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2.6: Mazoezi ya Mapitio ya Sura ya 2

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    178950

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    Kweli au Uongo. Katika mazoezi ya 1 - 4, thibitisha jibu lako kwa ushahidi au mfano wa kukabiliana.

    1) kazi ina kuwa kuendelea katika\(x=a\) kama\(\displaystyle \lim_{x→a}f(x)\) ipo.

    2) Unaweza kutumia utawala wa quotient kutathmini\(\displaystyle \lim_{x→0}\frac{\sin x}{x}\).

    Jibu
    Uongo, kwa kuwa hatuwezi kuwa\(\displaystyle \lim_{x→0}x=0\) katika denominator.

    3) Ikiwa kuna asymptote ya wima\(x=a\) kwa kazi\(f(x)\), basi\(f\) haijulikani kwa uhakika\(x=a\).

    4) Ikiwa\(\displaystyle \lim_{x→a}f(x)\) haipo, basi\(f\) haijulikani wakati huo\(x=a\).

    Jibu
    Uongo. Kusitisha kuruka kunawezekana.

    5) Kutumia grafu, pata kila kikomo au ueleze kwa nini kikomo haipo.

    a.\(\displaystyle \lim_{x→−1}f(x)\)

    b.\(\displaystyle \lim_{x→1}f(x)\)

    c.\(\displaystyle \lim_{x→0^+}f(x)\)

    d.\(\displaystyle \lim_{x→2}f(x)\)

    Grafu ya kazi ya kipande na makundi kadhaa. Ya kwanza ni kupungua kwa concave up Curve zilizopo kwa x <-1. Inakaribia kwenye mduara wazi (-1, 1). Ya pili ni kazi inayoongezeka ya mstari kuanzia saa (-1, -2) na kuishia saa (0, -1). Ya tatu ni kuongezeka concave chini Curve zilizopo kutoka mduara wazi katika (0,0) kwa mduara wazi katika (1,1). Ya nne ni mduara uliofungwa (1, -1). Ya tano ni mstari usio na mteremko uliopo kwa x 1, kuanzia kwenye mduara wazi saa (1,1)." style="width: 192px; height: 272px;" width="192px" height="272px" src="https://math.libretexts.org/@api/dek...02_05_207.jpeg">

    Katika mazoezi 6 - 15, tathmini kikomo algebraically au kuelezea kwa nini kikomo haipo.

    6)\(\displaystyle \lim_{x→2}\frac{2x^2−3x−2}{x−2}\)

    Jibu
    \(5\)

    7)\(\displaystyle \lim_{x→0}3x^2−2x+4\)

    8)\(\displaystyle \lim_{x→3}\frac{x^3−2x^2−1}{3x−2}\)

    Jibu
    \(8/7\)

    9)\(\displaystyle \lim_{x→π/2}\frac{\cot x}{\cos x}\)

    10)\(\displaystyle \lim_{x→−5}\frac{x^2+25}{x+5}\)

    Jibu
    DNE

    11)\(\displaystyle \lim_{x→2}\frac{3x^2−2x−8}{x^2−4}\)

    12)\(\displaystyle \lim_{x→1}\frac{x^2−1}{x^3−1}\)

    Jibu
    \(2/3\)

    13)\(\displaystyle \lim_{x→1}\frac{x^2−1}{\sqrt{x}−1}\)

    14)\(\displaystyle \lim_{x→4}\frac{4−x}{\sqrt{x}−2}\)

    Jibu
    \(−4\)

    15)\(\displaystyle \lim_{x→4}\frac{1}{\sqrt{x}−2}\)

    Katika mazoezi 16 - 17, tumia theorem itapunguza ili kuthibitisha kikomo.

    16)\(\displaystyle \lim_{x→0}x^2\cos(2πx)=0\)

    Jibu
    Tangu\(−1≤\cos(2πx)≤1\), basi\(−x^2≤x^2\cos(2πx)≤x^2\). Tangu\(\displaystyle \lim_{x→0}x^2=0=\lim_{x→0}−x^2\), inafuata kwamba\(\displaystyle \lim_{x→0}x^2\cos(2πx)=0\).

    17)\(\displaystyle \lim_{x→0}x^3\sin\left(\frac{π}{x}\right)=0\)

    18) Tambua kikoa kama kazi\(f(x)=\sqrt{x−2}+xe^x\) inaendelea juu ya uwanja wake.

    Jibu
    \([2,∞]\)

    Katika mazoezi 19 - 20, tambua thamani ya kazi\(c\) hiyo inabakia kuendelea. Chora kazi yako inayosababisha ili kuhakikisha inaendelea.

    19)\(f(x)=\begin{cases}x^2+1, & \text{if } x>c\\2^x, & \text{if } x≤c\end{cases}\)

    20)\(f(x)=\begin{cases}\sqrt{x+1}, & \text{if } x>−1\\x^2+c, & \text{if } x≤−1\end{cases}\)

    Katika mazoezi 21 - 22, tumia ufafanuzi sahihi wa kikomo ili kuthibitisha kikomo.

    21)\(\displaystyle \lim_{x→1}\,(8x+16)=24\)

    22)\(\displaystyle \lim_{x→0}x^3=0\)

    Jibu
    \(δ=\sqrt[3]{ε}\)

    23) Mpira unatupwa hewa na nafasi ya wima hutolewa na\(x(t)=−4.9t^2+25t+5\). Matumizi kati Theorem Thamani kuonyesha kwamba mpira lazima nchi juu ya ardhi wakati mwingine kati ya 5 sec na sec 6 baada ya kutupa.

    24) Chembe inayohamia kando ya mstari ina uhamisho kulingana na kazi\(x(t)=t^2−2t+4\), ambapo\(x\) inapimwa kwa mita na\(t\) inapimwa kwa sekunde. Kupata kasi ya wastani juu ya kipindi cha muda\(t=[0,2]\).

    Jibu
    \(0\)m/sec

    25) Kutoka kwa mazoezi ya awali, tathmini kasi ya papo hapo\(t=2\) kwa kuangalia kasi ya wastani ndani ya\(t=0.01\) sec.