Co-aontar tansaint
Obraichidh sinn a-mach caisead tansaint don lùb le diofarachadh.
Gus co-aontar tansaint obrachadh a-mach, bidh sinn:
- A' diofarachadh co-aontar na lùib
- Ag ionadachadh an luach \(x\) a-steach dhan cho-aontar dhiofaraichte gus an caisead obrachadh a-mach
- Ag ionadachadh an luach \(x\) a-steach do cho-aontar tùsail na lùib gus an co-chomharra-y obrachadh a-mach
- Ag ionadachadh na puing air an loidhne agad agus an caisead \(y - b = m(x - a)\)
Eisimpleir 1
Obraich a-mach co-aontar an tansaint don lùb \(y = \frac{1}{8}{x^3} - 3\sqrt x\) aig a' phuing far a bheil \(x = 4\).
Fuasgladh
Feumaidh sinn an toiseach co-aontar na lùib a chur dhan riochd as urrainn dhuinn a dhiofarachadh.
\(y = \frac{1}{8}{x^3} - 3{x^{\frac{1}{2}}}\)
\(\frac{{dy}}{{dx}} = \frac{3}{8}{x^2} - \frac{3}{2}{x^{ - \frac{1}{2}}}\)
\(\frac{{dy}}{{dx}} = \frac{{3{x^2}}}{8} - \frac{3}{{2\sqrt x }}\)
Nuair a tha \(x = 4\),
\(m = \frac{{3{{(4)}^2}}}{8} - \frac{3}{{2\sqrt 4 }}\)
\(= \frac{{3 \times 16}}{8} - \frac{3}{{2 \times 2}}\)
\(= 6 - \frac{3}{4}\)
\(= \frac{{21}}{4}\)
Co-chomharra-y nuair a tha \(x = 4\),
\(y = \frac{1}{8}{(4)^3} - 3{(4)^{\frac{1}{2}}}\)
\(= \frac{{64}}{8} - 3\sqrt 4\)
\(= 8 - 6\)
\(= 2\)
Co-aontar an tansaint nuair a tha \(a = 4,\,b = 2,\,m = \frac{{21}}{4}\)
\(y - b = m(x - a)\)
\(y - 2 = \frac{{21}}{4}(x - 4)\)
\(4(y - 2) = 21(x - 4)\)
\(4y - 8 = 21x - 84\)
\(4y = 21x - 76\)
\(21x - 4y - 76 = 0\)
Mar sin 's e co-aontar an tansaint \(21x - 4y - 76 = 0\)
Faodaidh tu cuideachd an dòigh seo a chleachdadh gus puing-coinneachaidh an tansaint air lùb a lorg nuair a tha fios agad air co-aontar na lùib agus caisead an tansaint.
Eisimpleir 2
Obraich a-mach a' phuing-coinneachaidh eadar an tansaint agus lùb leis a' cho-aontar \(y = 5{x^2} - 2x + 3\) nuair as e an caisead \(\frac{4}{3}\).
Fuasgladh
An toiseach, feumaidh sinn co-aontar na lùib a dhiofarachadh gus abairt fhaighinn dhan chaisead.
\(\frac{{dy}}{{dx}} = 10x - 2\)
Bhiomaid a-nis gu h-àbhaisteach ag ionadachadh a-steach luach \(x\) gus an caisead obrachadh a-mach, ach an turas seo chan eil fios againn air luach \(x\). Ach tha fios againn ge-tà gur e \(\frac{4}{3}\) am freagairt agus mar sin faodaidh sinn seo a chur co-ionann ris an abairt dhiofaraichte agus an co-aontar fhuasgladh gus \(x\) obrachadh a-mach.
\(10x - 2 = \frac{4}{3}\)
\(10x = \frac{{10}}{3}\)
\(x = \frac{1}{3}\)
Thèid againn a-nis air co-chomharra-y obrachadh a-mach le bhith ag ionadachadh seo a-steach dhan cho-aontar thùsail.
\(y = 5{\left( {\frac{1}{3}} \right)^2} - 2\left( {\frac{1}{3}} \right) + 3\)
\(y = 5\left( {\frac{1}{9}} \right) - \frac{2}{3} + 3\)
\(y = \frac{5}{9} - \frac{6}{9} + \frac{{27}}{9}\)
\(y = \frac{{26}}{9}\)
Mar sin 's e \(\left( {\frac{1}{3},\frac{{26}}{9}} \right)\) a' phuing-trasnaidh.
