3.3E: Mazoezi ya Sehemu
- Page ID
- 178918
Katika mazoezi 1 - 12, tafuta\(f'(x)\) kwa kila kazi.
1)\(f(x)=x^7+10\)
2)\(f(x)=5x^3−x+1\)
- Jibu
- \(f'(x)=15x^2−1\)
3)\(f(x)=4x^2−7x\)
4)\(f(x)=8x^4+9x^2−1\)
- Jibu
- \(f'(x) = 32x^3+18x\)
5)\(f(x)=x^4+2x\)
6)\(f(x)=3x\left(18x^4+\dfrac{13}{x+1}\right)\)
- Jibu
- \(f'(x) = 270x^4+\dfrac{39}{(x+1)^2}\)
7)\(f(x)=(x+2)(2x^2−3)\)
8)\(f(x)=x^2\left(\dfrac{2}{x^2}+\dfrac{5}{x^3}\right)\)
- Jibu
- \(f'(x) = \dfrac{−5}{x^2}\)
9)\(f(x)=\dfrac{x^3+2x^2−4}{3}\)
10)\(f(x)=\dfrac{4x^3−2x+1}{x^2}\)
- Jibu
- \(f'(x) = \dfrac{4x^4+2x^2−2x}{x^4}\)
11)\(f(x)=\dfrac{x^2+4}{x^2−4}\)
12)\(f(x)=\dfrac{x+9}{x^2−7x+1}\)
- Jibu
- \(f'(x) = \dfrac{−x^2−18x+64}{(x^2−7x+1)^2}\)
Katika mazoezi 13 - 16, pata usawa wa mstari wa tangent\(T(x)\) kwenye grafu ya kazi iliyotolewa kwenye hatua iliyoonyeshwa. Tumia calculator ya graphing ili kuchora kazi na mstari wa tangent.
13) [T]\(y=3x^2+4x+1\) katika\((0,1)\)
14) [T]\(y=2\sqrt{x}+1\) katika\((4,5)\)
- Jibu
-
\(T(x)=\frac{1}{2}x+3\)
15) [T]\(y=\dfrac{2x}{x−1}\) katika\((−1,1)\)
16) [T]\(y=\dfrac{2}{x}−\dfrac{3}{x^2}\) katika\((1,−1)\)
- Jibu
-
\(T(x)=4x−5\)
Katika mazoezi 17 - 20, kudhani kwamba\(f(x)\) na wote\(g(x)\) ni kazi tofauti kwa wote\(x\). Pata derivative ya kila kazi\(h(x)\).
17)\(h(x)=4f(x)+\dfrac{g(x)}{7}\)
18)\(h(x)=x^3f(x)\)
- Jibu
- \(h'(x)=3x^2f(x)+x^3f′(x)\)
19)\(h(x)=\dfrac{f(x)g(x)}{2}\)
20)\(h(x)=\dfrac{3f(x)}{g(x)+2}\)
- Jibu
- \(h'(x)=\dfrac{3f′(x)(g(x)+2)−3f(x)g′(x)}{(g(x)+2)^2}\)
Kwa mazoezi 21 - 24, kudhani kuwa\(f(x)\) na\(g(x)\) ni kazi zote mbili tofauti na maadili kama ilivyoelezwa katika meza ifuatayo. Tumia meza ifuatayo ili kuhesabu derivatives zifuatazo.
\(x\) | 1 | 2 | 3 | 4 |
\(f(x)\) | 3 | 5 | -2 | 0 |
\(g(x)\) | 2 | 3 | -4 | 6 |
\(f′(x)\) | -1 | 7 | 8 | 1-3 |
\(g′(x)\) | 4 | 1 | 2 | 9 |
21) Pata\(h′(1)\) kama\(h(x)=x f(x)+4g(x)\).
22) Pata\(h′(2)\) kama\(h(x)=\dfrac{f(x)}{g(x)}\).
- Jibu
- \(h'(2) =\frac{16}{9}\)
23) Tafuta\(h′(3)\) kama\(h(x)=2x+f(x)g(x)\).
24) Tafuta\(h′(4)\) kama\(h(x)=\dfrac{1}{x}+\dfrac{g(x)}{f(x)}\).
- Jibu
- \(h'(4)\)haijafafanuliwa.
Katika mazoezi 25 - 27, tumia takwimu zifuatazo ili kupata derivatives zilizoonyeshwa, ikiwa zipo.
25) Hebu\(h(x)=f(x)+g(x)\). Kupata
a)\(h′(1)\),
b)\(h′(3)\), na
c)\(h′(4)\).
26) Hebu\(h(x)=f(x)g(x).\) Tafuta
a)\(h′(1),\)
b)\(h′(3)\), na
c)\(h′(4).\)
- Jibu
- a.\(h'(1) = 2\),
b.\(h'(3)\) haipo,
c.\(h'(4) = 2.5\)
27) Hebu\(h(x)=\dfrac{f(x)}{g(x)}.\) Tafuta
a)\(h′(1),\)
b)\(h′(3)\), na
c)\(h′(4).\)
Katika mazoezi 28 - 31,
a) kutathmini\(f′(a)\), na
b) grafu kazi\(f(x)\) na mstari wa tangent saa\(x=a\).
28) [T]\(f(x)=2x^3+3x−x^2, \quad a=2\)
- Jibu
-
a. 23
b.\(y=23x−28\)
29) [T]\(f(x)=\dfrac{1}{x}−x^2, \quad a=1\)
30) [T]\(f(x)=x^2−x^{12}+3x+2, \quad a=0\)
- Jibu
-
a.\(3\)
b.\(y=3x+2\)
31) [T]\(f(x)=\dfrac{1}{x}−x^{2/3}, \quad a=−1\)
32) Pata usawa wa mstari wa tangent kwenye grafu ya\(f(x)=2x^3+4x^2−5x−3\) saa\(x=−1.\)
- Jibu
- \(y=−7x−3\)
33) Pata usawa wa mstari wa tangent kwenye grafu ya\(f(x)=x^2+\dfrac{4}{x}−10\) saa\(x=8\).
34) Pata usawa wa mstari wa tangent kwenye grafu ya\(f(x)=(3x−x^2)(3−x−x^2)\) saa\(x=1\).
- Jibu
- \(y=−5x+7\)
35) Kupata uhakika juu ya grafu ya\(f(x)=x^3\) vile kwamba line tangent katika hatua hiyo ina\(x\) -intercept ya\((6,0)\).
36) Kupata equation ya mstari kupita kwa njia ya uhakika\(P(3,3)\) na tangent kwa grafu ya\(f(x)=\dfrac{6}{x−1}\).
- Jibu
- \(y=−\frac{3}{2}x+\frac{15}{2}\)
37) Tambua pointi zote kwenye grafu ambayo mteremko wa mstari wa tangent ni\(f(x)=x^3+x^2−x−1\)
a. usawa
b. -1.
38) Pata polynomial quadratic kama hiyo\(f(1)=5,\; f′(1)=3\) na\(f''(1)=−6.\)
- Jibu
- \(y=−3x^2+9x−1\)
39) Gari inayoendesha gari kando ya barabara kuu na trafiki imesafiri\(s(t)=t^3−6t^2+9t\) mita kwa\(t\) sekunde.
a Kuamua wakati kwa sekunde wakati kasi ya gari ni 0.
b Kuamua kasi ya gari wakati kasi ni 0.
40) [T] sill kuogelea pamoja mstari wa moja kwa moja ina alisafiri\(s(t)=\dfrac{t^2}{t^2+2}\) miguu katika\(t\)
sekunde. Kuamua kasi ya herring wakati imesafiri sekunde 3.
- Jibu
- \(\frac{12}{121}\)au 0.0992 ft/s
41) Idadi ya watu katika mamilioni ya flounder ya arctic katika Bahari ya Atlantiki inatokana na kazi\(P(t)=\dfrac{8t+3}{0.2t^2+1}\), ambapo\(t\) hupimwa kwa miaka.
a Tambua idadi ya watu ya awali ya flounder.
b Kuamua\(P′(10)\) na kutafsiri kwa ufupi matokeo.
42) [T] Mkusanyiko wa antibiotiki katika\(t\) masaa ya damu baada ya kuingizwa hutolewa na kazi\(C(t)=\dfrac{2t^2+t}{t^3+50}\), ambapo\(C\) hupimwa kwa miligramu kwa lita moja ya damu.
Kupata kiwango cha mabadiliko ya\(C(t).\)
b Kuamua kiwango cha mabadiliko kwa\(t=8,12,24\), na\(36\).
c. kuelezea kwa kifupi kile kinachoonekana kuwa kinatokea kama idadi ya masaa inavyoongezeka.
- Jibu
- a.\(\dfrac{−2t^4−2t^3+200t+50}{(t^3+50)^2}\)
b.\(−0.02395\) mg/L-hr,\(−0.01344\) mg/L-hr,\(−0.003566\)\(−0.001579\) mg/L-hr, mg/L-hr
c. kiwango ambacho mkusanyiko wa madawa ya kulevya katika damu hupungua kwa 0 kama ongezeko la muda.
43) Mchapishaji wa kitabu ana kazi ya gharama iliyotolewa na\(C(x)=\dfrac{x^3+2x+3}{x^2}\), wapi\(x\) idadi ya nakala za kitabu kwa maelfu na\(C\) ni gharama, kwa kitabu, kipimo kwa dola. Tathmini\(C′(2)\) na kuelezea maana yake.
44) [T] Kwa mujibu wa sheria ya Newton ya gravitation zima, nguvu\(F\) kati ya miili miwili ya molekuli ya mara kwa mara\(m_1\) na\(m_2\) inatolewa na formula\(F=\dfrac{Gm_1m_2}{d^2}\), ambapo\(G\) ni mara kwa mara mvuto na\(d\) ni umbali kati ya miili.
a Tuseme kwamba\(G,m_1,\) na\(m_2\) ni mara kwa mara. Pata kiwango cha mabadiliko ya nguvu\(F\) kwa heshima na umbali\(d\).
pata kiwango cha mabadiliko ya nguvu\(F\) na mara kwa mara ya mvuto\(G=6.67×10^{−11} \text{Nm}^2/\text{kg}^2\), juu ya miili miwili mita 10 mbali, kila mmoja na uzito wa kilo 1000.
- Jibu
- a.\(F'(d)=\dfrac{−2Gm_1m_2}{d_3}\)
b.\(−1.33×10^{−7}\) N/m