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10.5: Mazoezi ya Mapitio ya Sura ya 10

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  • Page ID
    178101

    • Edwin “Jed” Herman & Gilbert Strang
    • OpenStax
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    Kweli au Uongo? Katika mazoezi ya 1 - 4, thibitisha jibu lako kwa ushahidi au mfano wa kukabiliana.

    1) Ikiwa radius ya kuunganisha kwa mfululizo wa nguvu\(\displaystyle \sum_{n=0}^∞a_nx^n\) ni\(5\), basi radius ya kuunganishwa kwa mfululizo pia\(\displaystyle \sum_{n=1}^∞na_nx^{n−1}\) ni\(5\).

    Jibu
    Kweli

    2) Power mfululizo inaweza kutumika kuonyesha kwamba derivative ya\(e^x\) ni\(e^x\). (Kidokezo: Kumbuka kwamba\(\displaystyle e^x=\sum_{n=0}^∞\frac{1}{n!}x^n.\))

    3) Kwa maadili madogo ya\(x,\)\(\sin x ≈ x.\)

    Jibu
    Kweli

    4) Radius ya muunganiko kwa mfululizo Maclaurin ya\(f(x)=3^x\) ni\(3\).

    Katika mazoezi ya 5 - 8, tafuta radius ya kuunganisha na muda wa kuunganishwa kwa mfululizo uliopewa.

    5)\(\displaystyle \sum_{n=0}^∞n^2(x−1)^n\)

    Jibu
    ROC:\(1\); IOC:\((0,2)\)

    6)\(\displaystyle \sum_{n=0}^∞\frac{x^n}{n^n}\)

    7)\(\displaystyle \sum_{n=0}^∞\frac{3nx^n}{12^n}\)

    Jibu
    ROC:\(12;\) IOC:\((−16,8)\)

    8)\(\displaystyle \sum_{n=0}^∞\frac{2^n}{e^n}(x−e)^n\)

    Katika mazoezi 9 - 10, pata uwakilishi wa mfululizo wa nguvu kwa kazi iliyotolewa. Kuamua radius ya muunganiko na muda wa muunganiko kwa mfululizo huo.

    9)\(f(x)=\dfrac{x^2}{x+3}\)

    Jibu
    \(\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{3^{n+1}}x^n;\)ROC:\(3\); IOC:\((−3,3)\)

    10)\(f(x)=\dfrac{8x+2}{2x^2−3x+1}\)

    Katika mazoezi 11 - 12, pata mfululizo wa nguvu kwa kazi iliyotolewa kwa kutumia upambanuzi wa muda au ushirikiano.

    11)\(f(x)=\tan^{−1}(2x)\)

    Jibu
    ushirikiano:\(\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{2n+1}(2x)^{2n+1}\)

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

    Katika mazoezi 13 - 14, tathmini upanuzi wa mfululizo wa Taylor wa shahada nne kwa kazi iliyotolewa katika hatua maalum. ni makosa katika makadirio gani?

    13)\(f(x)=x^3−2x^2+4, \quad a=−3\)

    Jibu
    \(p_4(x)=(x+3)^3−11(x+3)^2+39(x+3)−41;\)sahihi

    14)\(f(x)=e^{1/(4x)}, \quad a=4\)

    Katika mazoezi 15 - 16, tafuta mfululizo wa Maclaurin kwa kazi iliyotolewa.

    15)\(f(x)=\cos(3x)\)

    Jibu
    \(\displaystyle \sum_{n=0}^∞\frac{(−1)^n(3x)^{2n}}{2n!}\)

    16)\(f(x)=\ln(x+1)\)

    Katika mazoezi 17 - 18, tafuta mfululizo wa Taylor kwa thamani iliyotolewa.

    17)\(f(x)=\sin x, \quad a=\frac{π}{2}\)

    Jibu
    \(\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{(2n)!}\left(x−\frac{π}{2}\right)^{2n}\)

    18)\(f(x)=\dfrac{3}{x},\quad a=1\)

    Katika mazoezi 19 - 20, tafuta mfululizo wa Maclaurin kwa kazi iliyotolewa.

    19)\(f(x)=e^{−x^2}−1\)

    Jibu
    \(\displaystyle \sum_{n=1}^∞\frac{(−1)^n}{n!}x^{2n}\)

    20)\(f(x)=\cos x−x\sin x\)

    Katika mazoezi 21 - 23, tafuta mfululizo wa Maclaurin\(F(x)=∫^x_0f(t)dt\) kwa kuunganisha mfululizo wa Maclaurin wa\(f(x)\) muda kwa muda.

    21)\(f(x)=\dfrac{\sin x}{x}\)

    Jibu
    \(\displaystyle F(x)=\sum_{n=0}^∞\frac{(−1)^n}{(2n+1)(2n+1)!}x^{2n+1}\)

    22)\(f(x)=1−e^x\)

    23) Tumia mfululizo wa nguvu ili kuthibitisha formula ya Euler:\(e^{ix}=cosx+isinx\)

    Jibu
    Majibu yanaweza kutofautiana.

    Mazoezi 24 - 26 fikiria matatizo ya malipo ya annuity.

    24) Kwa annuities na thamani ya sasa ya\($1\) milioni, mahesabu ya malipo ya kila mwaka kutokana zaidi ya\(25\) miaka kuchukua riba ya\(1\%,5\%\), na\(10\%.\)

    25) Mshindi wa bahati nasibu ana annuity ambayo ina thamani ya sasa ya\($10\) milioni. Nini riba ingekuwa wanahitaji kuishi juu ya malipo ya daima ya kila mwaka ya\($250,000\)?

    Jibu
    \(2.5\%\)

    26) Mahesabu muhimu ya sasa thamani ya annuity ili kusaidia payouts ya kila mwaka ya\($15,000\) kutolewa zaidi ya\(25\) miaka kuchukua riba ya\(1\%,5\%\), na\(10\%.\)