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ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
ÌÀÒÅÌÀÒÈÊ-2Îëîí õóâüñàã÷òàé ôóíêöèéí ³íäýñ
Ä. Áàòò°ð
ÎËÎÍ ÓËÑÛÍ ÓËÀÀÍÁÀÀÒÀÐÛÍ ÈÕ ÑÓÐÃÓÓËÜ
2016.01.15
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
1 Îëîí õóâüñàã÷òàé ôóíêö (ÎÕÔ)
(ÎÕÔ)-èéí òîäîðõîéëîãäîõ ìóæ
(ÎÕÔ)-èéí òóõàéí áà á³òýí °°ð÷ë°ëò
2 Õî¼ð õóâüñàã÷òàé ôóíêöèéí õÿçãààð, òàñðàëòã³é ÷àíàð
Äàâõàð õÿçãààð
Äàðààëñàí õÿçãààð
(ÎÕÔ)-èéí òàñðàëòã³é ÷àíàð
3 (ÎÕÔ)-èéí òóõàéí óëàìæëàë áà á³òýí äèôôåðåíèéë
(ÎÕÔ)-èéí òóõàéí óëàìæëàë
(ÎÕÔ)-èéí á³òýí äèôåðåíöèàë
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òîäîðõîéëîãäîõ ìóæ
Òîäîðõîéëò
Õýðýâ áèå áèåíýýñýý ³ë õàìààðàí °°ð÷ë°ãä°õ x , y -ãýñýíõóâüñàõ õýìæèãäýõ³³íèé ýðýìáýëýãäñýí õîñ óòãà (x ; y)-ýýñòîãòîõ (D) ìóæèéí M(x ; y) öýã á³õýíä ÿìàð íýãýí õóóëü,
ä³ðìýýð òîäîðõîé íýã áîäèò òîî z-èéã õàðãàëçóóëæ áàéâàë
z-ûã x , y -õóâüñàã÷ààñ õàìààðñàí, D ìóæ äýýð
òîäîðõîéëîãäñîí íýãýí óòãàò õî¼ð õóâüñàã÷òàé ôóíêö ãýýä
z = f (x ; y) z = φ(x ; y)
ãýõ ìýòýýð òýìäýãëýíý.
ÎÕÔ-ã °ã°õ àðãóóä
1 Õ³ñíýãòýýð
2 Àíàëèòèê àðãààð (ìàòåìàòèêèéí
òîìú¼îíû òóñëàìæòàéãààð)
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òîäîðõîéëîãäîõ ìóæ
Òîäîðõîéëò
Õýðýâ áèå áèåíýýñýý ³ë õàìààðàí °°ð÷ë°ãä°õ x , y -ãýñýíõóâüñàõ õýìæèãäýõ³³íèé ýðýìáýëýãäñýí õîñ óòãà (x ; y)-ýýñòîãòîõ (D) ìóæèéí M(x ; y) öýã á³õýíä ÿìàð íýãýí õóóëü,
ä³ðìýýð òîäîðõîé íýã áîäèò òîî z-èéã õàðãàëçóóëæ áàéâàë
z-ûã x , y -õóâüñàã÷ààñ õàìààðñàí, D ìóæ äýýð
òîäîðõîéëîãäñîí íýãýí óòãàò õî¼ð õóâüñàã÷òàé ôóíêö ãýýä
z = f (x ; y) z = φ(x ; y)
ãýõ ìýòýýð òýìäýãëýíý.
ÎÕÔ-ã °ã°õ àðãóóä
1 Õ³ñíýãòýýð
2 Àíàëèòèê àðãààð (ìàòåìàòèêèéí
òîìú¼îíû òóñëàìæòàéãààð)
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òîäîðõîéëîãäîõ ìóæ
Òîäîðõîéëò
Õýðýâ áèå áèåíýýñýý ³ë õàìààðàí °°ð÷ë°ãä°õ x , y -ãýñýíõóâüñàõ õýìæèãäýõ³³íèé ýðýìáýëýãäñýí õîñ óòãà (x ; y)-ýýñòîãòîõ (D) ìóæèéí M(x ; y) öýã á³õýíä ÿìàð íýãýí õóóëü,
ä³ðìýýð òîäîðõîé íýã áîäèò òîî z-èéã õàðãàëçóóëæ áàéâàë
z-ûã x , y -õóâüñàã÷ààñ õàìààðñàí, D ìóæ äýýð
òîäîðõîéëîãäñîí íýãýí óòãàò õî¼ð õóâüñàã÷òàé ôóíêö ãýýä
z = f (x ; y) z = φ(x ; y)
ãýõ ìýòýýð òýìäýãëýíý.
ÎÕÔ-ã °ã°õ àðãóóä
1 Õ³ñíýãòýýð
2 Àíàëèòèê àðãààð (ìàòåìàòèêèéí
òîìú¼îíû òóñëàìæòàéãààð)
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òîäîðõîéëîãäîõ ìóæ
Æèøýý
x
y0 1 2 3 4 5
1 0 1 2 3 4 5
2 0 2 4 6 8 10
3 0 3 6 9 12 15
4 0 4 8 12 16 20
z = x2 + y2
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òîäîðõîéëîãäîõ ìóæ
Òîäîðõîéëò
Àíàëèòèê àðãààð òîäîðõîéëîãäñîí z = f (x ; y) ôóíêöèéíóòãûã òîäîðõîé áîäèò òîî áàéëãàõ á³õ (x ; y)-õîñ óòãóóäûíîëîíëîãèéã z = f (x ; y) ôóíêöèéí òîäîðõîéëîãäîõ ìóæ ãýíý.
Òîäîðõîéëò
Òîäîðõîéëîãäîõ ìóæèéí (x ; y)-õîñ óòãà á³õýíä õàðãàëçàõ
z-ûí á³õ óòãóóäûí îëîíëîãèéã ôóíêöèéí óòãûí ìóæ ãýíý.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òîäîðõîéëîãäîõ ìóæ
Òîäîðõîéëò
Àíàëèòèê àðãààð òîäîðõîéëîãäñîí z = f (x ; y) ôóíêöèéíóòãûã òîäîðõîé áîäèò òîî áàéëãàõ á³õ (x ; y)-õîñ óòãóóäûíîëîíëîãèéã z = f (x ; y) ôóíêöèéí òîäîðõîéëîãäîõ ìóæ ãýíý.
Òîäîðõîéëò
Òîäîðõîéëîãäîõ ìóæèéí (x ; y)-õîñ óòãà á³õýíä õàðãàëçàõ
z-ûí á³õ óòãóóäûí îëîíëîãèéã ôóíêöèéí óòãûí ìóæ ãýíý.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òîäîðõîéëîãäîõ ìóæ
Æèøýý
z = 4x − y ôóíêö íü äóðûí (x ; y)-õîñ óòãà á³õýí äýýðòîäîðõîé áîäèò óòãàòàé áàéõ òóë ò³³íèé òîäîðõîéëîãäîõìóæ íü õàâòãàéí á³õ öýã³³ä áàéíà.
z =√
9− x2 − y2 ôóíêö íü 9− x2 − y2 ≥ 0 áóþó x2 + y2 ≤ 9í°õöëèéã õàíãàõ (x ; y)-ûí õîñ óòãà á³õýí äýýðòîäîðõîéëîãäîíî.
x2 + y2 ≤ 9 íü O(0; 0) öýã äýýð ò°âòýé R = 3 ðàäèóñòàéäóãóéí äîòîðõè á³õ öýã³³ä áîëîí x2 + y2 = 9 òîéðãèéíöýã³³ä áàéíà. Èéìä z =
√9− x2 − y2 ôóíêöèéí
òîäîðõîéëîãäîõ ìóæ íü áèò³³ ìóæ áàéíà.
z = ln(x + y) ôóíêöèéí òîäîðõîéëîãäîõ ìóæ íü x + y > 0áóþó y > −x í°õöëèéã õàíãàñàí (x ; y)-õîñ óòãà á³õýí äýýðòîäîðõîéëîãäîíî. Ýíýõ³³ (x ; y)-õîñ óòãóóäûí îëîíëîã íüy = −x-øóëóóíààñ äýýø îðøèõ õàâòãàéí á³õ öýã³³äèéíîëîíëîã áàéíà.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òîäîðõîéëîãäîõ ìóæ
Æèøýý
z = 4x − y ôóíêö íü äóðûí (x ; y)-õîñ óòãà á³õýí äýýðòîäîðõîé áîäèò óòãàòàé áàéõ òóë ò³³íèé òîäîðõîéëîãäîõìóæ íü õàâòãàéí á³õ öýã³³ä áàéíà.
z =√
9− x2 − y2 ôóíêö íü 9− x2 − y2 ≥ 0 áóþó x2 + y2 ≤ 9í°õöëèéã õàíãàõ (x ; y)-ûí õîñ óòãà á³õýí äýýðòîäîðõîéëîãäîíî.
x2 + y2 ≤ 9 íü O(0; 0) öýã äýýð ò°âòýé R = 3 ðàäèóñòàéäóãóéí äîòîðõè á³õ öýã³³ä áîëîí x2 + y2 = 9 òîéðãèéíöýã³³ä áàéíà. Èéìä z =
√9− x2 − y2 ôóíêöèéí
òîäîðõîéëîãäîõ ìóæ íü áèò³³ ìóæ áàéíà.
z = ln(x + y) ôóíêöèéí òîäîðõîéëîãäîõ ìóæ íü x + y > 0áóþó y > −x í°õöëèéã õàíãàñàí (x ; y)-õîñ óòãà á³õýí äýýðòîäîðõîéëîãäîíî. Ýíýõ³³ (x ; y)-õîñ óòãóóäûí îëîíëîã íüy = −x-øóëóóíààñ äýýø îðøèõ õàâòãàéí á³õ öýã³³äèéíîëîíëîã áàéíà.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òîäîðõîéëîãäîõ ìóæ
Æèøýý
z = 4x − y ôóíêö íü äóðûí (x ; y)-õîñ óòãà á³õýí äýýðòîäîðõîé áîäèò óòãàòàé áàéõ òóë ò³³íèé òîäîðõîéëîãäîõìóæ íü õàâòãàéí á³õ öýã³³ä áàéíà.
z =√
9− x2 − y2 ôóíêö íü 9− x2 − y2 ≥ 0 áóþó x2 + y2 ≤ 9í°õöëèéã õàíãàõ (x ; y)-ûí õîñ óòãà á³õýí äýýðòîäîðõîéëîãäîíî.
x2 + y2 ≤ 9 íü O(0; 0) öýã äýýð ò°âòýé R = 3 ðàäèóñòàéäóãóéí äîòîðõè á³õ öýã³³ä áîëîí x2 + y2 = 9 òîéðãèéíöýã³³ä áàéíà. Èéìä z =
√9− x2 − y2 ôóíêöèéí
òîäîðõîéëîãäîõ ìóæ íü áèò³³ ìóæ áàéíà.
z = ln(x + y) ôóíêöèéí òîäîðõîéëîãäîõ ìóæ íü x + y > 0áóþó y > −x í°õöëèéã õàíãàñàí (x ; y)-õîñ óòãà á³õýí äýýðòîäîðõîéëîãäîíî. Ýíýõ³³ (x ; y)-õîñ óòãóóäûí îëîíëîã íüy = −x-øóëóóíààñ äýýø îðøèõ õàâòãàéí á³õ öýã³³äèéíîëîíëîã áàéíà.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òîäîðõîéëîãäîõ ìóæ
Ñàíàìæ
ÎÕÔ-èéí òîäîðõîéëîãäîõ ìóæ íü êîîäèíàòûí õàâòãàé,
ýñâýë ò³³íèé õýñýã áàéäàã.
6
-
y
x
y = −x
0
1-ð çóðàã
Çóðàã: Êîîðäèíàòûí õàìòãàé
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òîäîðõîéëîãäîõ ìóæ
Òîäîðõîéëò
Áèå áèåíýýñýý ³ë õàìààðàõ x1, x2, ..., xn-ãýñýí n-øèðõýãõóâüñàã÷óóäûí óòãóóäààñ òîãòîõ D ìóæèéí (x1, x2, ..., xn)óòãà á³õýíä ÿìàð íýãýí õóóëü ä³ðìýýð z-ãýñýí õóâüñàõ
õýìæèãäýõ³³íèé òîäîðõîé íýã óòãûã õàðãàëçóóëæ áîëæ
áàéâàë z-ûã D-ìóæ äýýð òîäîðõîéëîãäñîí (x1, x2, ..., xn)õóâüñàã÷ààñ õàìààðñàí n-õóâüñàã÷òàé íýãýí óòãàò ôóíêö ãýæ
íýðëýýä
z = f (x1; x2; ...; xn) áóþó z = φ(x1; x2; ...; xn)
ãýõ ìýòýýð òýìäýãëýíý. Åð°íõèéä°° õî¼ð áà ò³³íýýñ äýýø
òîîíû õóâüñàã÷ààñ õàìààðñàí ôóíêöèéã îëîí õóâüñàã÷òàé
ôóíêö ãýíý.
Æèøýýëáýë W = x2+y2+z2√
1+t2ôóíêö íü x , y , z , t ãýñýí ä°ðâ°í
õóâüñàã÷òàé ôóíêö þì.
Ä°ð°â áà ò³³íýýñ äýýø õóâüñàã÷òàé ôóíêöèéí
òîäîðõîéëîãäîõ ìóæèéã ãåîìåòðèéí àðãààð ä³ðñëýõ
áîëîìæã³é.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òóõàéí áà á³òýí °°ð÷ë°ëò
Òîäîðõîéëò
z = f (x ; y), (x ; y) ∈ D õî¼ð õóâüñàã÷òàé ôóíêö àâúÿ. Ýíý
ôóíêöèéí x-õóâüñàã÷èä ∆x-°°ð÷ë°ëòèéã (x + ∆x ; y) ∈ Dáàéõààð °ã÷ y -ûã õýâýýð áàéëãàâàë z-ôóíêö °°ð÷ë°ãä°õ
á°ã°°ä ò³³íä õàðãàëçàõ °°ð÷ë°ëò
∆xz = f (x + ∆x ; y)− f (x ; y) (1)
-èéã z-ôóíêöèéí x-ýýð àâñàí òóõàéí °°ð÷ë°ëò ãýíý. �³íèé
àäèëààð z-ýýñ y -ýýð àâñàí òóõàéí °°ð÷ë°ëòèéã
∆yz = f (x ; y + ∆y)− f (x ; y)
ãýæ òîäîðõîéëæ áîëíî.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òóõàéí áà á³òýí °°ð÷ë°ëò
Òîäîðõîéëò
Õýðýâ z = f (x ; y) ôóíêöèéí x àðãóìåíòàä ∆x , y -àðãóìåíòàä∆y -°°ð÷ë°ëòèéã íýãýí çýðýã (x + ∆x ; y + ∆y) ∈ D áàéõààð
°ã°õ°ä ãàðàõ ôóíêöèéí °°ð÷ë°ëò
∆z = f (x + ∆x ; y + ∆y)− f (x ; y)
-èéã z = f (x ; y)-èéí á³òýí °°ð÷ë°ëò ãýíý.
Ñàíàìæ
∆z 6= ∆xz + ∆yz áàéæ áîëíî.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òóõàéí áà á³òýí °°ð÷ë°ëò
Æèøýý
z = x2 · y ôóíêöèéí õóâüä
∆xz = (x + ∆x)2y − x2y = x2y + 2x ·∆x · y + (∆x)2y − x2y == (2x + ∆x) ·∆x · y
∆yz = x2(y + ∆y)− x2y = x2 ·∆y∆z = (x + ∆x)2(y + ∆y)− x2y
= [x2 + 2x∆x + (∆x)2](y + ∆y)− x2y= 2x ·∆x · y + (∆x)2y + x2∆y + x2∆y
+2x∆y∆x + (∆x)2∆y
áàéõ áà∆z 6= ∆xz + ∆yz
áàéíà.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òóõàéí áà á³òýí °°ð÷ë°ëò
1-èéí �ðã°òã°ë
�³íèé àäèëààð z = f (x1; x2; ...; xn) ôóíêöèéí òóõàéí áà
á³òýí °°ð÷ë°ëòèéã áè÷âýë
∆x1 = f (x1 + ∆x1, x2, ..., xn)− f (x1, x2, ..., xn)∆x2 = f (x1, x2 + ∆x2, ..., xn)− f (x1, x2, ..., xn)....................................∆xn = f (x1, x2, ..., xn + ∆xn)− f (x1, x2, ..., xn)∆z = f (x1 + ∆x1; x2 + ∆x2; ...; xn + ∆xn)− f (x1, x2, ..., xn)
áàéíà.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
Äàâõàð õÿçãààð
Õàâòãàéí M0 áà M öýãèéí õîîðîíäàõü çàéã
ρ(M,M0) =√
(x − x0)2 + (y − y0)2
ãýæ òýìäýãëüå.
Òîäîðõîéëò
M0(x0, y0) öýãèéí õóâüä ρ(M,M0) < ε í°õö°ëèéã õàíãàñàíM(x , y) öýã³³äèéí îëîíëîãèéã M0(x0, y0) öýãèéí ε îð÷èíãýíý.
M0(x0, y0) öýãèéí îð÷èí ãýäýã
íü ãåîìåòðèéí ³³äíýýñ M0(x0, y0)öýã äýýð ò°âòýé ε ðàäèóñòàé
äóãóéí á³õ öýã³³äèéí îëîíëîã
þì.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
Äàâõàð õÿçãààð
Õàâòãàéí M0 áà M öýãèéí õîîðîíäàõü çàéã
ρ(M,M0) =√
(x − x0)2 + (y − y0)2
ãýæ òýìäýãëüå.
Òîäîðõîéëò
M0(x0, y0) öýãèéí õóâüä ρ(M,M0) < ε í°õö°ëèéã õàíãàñàíM(x , y) öýã³³äèéí îëîíëîãèéã M0(x0, y0) öýãèéí ε îð÷èíãýíý.
M0(x0, y0) öýãèéí îð÷èí ãýäýã
íü ãåîìåòðèéí ³³äíýýñ M0(x0, y0)öýã äýýð ò°âòýé ε ðàäèóñòàé
äóãóéí á³õ öýã³³äèéí îëîíëîã
þì.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
Äàâõàð õÿçãààð
Òîäîðõîéëò
∀ε > 0 òîî àâàõàä ρ(M,M0) < δ áàéõ M(x , y) öýã³³äèéíõóâüä
|f (M)− A| < ε
òýíöýòãýë áèø áèåëýãäýæ áàéõààð δ = δ(ε) > 0 òîî îëäîæ
áàéâàë A− const òîîã M → M0(x0, y0) ³åèéí f (x , y)ôóíêöèéí äàâõàð õÿçãààð ãýæ íýðëýýä
limM→M0
f (M) = A áóþó limx → x0y → y0
f (x , y) = A (2)
ãýæ òýìäýãëýíý.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
Äàâõàð õÿçãààð
Æèøýý
limx → 0y → 0
a−√
a2−xyxy õÿçãààðûã îë.
C limx → 0y → 0
a−√
a2−xyxy = lim
x → 0y → 0
(a−√
a2−xy)(a+√
a2−xy)
xy ·(a+√
a2−xy)
= limx → 0y → 0
xy
xy ·(a+√
a2−xy)= lim
x → 0y → 0
1
(a+√
a2−xy)= 1
2a .B
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
Äàâõàð õÿçãààð
Æèøýý
limx → 0y → 0
a−√
a2−xyxy õÿçãààðûã îë.
C limx → 0y → 0
a−√
a2−xyxy = lim
x → 0y → 0
(a−√
a2−xy)(a+√
a2−xy)
xy ·(a+√
a2−xy)
= limx → 0y → 0
xy
xy ·(a+√
a2−xy)= lim
x → 0y → 0
1
(a+√
a2−xy)= 1
2a .B
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
Äàðààëñàí õÿçãààð
Òîäîðõîéëò
z = f (x , y), x ∈ X , y ∈ Y ôóíêöèéí õóâüä y -ûã áýõëýýä x → a³åèéí f (x , y) ôóíêöèéí õÿçãààð íü îðøèí áàéâàë òýð íüåð°íõèéä°° y -ýýñ õàìààðñàí ôóíêö áàéíà.
limx→a
f (x , y) = φ(y).
Ýíý ôóíêöýýñ y → b ³åèéí õÿçãààð àâáàë
limy→b
φ(y) = limy→b
( limx→a
f (x , y)) (3)
áîëíî. (4)-èéã äàðààëñàí õÿçãààð ãýæ íýðëýíý.�³íèé àäèëààð
limx→a
limy→b
f (x , y) = limx→a
ψ(x) (4)
ãýñýí äàðààëñàí õÿçãààðûí òóõàé ÿðüæ áîëíî.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
Äàðààëñàí õÿçãààð
Æèøýý
f (x , y) = 5x−3y+x2+y2
x+y ôóíêöèéí x → 0, y → 0 ³åèéí á³õ
áîëîìæò äàðààëñàí õÿçãààðûã îë.
C
limy→0
limx→0
5x − 3y + x2 + y2
x + y= lim
y→0(y − 3) = −3
limx→0
limy→0
5x − 3y + x2 + y2
x + y= lim
x→0
5x + x2
x= 5.B
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
Äàðààëñàí õÿçãààð
Æèøýý
f (x , y) = 5x−3y+x2+y2
x+y ôóíêöèéí x → 0, y → 0 ³åèéí á³õ
áîëîìæò äàðààëñàí õÿçãààðûã îë.
C
limy→0
limx→0
5x − 3y + x2 + y2
x + y= lim
y→0(y − 3) = −3
limx→0
limy→0
5x − 3y + x2 + y2
x + y= lim
x→0
5x + x2
x= 5.B
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òàñðàëòã³é ÷àíàð
Òîäîðõîéëò
Õýðýâ M0(x0, y0) öýã íü f (x , y) ôóíêöèéí òîäîðõîéëîãäîõ ìóæèéíöýã áàéõ áà
limx → x0y → y0
f (x , y) = f (x0, y0) (5)
òýíöýòãýë áèåëýãäýæ áàéâàë z = f (x , y) ôóíêöèéã M0(x0, y0) öýãäýýð òàñðàëòã³é ôóíêö ãýíý.Áèä x = x0 + ∆x , y = y0 + ∆y ãýâýë (6)-èéã
lim∆x → 0∆y → 0
f (x0 + ∆x , y0 + ∆y) = f (x0, y0) (6)
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òóõàéí óëàìæëàë
Òîäîðõîéëò
z = f (x ; y) ôóíêöèéí M0(x0; y0) öýã äýýðõ x-ýýð àâñàí
òóõàéí °°ð÷ë°ëòèéã àðãóìåíò x-èéí °°ð÷ë°ëò ∆x-ä
õàðüöóóëñàí õàðüöàà∆xz
∆x-ààñ ∆x → 0 ³åèéí õÿçãààð àâàõàä
ò°ãñã°ë°ã òîî ãàðàõ áîë ýíý õÿçãààðûã f (x ; y) ôóíêöýýñM0(x0; y0) öýã äýýðõ x-ýýð àâñàí òóõàéí óëàìæëàë ãýýä
∂z
∂x;
∂f (x0; y0)
∂x; z ′x ; f ′x(x0; y0)
ãýæ òýìäýãëýíý.
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òóõàéí óëàìæëàë
Òîäîðõîéëò
�°ð°°ð õýëáýë:
∂z
∂x= lim
∆→0
∆xz
∆x= lim
∆x→0
f (x0 + ∆x ; y0)− f (x0; y0)
∆x(7)
∂z
∂y= lim
∆→0
∆yz
∆y= lim
∆y→0
f (x0; y0 + ∆y)− f (x0; y0)
∆y(8)
Æèøýý
z = x2y2 + ln(5x + 4y) + 1 ôóíêöýýñ x , y -àðãóìåíòóóäààðàâñàí òóõàéí óëàìæëàëóóäûã îë.
C ∂z∂x = 2xy2 + 5
5x+4y ; ∂z∂y = 2x2y + 4
5x+4y B
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òóõàéí óëàìæëàë
Òîäîðõîéëò
�°ð°°ð õýëáýë:
∂z
∂x= lim
∆→0
∆xz
∆x= lim
∆x→0
f (x0 + ∆x ; y0)− f (x0; y0)
∆x(7)
∂z
∂y= lim
∆→0
∆yz
∆y= lim
∆y→0
f (x0; y0 + ∆y)− f (x0; y0)
∆y(8)
Æèøýý
z = x2y2 + ln(5x + 4y) + 1 ôóíêöýýñ x , y -àðãóìåíòóóäààðàâñàí òóõàéí óëàìæëàëóóäûã îë.
C ∂z∂x = 2xy2 + 5
5x+4y ; ∂z∂y = 2x2y + 4
5x+4y B
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òóõàéí óëàìæëàë
Òîäîðõîéëò
�°ð°°ð õýëáýë:
∂z
∂x= lim
∆→0
∆xz
∆x= lim
∆x→0
f (x0 + ∆x ; y0)− f (x0; y0)
∆x(7)
∂z
∂y= lim
∆→0
∆yz
∆y= lim
∆y→0
f (x0; y0 + ∆y)− f (x0; y0)
∆y(8)
Æèøýý
z = x2y2 + ln(5x + 4y) + 1 ôóíêöýýñ x , y -àðãóìåíòóóäààðàâñàí òóõàéí óëàìæëàëóóäûã îë.
C ∂z∂x = 2xy2 + 5
5x+4y ; ∂z∂y = 2x2y + 4
5x+4y B
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí òóõàéí óëàìæëàë
Òîäîðõîéëò
�°ð°°ð õýëáýë:
∂z
∂x= lim
∆→0
∆xz
∆x= lim
∆x→0
f (x0 + ∆x ; y0)− f (x0; y0)
∆x(7)
∂z
∂y= lim
∆→0
∆yz
∆y= lim
∆y→0
f (x0; y0 + ∆y)− f (x0; y0)
∆y(8)
Æèøýý
z = x2y2 + ln(5x + 4y) + 1 ôóíêöýýñ x , y -àðãóìåíòóóäààðàâñàí òóõàéí óëàìæëàëóóäûã îë.
C ∂z∂x = 2xy2 + 5
5x+4y ; ∂z∂y = 2x2y + 4
5x+4y B
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí á³òýí äèôåðåíöèàë
Òîäîðõîéëò
Õýðýâ z = f (x ; y) ôóíêöèéí M0(x0; y0) öýã äýýðõ á³òýí°°ð÷ë°ëò íü (2) õýëáýðòýé áè÷èãäýõ áîë z = f (x ; y)-èéãM0(x0; y0) öýã äýýð äèôôåðåíöèàë÷ëàãäàõ ôóíêö ãýæ íýðëýõ
áà ýíýõ³³ °°ð÷ë°ëòèéí ∆x , ∆y -òàé õàðüöóóëàõàä øóãàìàí
áàéõ õýñãèéã f (x ; y) ôóíêöèéí á³òýí äèôôåðåíöèàë ãýíý.
dz =∂f
∂xdx +
∂f
∂ydy (9)
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí á³òýí äèôåðåíöèàë
Òîäîðõîéëò (9)-èéí °ðã°òã°ë
z = f (x1; x2; ...; xn) ãýñýí n õóâüñàã÷òàé ôóíêöèéí á³òýí
äèôôåðåíöèàë:
dz =∂f
∂x1dx1 +
∂f
∂x2dx2 + · · ·+ ∂f
∂xndxn
áàéíà.
Æèøýý
u = xy − yx + zx + z2 ôóíêöèéí á³òýí äèôôåðåíöèàëûã îë.
C ∂u∂x = y + y
x2 + z , ∂u∂y = x − 1
x ,∂u∂z = x + 2z .
dz = (y +y
x2+ z)dx + (x − 1
x)dy + (x + 2z)dz .B
ÌÀÒÅÌÀÒÈÊ-2
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíòîäîðõîéëîãäîõìóæ(ÎÕÔ)-èéíòóõàéí áàá³òýí°°ð÷ë°ëò
Õî¼ðõóâüñàã÷òàéôóíêöèéíõÿçãààð,òàñðàëòã³é÷àíàð
ÄàâõàðõÿçãààðÄàðààëñàíõÿçãààð(ÎÕÔ)-èéíòàñðàëòã³é÷àíàð
(ÎÕÔ)-èéíòóõàéíóëàìæëàëáà á³òýíäèôôåðåíèéë
(ÎÕÔ)-èéíòóõàéíóëàìæëàë(ÎÕÔ)-èéíá³òýíäèôåðåíöèàë
(ÎÕÔ)-èéí á³òýí äèôåðåíöèàë
Òîäîðõîéëò (9)-èéí °ðã°òã°ë
z = f (x1; x2; ...; xn) ãýñýí n õóâüñàã÷òàé ôóíêöèéí á³òýí
äèôôåðåíöèàë:
dz =∂f
∂x1dx1 +
∂f
∂x2dx2 + · · ·+ ∂f
∂xndxn
áàéíà.
Æèøýý
u = xy − yx + zx + z2 ôóíêöèéí á³òýí äèôôåðåíöèàëûã îë.
C ∂u∂x = y + y
x2 + z , ∂u∂y = x − 1
x ,∂u∂z = x + 2z .
dz = (y +y
x2+ z)dx + (x − 1
x)dy + (x + 2z)dz .B