)=\sin x\cos x\)

14)\(f(x)=\sin^2(x)\cos(x)\)

Jibu: \(F(x)=\frac{1}{3}\sin^3(x)+C\)

15)\(f(x)=0\)

16)\(f(x)=\frac{1}{2}\csc^2 x+\dfrac{1}{x^2}\)

Jibu: \(F(x)=−\frac{1}{2}\cot x −\dfrac{1}{x}+C\)

17)\(f(x)=\csc x\cot x+3x\)

18)\(f(x)=4\csc x\cot x−\sec x\tan x\)

Jibu: \(F(x)=−\sec x−4\csc x+C\)

19)\(f(x)=8(\sec x)\big(\sec x−4\tan x\big)\)

20)\(f(x)=\frac{1}{2}e^{−4x}+\sin x\)

Jibu: \(F(x)=−\frac{1}{8}e^{−4x}−\cos x+C\)

Kwa mazoezi 21 - 29, tathmini muhimu.

21)\(\displaystyle ∫(−1)\,dx\)

22)\(\displaystyle ∫\sin x\,dx\)

Jibu: \(\displaystyle ∫\sin x\,dx = −\cos x+C\)

23)\(\displaystyle ∫\big(4x+\sqrt{x}\big)\,dx\)

24)\(\displaystyle ∫\frac{3x^2+2}{x^2}\,dx\)

Jibu: \(\displaystyle ∫\frac{3x^2+2}{x^2}\,dx=3x−\frac{2}{x}+C\)

25)\(\displaystyle ∫\big(\sec x\tan x+4x\big)\,dx\)

26)\(\displaystyle ∫\big(4\sqrt{x}+\sqrt[4]{x}\big)\,dx\)

Jibu: \(\displaystyle ∫\big(4\sqrt{x}+\sqrt[4]{x}\big)\,dx=\frac{8}{3}x^{3/2}+\frac{4}{5}x^{5/4}+C\)

27)\(\displaystyle ∫\left(x^{−1/3}−x^{2/3}\right)\,dx\)

28)\(\displaystyle ∫\frac{14x^3+2x+1}{x^3}\,dx\)

Jibu: \(\displaystyle ∫\frac{14x^3+2x+1}{x^3}\,dx=14x−\frac{2}{x}−\frac{1}{2x^2}+C\)

29)\(\displaystyle ∫\big(e^x+e^{−x}\big)\,dx\)

Katika mazoezi 30 - 34, tatua tatizo la thamani ya awali.

30)\(f′(x)=x^{−3},\quad f(1)=1\)

Jibu: \(f(x)=−\dfrac{1}{2x^2}+\dfrac{3}{2}\)

31)\(f′(x)=\sqrt{x}+x^2,\quad f(0)=2\)

32)\(f′(x)=\cos x+\sec^2(x),\quad f(\frac{π}{4})=2+\frac{\sqrt{2}}{2}\)

Jibu: \(f(x)=\sin x+\tan x+1\)

33)\(f′(x)=x^3−8x^2+16x+1,\quad f(0)=0\)

34)\(f′(x)=\dfrac{2}{x^2}−\dfrac{x^2}{2},\quad f(1)=0\)

Jibu: \(f(x)=−\frac{1}{6}x^3−\dfrac{2}{x}+\dfrac{13}{6}\)

Katika mazoezi 35 - 39, pata kazi mbili zinazowezekana\(f\) kutokana na derivatives ya pili au ya tatu

35)\(f''(x)=x^2+2\)

36)\(f''(x)=e^{−x}\)

Jibu: Majibu yanaweza kutofautiana; jibu moja linalowezekana ni\(f(x)=e^{−x}\)

37)\(f''(x)=1+x\)

38)\(f'''(x)=\cos x\)

Jibu: Majibu yanaweza kutofautiana; jibu moja linalowezekana ni\(f(x)=−\sin x\)

39)\(f'''(x)=8e^{−2x}−\sin x\)

40) Gari inaendeshwa kwa kiwango cha\(40\) mph wakati breki zinatumika. gari decelerates kwa kiwango cha mara kwa mara ya\(10\, \text{ft/sec}^2\). Muda gani kabla ya gari kuacha?

Jibu: \(5.867\)sekunde

41) Katika tatizo lililotangulia, hesabu jinsi gari linasafiri wakati inachukua kuacha.

42) Wewe ni kuunganisha kwenye barabara kuu, kuongeza kasi kwa kiwango cha mara kwa mara ya\(12\, \text{ft/sec}^2\). Inachukua muda gani kufikia kasi ya kuunganisha kwa\(60\) mph?

Jibu: \(7.333\)sekunde

43) Kulingana na tatizo la awali, gari linasafiri umbali gani kufikia kasi ya kuunganisha?

44) Kampuni ya gari inataka kuhakikisha mfano wake mpya zaidi unaweza kuacha katika\(8\) sec wakati wa kusafiri kwa\(75\) mph. Ikiwa tunadhani kupungua kwa mara kwa mara, pata thamani ya kupungua kwa kasi ambayo inakamilisha hili.

Jibu: \(13.75\, \text{ft/sec}^2\)

45) Kampuni ya gari inataka kuhakikisha mfano wake mpya zaidi unaweza kuacha chini ya\(450\) ft wakati wa kusafiri kwa\(60\) mph. Ikiwa tunadhani kupungua kwa mara kwa mara, pata thamani ya kupungua kwa kasi ambayo inakamilisha hili.

Katika mazoezi 46-51, kupata antiderivative ya kazi, kuchukua\(F(0)=0.\)

46) [T]\(\quad f(x)=x^2+2\)

Jibu: \(F(x)=\frac{1}{3}x^3+2x\)

47) [T]\(\quad f(x)=4x−\sqrt{x}\)

48) [T]\(\quad f(x)=\sin x+2x\)

Jibu: \(F(x)=x^2−\cos x+1\)

49) [T]\(\quad f(x)=e^x\)

50) [T]\(\quad f(x)=\dfrac{1}{(x+1)^2}\)

Jibu: \(F(x)=−\dfrac{1}{x+1}+1\)

51) [T]\(\quad f(x)=e^{−2x}+3x^2\)

Katika mazoezi 52 - 55, onyesha kama taarifa hiyo ni ya kweli au ya uongo. Aidha kuthibitisha ni kweli au kupata counterexample kama ni uongo.

52) Kama\(f(x)\) ni antiderivative ya\(v(x)\), basi\(2f(x)\) ni antiderivative ya\(2v(x).\)

Jibu: Kweli

53) Kama\(f(x)\) ni antiderivative ya\(v(x)\), basi\(f(2x)\) ni antiderivative ya\(v(2x).\)

54) Kama\(f(x)\) ni antiderivative ya\(v(x),\) basi\(f(x)+1\) ni antiderivative ya\(v(x)+1.\)

Jibu: Uongo

55) Kama\(f(x)\) ni antiderivative ya\(v(x)\), basi\((f(x))^2\) ni antiderivative ya\((v(x))^2.\)