Mienzaniso yeMibvunzo Inotsanangura Kushandiswa kweMiganhu yeBasa
Muganho webasa ipfungwa huru mukuverenga, inowanzoshandiswa kuona maitiro ebasa parinosvika padanho rakati. Mumasvomhu, kunyanya kuverenga, kunzwisisa muganho webasa kwakakosha pakusimbisa hwaro hwemamwe mafungiro akadai semaderivatives ne integrals. Chinyorwa chino chichataura nezvematambudziko emuenzaniso uye chichakurukura mashandisirwo emabasa emuganhu kuti tinzwisise zvakadzama nyaya iyi.
Nhanganyaya kuMiganhu Yebasa
Muganho webasa unotsanangura kukosha kunosvika pabasa sezvo shanduko ichisvika pamutengo wakati. Kune mhando mbiri dzemiganhu dzinowanzo kurukurwa: miganhu yerutivi rumwe (muganhu wekuruboshwe uye muganhu weruoko rwerudyi) uye miganhu yemativi maviri. Ruzivo rwakazara rwemuganhu webasa \( f(x) \) se \( x \) rinosvika \( a \) nderwekuti:
\[
\lim_{x \to a} f(x)
\]
Muenzaniso Mubvunzo 1: Muganho wekutanga
Mubvunzo:
Sarudza kukosha kwe \(\lim_{x \to 2} (3x + 1)\).
Kukurukurirana:
Uyu muenzaniso we muganho wekutanga apo basa \( f(x) = 3x + 1 \) riri basa rakatsetseka rinoenderera mberi munzvimbo yaro yese. Zvadaro tinogona kutsiva zvakananga kukosha kwe \( x = 2 \) mubasa racho.
\[
\lim_{x \kusvika 2} (3x + 1) = 3(2) + 1 = 6 + 1 = 7
\]
Saka, \(\lim_{x \to 2} (3x + 1) = 7\).
Muenzaniso Mubvunzo 2: Muganho neKupatsanura neZero
Mubvunzo:
Sarudza kukosha kwe \(\lim_{x \to 3} \frac{x^2 – 9}{x – 3}\).
Kukurukurirana:
Kana tikaisa zvakananga \( x = 3 \) mubasa, tinowana fomu risingazivikanwe \(\frac{0}{0}\). Saka, tinofanira kutanga taita kuti basa rive nyore.
Cherechedza kuti nhamba \( x^2 – 9 \) imhando yequadratic inogona kuverengerwa muzvikamu zvina:
\[
x^2 – 9 = (x – 3)(x + 3)
\]
Saka, basa rekutanga rinogona kunyorwazve seizvi:
\[
\frac{x^2 – 9}{x – 3} = \frac{(x – 3)(x + 3)}{x – 3}
\]
Kubva pano, tinogona kurerutsa nekudzima \( x – 3 \) munumerator nedenominator, chero bedzi \( x \neq 3 \):
\[
\frac{(x – 3)(x + 3)}{x – 3} = x + 3
\]
Iye zvino tinogona kuverenga zvakananga muganho nekutsiva \( x = 3 \):
\[
\lim_{x \kusvika 3} (x + 3) = 3 + 3 = 6
\]
Saka, \(\lim_{x \to 3} \frac{x^2 – 9}{x – 3} = 6\).
Muenzaniso 3: Miganhu ine Mashandiro Ezvikamu
Mubvunzo:
Tsvaga kukosha kwe \(\lim_{x \to 1} \frac{\sqrt{x + 3} – 2}{x – 1}\).
Kukurukurirana:
Kana tikaisa zvakananga \( x = 1 \) mubasa, tinowana fomu risingazivikanwe \(\frac{0}{0}\). Kuti tigadzirise izvi, tinofanira kurerutsa basa racho. Imwe nzira ndeyekugadzirisa nhamba.
Tinowanza nhamba nedhinominator ne conjugate yenhamba:
\[
\frac{\sqrt{x + 3} – 2}{x – 1} \cdot \frac{\sqrt{x + 3} + 2}{\sqrt{x + 3} + 2}
\]
Zvadaro tinowana:
\[
\frac{(\sqrt{x + 3} – 2)(\sqrt{x + 3} + 2)}{(x – 1)(\sqrt{x + 3} + 2)} = \frac{(x + 3) – 4}{(x – 1)(\sqrt{x + 3} + 2)}
\]
Nyoresa nhamba:
\[
x + 3 – 4 = x – 1
\]
Kuti:
\[
\frac{x – 1}{(x – 1)(\sqrt{x + 3} + 2)} = \frac{1}{\sqrt{x + 3} + 2}
\]
Iye zvino tinogona kuverenga muganho nekutsiva \( x = 1 \):
\[
\lim_{x \kusvika 1} \frac{1}{\sqrt{x + 3} + 2} = \frac{1}{\sqrt{1 + 3} + 2} = \frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}
\]
Saka, \(\lim_{x \to 1} \frac{\sqrt{x + 3} – 2}{x – 1} = \frac{1}{4}\).
Muenzaniso Mubvunzo 4: Miganhu neTrigonometry
Mubvunzo:
Sarudza kukosha kwe \(\lim_{x \to 0} \frac{\sin(3x)}{x}\).
Kukurukurirana:
Tinoziva kuti pamiganho yekutanga yetrigonometry, pane miganho inotevera inozivikanwa:
\[
\lim_{x \to 0} \frac{\sin(x)}{x} = 1
\]
Padambudziko iri, tinofanira kuribatanidza nechimiro ichocho chekutanga. Cherechedza kuti \( 3x \) ndiyo nharo ye sine. Tinogona kuratidza muganho nekuushandura seizvi:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{x \to 0} \frac{\sin(3x)}{3x} \cdot 3
\]
Nekuti \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \) na \( u = 3x \), saka:
\[
\lim_{x \to 0} \frac{\sin(3x)}{3x} = 1
\]
Saka:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = 1 \cdot 3 = 3
\]
Saka, \(\lim_{x \to 0} \frac{\sin(3x)}{x} = 3\).
Mhedziso
Chinyorwa chino chakurukura matambudziko akati wandei emuenzaniso uye chakurukura nezvekushandiswa kwemiganhu yebasa mukuverenga. Mumubvunzo wega wega, hurukuro inotanga nekuona chimiro chinowanikwa kana uchitsiva mavalues uye wozotsvaga nzira dzekurerutsa kana kururamisa basa racho. Kunzwisisa miganhu yebasa uye maitiro ekuigadzirisa kwakakosha pakuziva pfungwa dzepamusoro dzemasvomhu, dzakadai sedzinobva kune mamwe uye integrals. Nekudzidzira kwakasimba, kunzwisisa kwako miganhu yebasa kuchawedzera kusimba.
