3.R: Kazi nyingi na za busara (Tathmini)
- Page ID
- 181265
3.1 Idadi tata
Fanya operesheni iliyoonyeshwa na namba tata.
1)\((4+3 i)+(-2-5 i)\)
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\(2-2 i\)
2)\((6-5 i)-(10+3 i)\)
3)\((2-3 i)(3+6 i)\)
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\(24+3 i\)
4)\(\dfrac{2-i}{2+i}\)
Tatua equations zifuatazo juu ya mfumo wa namba tata.
5)\(x^{2}-4 x+5=0\)
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\(\{2+i, 2-i\}\)
6)\(x^{2}+2 x+10=0\)
3.2 Kazi za Quadratic
Kwa mazoezi 1-2, weka kazi ya quadratic katika fomu ya kawaida. Kisha, fanya vertex na axes intercepts. Hatimaye, graph kazi.
1)\(f(x)=x^{2}-4 x-5\)
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\(f(x)=(x-2)^{2}-9\)vertex\((2,-9)\), inakataza\((5,0); (-1,0); (0,-5)\)
2)\(f(x)=-2 x^{2}-4 x\)
Kwa matatizo 3-4, pata equation ya kazi ya quadratic kwa kutumia taarifa iliyotolewa.
3) Vertex ni\((-2,3)\) na uhakika kwenye grafu ni\((3,6)\).
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\(f(x)=\dfrac{3}{25}(x+2)^{2}+3\)
4) Vertex ni\((-3,6.5)\) na uhakika kwenye grafu ni\((2,6)\).
Jibu maswali yafuatayo.
5) Mpango wa mstatili wa ardhi unapaswa kuingizwa na uzio. Upande mmoja ni kando ya mto na hivyo hauhitaji uzio. Ikiwa uzio wa jumla unaopatikana ni\(600\) mita, pata vipimo vya njama ili uwe na eneo la juu.
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\(300\)mita kwa\(150\) mita, upande tena sambamba na mto.
6) Kitu kilichopangwa kutoka chini kwa pembe ya\(45\) shahada na kasi ya awali ya\(120\) miguu kwa pili ina urefu,\(h\), kwa upande wa umbali wa usawa uliosafiri\(x\),
3.3 Kazi za Nguvu na Kazi za Polynomial
Kwa mazoezi 1-3, onyesha kama kazi ni kazi ya polynomial na, ikiwa ni hivyo, kutoa shahada na mgawo wa kuongoza.
1)\(f(x)=4 x^{5}-3 x^{3}+2 x-1\)
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Ndiyo\(\text{degree} = 5\),\(\text{leading coefficient} = 4\)
2)\(f(x)=5^{x+1}-x^{2}\)
3)\(f(x)=x^{2}\left(3-6 x+x^{2}\right)\)
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Ndiyo\(\text{degree} = 4\),\(\text{leading coefficient} = 1\)
Kwa mazoezi 4-6, tambua tabia ya mwisho ya kazi ya polynomial.
4)\(f(x)=2 x^{4}+3 x^{3}-5 x^{2}+7\)
5)\(f(x)=4 x^{3}-6 x^{2}+2\)
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Kama\(x \rightarrow-\infty, f(x) \rightarrow-\infty \), kama\(x \rightarrow \infty, f(x) \rightarrow \infty\)
6)\(f(x)=2 x^{2}\left(1+3 x-x^{2}\right)\)
3.4 Grafu ya Kazi za Polynomial
Kwa mazoezi 1-3, tafuta zero zote za kazi ya polynomial, akibainisha kuzidisha.
1)\(f(x)=(x+3)^{2}(2 x-1)(x+1)^{3}\)
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\(-3\)na wingi\(2\),\(-\dfrac{1}{2}\) na msururu\(1\),\(-1\) na wingi\(3\)
2)\(f(x)=x^{5}+4 x^{4}+4 x^{3}\)
3)\(f(x)=x^{3}-4 x^{2}+x-4\)
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\(4\)na msururu\(1\)
Kwa mazoezi 4-5, kulingana na grafu iliyotolewa, tambua zero za kazi na uingizaji wa kumbuka.
4)
5)
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\(\dfrac{1}{2}\)na wingi\(1\),\(3\) na wingi\(3\)
6) Tumia Theorem ya Thamani ya Kati ili kuonyesha kwamba angalau sifuri moja iko kati\(2\) na\(3\) kwa kazi\(f(x)=x^{3}-5 x+1\)
3.5 Kugawanya Polynomials
Kwa mazoezi 1-2, tumia mgawanyiko mrefu ili kupata quotient na salio.
1)\(\dfrac{x^{3}-2 x^{2}+4 x+4}{x-2}\)
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\(x^{2}+4\)na salio\(12\)
2)\(\dfrac{3 x^{4}-4 x^{2}+4 x+8}{x+1}\)
Kwa mazoezi 3-6, tumia mgawanyiko wa synthetic ili kupata quotient. Ikiwa mgawanyiko ni sababu, kisha uandike fomu iliyosababishwa.
3)\(\dfrac{x^{2}-2 x^{2}+5 x-1}{x+3}\)
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\(x^{2}-5 x+20-\dfrac{61}{x+3}\)
4)\(\dfrac{x^{2}+4 x+10}{x-3}\)
5)\(\dfrac{2 x^{3}+6 x^{2}-11 x-12}{x+4}\)
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\(2 x^{2}-2x-3\), hivyo faktored fomu ni\((x+4)\left(2 x^{2}-2x-3\right)\)
6)\(\dfrac{3 x^{4}+3 x^{3}+2 x+2}{x+1}\)
3.6 Zero za Kazi za Polynomial
Kwa mazoezi 1-4, tumia Theorem ya Zero ya Mantiki ili kukusaidia kutatua equation ya polynomial.
1)\(2 x^{3}-3 x^{2}-18 x-8=0\)
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\(\left\{-2,4,-\dfrac{1}{2}\right\}\)
2)\(3x^{3}+11 x^{2}+8 x-4=0\)
3)\(2 x^{4}-17 x^{3}+46 x^{2}-43 x+12=0\)
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\(\left\{1,3,4, \dfrac{1}{2}\right\}\)
4)\(4 x^{4}+8 x^{3}+19 x^{2}+32 x+12=0\)
Kwa mazoezi 5-6, tumia Utawala wa Descartes wa Ishara ili kupata idadi inayowezekana ya ufumbuzi mzuri na hasi.
5)\(x^{3}-3 x^{2}-2 x+4=0\)
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\(0\)au\(2\) chanya,\(1\) hasi
6)\(2 x^{4}-x^{3}+4 x^{2}-5 x+1=0\)
3.7 Kazi za busara
Kwa kazi zifuatazo za busara 1-4, pata vipindi na asymptotes ya wima na ya usawa, na kisha uitumie kupiga grafu.
1)\(f(x)=\dfrac{x+2}{x-5}\)
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Intercepts\((-2,0)\) na\(\left(0,-\dfrac{2}{5}\right)\), Asymptotes\(x=5\) na\(y=1\)
2)\(f(x)=\dfrac{x^{2}+1}{x^{2}-4}\)
3)\(f(x)=\dfrac{3 x^{2}-27}{x^{2}-9}\)
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Inakataza\((3,0),(-3,0)\), na\(\left(0, \dfrac{27}{2}\right)\), Asymptotes\(x=1, x=-2, y=3\)
4)\(f(x)=\dfrac{x+2}{x^{2}-9}\)
Kwa mazoezi 5-6, pata asymptote ya slant.
5)\(f(x)=\dfrac{x^{2}-1}{x+2}\)
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\(y=x-2\)
6)\(f(x)=\dfrac{2 x^{3}-x^{2}+4}{x^{2}+1}\)
3.8 Inverses na Kazi kubwa
Kwa mazoezi 1-6, tafuta inverse ya kazi na uwanja uliotolewa.
1)\(f(x)=(x-2)^{2}, x \geq 2\)
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\(f^{-1}(x)=\sqrt{x}+2\)
2)\(f(x)=(x+4)^{2}-3, x \geq-4\)
3)\(f(x)=x^{2}+6 x-2, x \geq-3\)
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\(f^{-1}(x)=\sqrt{x+11}-3\)
4)\(f(x)=2 x^{3}-3\)
5)\(f(x)=\sqrt{4 x+5}-3\)
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\(f^{-1}(x)=\dfrac{(x+3)^{2}-5}{4}, x \geq-3\)
6)\(f(x)=\dfrac{x-3}{2 x+1}\)
3.9 Mfano Kutumia Tofauti
Kwa mazoezi 1-4, pata thamani isiyojulikana.
1)\(y\) inatofautiana moja kwa moja kama mraba wa\(x\). Kama wakati\(x=3, y=36\), kupata\(y\) kama\(x=4\).
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\(y=64\)
2)\(y\) inatofautiana inversely kama mizizi mraba ya\(x\). Kama wakati\(x=25, y=2\), kupata\(y\) kama\(x=4\).
3)\(y\) inatofautiana kwa pamoja kama mchemraba wa\(x\) na kama\(z\). Kama wakati\(x=1\) na\(z=2, y=6\), kupata\(y\) kama\(x=2\) na\(z=3\).
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\(y=72\)
4)\(y\) inatofautiana kwa pamoja kama\(x\) na mraba wa\(z\) na inversely kama mchemraba wa\(w\). Kama wakati\(x=3, z=4\), na\(w=2, y=48\), kujua\(y\) kama\(x=4, z=5\), na\(w=3\).
Kwa mazoezi 5-6, tatua tatizo la maombi.
5) Uzito wa kitu juu ya uso wa dunia hutofautiana kinyume na umbali kutoka katikati ya dunia. Ikiwa mtu hupima\(150\) paundi wakati akiwa juu ya uso wa dunia (\(3,960\)maili kutoka katikati), pata uzito wa mtu ikiwa ni\(20\) maili juu ya uso.
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\(148.5\)paundi
6) Kiasi\(V\) cha gesi bora kinatofautiana moja kwa moja na joto\(T\) na inversely na shinikizo\(P\). Silinda ina oksijeni kwenye joto la\(310\) digrii K na shinikizo la\(18\) anga kwa kiasi cha\(120\) lita. Pata shinikizo ikiwa kiasi kinapungua kwa\(100\) lita na joto huongezeka hadi\(320\) digrii K.
Mazoezi mtihani
Fanya operesheni iliyoonyeshwa au kutatua equation.
1)\((3-4 i)(4+2 i)\)
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\(20-10 i\)
2)\(\dfrac{1-4 i}{3+4 i}\)
3)\(x^{2}-4 x+13=0\)
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\(\{2+3 i, 2-3 i\}\)
4) Kutoa kiwango na mgawo wa kuongoza wa kazi ya polynomial ifuatayo. \[f(x)=x^{3}\left(3-6 x^{2}-2 x^{2}\right) \nonumber \]
Kuamua tabia ya mwisho ya kazi ya polynomial.
5)\(f(x)=8 x^{3}-3 x^{2}+2 x-4\)
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Kama\(x \rightarrow-\infty, f(x) \rightarrow-\infty\), kama\(x \rightarrow \infty, f(x) \rightarrow \infty\)
6)\(f(x)=-2 x^{2}\left(4-3 x-5 x^{2}\right)\)
7) Andika kazi ya quadratic katika fomu ya kawaida. Kuamua vertex na axes intercepts na grafu kazi. \[f(x)=x^{2}+2 x-8 \nonumber \]
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\(f(x)=(x+1)^{2}-9,\)vertex\((-1,-9),\) inakataza\((2,0); (-4,0); (0,-8)\)
8) Kutokana na taarifa kuhusu grafu ya kazi ya quadratic, pata equation yake: Vertex\((2,0)\) na uhakika kwenye grafu\((4,12)\)
Tatua tatizo la maombi yafuatayo.
9) Shamba la mstatili linapaswa kuingizwa na uzio. Mbali na uzio unaofungwa, uzio mwingine ni kugawanya shamba katika sehemu mbili, kukimbia sambamba na pande mbili. Ikiwa\(1,200\) miguu ya uzio inapatikana, pata eneo la juu ambalo linaweza kufungwa.
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\(60,000\)miguu ya mraba
Pata zero zote za kazi zifuatazo za polynomial, akibainisha kuzidisha.
10)\(f(x)=(x-3)^{3}(3 x-1)(x-1)^{2}\)
11)\(f(x)=2 x^{6}-12 x^{5}+18 x^{4}\)
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\(0\)na wingi\(4\),\(3\) na wingi\(2\)
12) Kulingana na grafu, tambua zero za kazi na kuzidisha.
13) Tumia mgawanyiko mrefu ili kupata quotient:\[\dfrac{2 x^{2}+3 x-4}{x+2} \nonumber \]
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\(2 x^{2}-4 x+11-\dfrac{26}{x+2}\)
Tumia mgawanyiko wa synthetic ili kupata quotient. Ikiwa mgawanyiko ni sababu, weka fomu iliyosababishwa.
14)\(\dfrac{x^{4}+3 x^{2}-4}{x-2}\)
15)\(\dfrac{2 x^{3}+5 x^{2}-7 x-12}{x+3}\)
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\(2 x^{2}-x-4\). Hivyo faktored fomu ni\((x+3)\left(2 x^{2}-x-4\right)\)
Tumia Theorem ya Zero ya Mantiki ili kukusaidia kupata zero za kazi za polynomial.
16)\(f(x)=2 x^{3}+5 x^{2}-6 x-9\)
17)\(f(x)=4 x^{4}+8 x^{3}+21 x^{2}+17 x+4\)
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\(-\dfrac{1}{2}\)(ina wingi\(2\)),\(\dfrac{-1+i \sqrt{15}}{2}\)
18)\(f(x)=4 x^{4}+16 x^{3}+13 x^{2}-15 x-18\)
19)\(f(x)=x^{5}+6 x^{4}+13 x^{3}+14 x^{2}+12 x+8\)
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\(-2\)(ina wingi\(3\)),\(\pm i\)
Kutokana na habari zifuatazo kuhusu kazi ya polynomial, tafuta kazi.
20) Ina sifuri mara mbili katika\(x=3\) na zeroes katika\(x=1\) na\(x=-2\). Yake\(y\) -intercept ni\((0,12)\).
21) Ina sifuri ya wingi\(3\) katika\(x=\dfrac{1}{2}\) na mwingine sifuri katika\(x=-3\). Ina uhakika\((1,8)\).
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\(f(x)=2(2 x-1)^{3}(x+3)\)
22) Tumia Utawala wa Descartes wa Ishara ili kuamua idadi inayowezekana ya ufumbuzi mzuri na hasi. \[8 x^{3}-21 x^{2}+6=0 \nonumber \]
Kwa kazi zifuatazo za busara, pata vipindi vya kuingilia na usawa na wima, na mchoro grafu.
23)\(f(x)=\dfrac{x+4}{x^{2}-2 x-3}\)
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Inakataza\((-4,0)\)\(\left(0,-\dfrac{4}{3}\right)\), Asymptotes\(x=3, x=-1, y=0\)
24)\(f(x)=\dfrac{x^{2}+2 x-3}{x^{2}-4}\)
25) Pata asymptote ya slant ya kazi ya busara. \[f(x)=\dfrac{x^{2}+3 x-3}{x-1} \nonumber \]
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\(y=x+4\)
Pata inverse ya kazi.
26)\(f(x)=\sqrt{x-2}+4\)
27)\(f(x)=3 x^{3}-4\)
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\(f^{-1}(x)=\sqrt[3]{\dfrac{x+4}{3}}\)
28)\(f(x)=\dfrac{2 x+3}{3 x-1}\)
Pata thamani isiyojulikana.
29)\(y\) inatofautiana inversely kama mraba wa\(x\) na wakati\(x=3, y=2\). Tafuta\(y\) kama\(x=1\).
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\(y=18\)
30)\(y\) inatofautiana kwa pamoja na\(x\) na mchemraba mzizi wa\(z\). Kama wakati\(x=27, y=12\), kupata\(y\) kama\(x=5\) na\(z=8\).
Tatua tatizo la maombi yafuatayo.
31) umbali mwili huanguka hutofautiana moja kwa moja kama mraba wa wakati unapoanguka. Ikiwa kitu kinaanguka\(64\) miguu kwa\(2\) sekunde, itachukua muda gani kuanguka\(256\) miguu?
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\(4\)sekunde