2.6: Mazoezi ya Mapitio ya Sura ya 2
- Page ID
- 178950
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.\(\displaystyle \lim_{x→2}f(x)\)
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.