Via tehnika kola
Via tehnika kola
Doboj
Vesna Mii
Doboj 2003
I ispitni rokGrupa I1. Izraunati korjen .
2.Ako je u razvoju koeficijenti drugog lana za 44 je manji od
koeficijenata treeg lana, odrediti koji lan ne sadri x.
3. Sastaviti jednainu ravni koja prolazi kroz presjek ravni
2x-3y-z+5=0 i x+3y+2z-2=0 i taku M(2,1,0).4. Rijeit i diskutovati
sistem jednaina:
5. Nacrtati grafik funkcije i ispitati tok: .
6.
7. Rijeiti diferencijalnu jednainu:
1.
2.
je koeficijent etvrtog lana, a etvrti lan ne sadri x.
3.
4.
Sistem je neodreen
Sistem je neodreen
5.
3.) Funkcija nije periodina
5.) Znak funkcije :
6.)
7.
8.
9.Grafik 6.
(*)
-
-
,
(1)+(3)
, , ,
Uvrstimo u (*)
,
7.
Grupa II
1. Pokazati da predstavlja krunicu sa centrom u taki i .2.
Rijeiti matrinu jednainu E jedinina matrica i .
3. Odrediti jednainu prave koja prolazi kroz taku M(1,1,0) i
paralelna je sa ravnima i .
4. Nacrtati grafik i ispitati tok funkcije : .
5. Rijeiti integral .
6. Rijeiti diferencijalnu jednainu : .7. Dokazati da je svaki
prirodan broj n :
djeljivo sa 7.
1.
2.
,
3.
prava p : (Uslov paralelnosti prave p i ravni , je )
Uzmimo n=-4 , l=5 , m=-7
4.
funkcija nije periodina
, nema nula
5.) Znak funkcije
6.)
7.) Znak prvog izvoda
5.
6.
7.
djeljivo sa 7
I korak : n=1
II korak : n=k (pretpostavimo da je funkcija tana, za n=k)
III korak : n=k+1
a to znai da je formula tana i za m=k+1.II ispitni rokGrupa
I
1. Izraunati x i y iz jednaina :
2. Rijeiti i diskutovati sistem :
3. Odrediti taku B simetrinu taki u odnosu na ravan
4. Ispitati i nacrtati grafik funkcije : .
5.
6.
1.
/
uvrstimo u prvu jednainu
2.
Sistem je neodreen
Sistem je neodreen
3.
vektor normale ravni
zadovoljava
jednainu ravni
(1)
(*)
uvrstimo u (1)
uvrstimo u (*)
je simetrina taka
je traena taka4.
1.) D:
2.) Nije ni parna ni neparna
3.) Nije periodina
4.)
5.)Znak funkcije:
Znak
5.
6.
Grupa II1. Ako je nai emu je jednako .
2. Rijeiti matrinu jednainu AX=B , X=?.
3. Sastaviti jednainu ravni koja prolazi kroz presjek ravni i i
kroz taku M (2,1,0).
4.
5. Nacrtati grafik funkcije i ispitati tok funkcije : .6.
1.
Dakle,
2.
3.
Traena ravan
4.
5.)
1.)
2.) funkcija nije ni parna ni neparna.
3.) Nije periodina.
4.) presjek s x-osom5.) Znak funkcije
6.) Prvi izvod funkcije
7.) Znak prvog izvoda
8.) Asimptote
kosa asimptota funkcije .
9.) Grafik
6.
III ispitni rokGrupa I
1.Nai redni broj onog lana razvoja binoma koja ne sadri a i b na
isti eksponent.
2.
. X=?
3.Sastaviti jednainu ravni koja prolazi kroz taku A (3,2,1) i
normalna je na ravnima i .4. Nacrtati grafik funkcije i ispitati
tok .5. .
6. Rijeiti diferencijalnu jednainu :
1.
Ako je redni broj traenog lana k+1 , onda je :
Poto po uslovu zadatka a i b imaju iste eksponente to je :
Znai, deseti lan po redu sadravae i a i b sa jednakim
eksponentima.
2.
Za
Za a=1 , ima beskonano rjeenja, a za a=-2, nema rjeenja.
3.
4.
1.) .
2.) Nije ni parna ni neparna funkcija.
3.) Nije periodina.
4.)
5.) Znak funkcije
6.) Prvi izvod funkcije
7.) Znak prvog izvoda
8.)Grafik5.
6.
Grupa II
1.Izraunati:
2.
3.Kako glasi jednaina ravni koja prolazi takama M(3,-5,1) i
N(4,1,2),a normalna je na ravan .4. Ispitati i nacrtati grafik
funkcije .
5. .
6.
1.
2.
Za sistem je neodreen, ima beskonano mnogo rjeenja.
3.
4.
1.)
2.) funkcija je neparna.3.) Nije periodina4.) presjek sa
x-osom5.)Znak funkcije
prevojna takaKosa asimptota
kosa asimptota
6.Grafik
5.
6.
IV Ispitni rokGrupa I
1. Rijeiti jednainu : .2. Odrediti koeficijente lanova koji
sadre u razvoju .3. Odrediti projekciju take A(2,1,2) na pravu
.
4. Nacrtati grafik funkcije .
5.
6. Rijeiti diferencijalnu jednainu .
1.Rijeiti
2.
I lan je
II lan je
je etvrti lan i on sadri .
3.
Uvrstimo u (*)
Taka .4.
1.)
2.)
funkcija nije parna ni neparna
3.)Funkcija nije periodina4.)
5.)Znak funkcije:
Funkcija pozitivna
(grafik iznad x-ose)
6.
Znak
5.
Uvrstimo
Dakle,
6.
Grupa II
1. Rijeiti jednainu : .
2. Odrediti za koje vrijednosti parametara a i b sistem
jednaina:
a.) ima samo jedno rjeenje,
b.) nema rjeenja,
c.) ima beskonano mnogo rjeenja.
3. Sastaviti jednainu prave koja prolazi kroz taku M(1,1,0) i
paralelna je sa ravnima i .
4. Nacrtati i ispitati funkcije : .
5. Rijeiti integral .
6. Rijeiti diferencijalnu jednainu .1.
2.
Diskusija
Za sistem ima jedinstveno rjeenje
Za , imamo sistem je neodreen
(ima beskonano mnogo rjeenja)
Za sistem je nemogu, nema rjeenja.
3.
4.
2.Nije ni parna ni neparna
3.Nije periodina Znak funkcije :
5.Izvod :
Znak
y=2 je asimptota
Grafik
5.
(1) , ,
(2)
(3)
Uvrstimo A i B u (2):
uvrstimo u (3)
6.
smjena:
Grupa III
1.Izraunati
2.Diskutovati rjeenja sistema jednaina u zavisnosti od parametra
a:
3.Kako glasi jednaina ravni koja prolazi takama a normalna je na
ravan .4. Grafik funkcije nacrtati i ispitati tok:
5.
6. Rijeiti diferencijalnu jednainu uz poetni uslov .
1.
2.
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3.
4.
1.
2.Nije ni parna ni neparna3.Nije periodina
4.
Znak y
, y=0 asimptota
5.
6.
Kako je y=1 za x=0 imamo tj. .
Traeno partikularno rjeenje je .
V Ispitni rokGrupa I
1. Rijeiti jednainu : .
2. Dokazati da je matrica :
Nula polinoma .
3. Odrediti presjenu taku pravih :
i
4. Nacrtati grafik funkcije : .5. Izraunati integral : .
6. Rijeiti jednainu : .
1.
2.
3.
Uslov presjeka pravih je ispunjen.
Uvrstimo x, y, z u (p2):
Uvrstimo u (*):
4.
3.) Funkcija nije periodina
4.) y=0 presjek s x osom
5.) Znak funkcije
Znak
5.
6.
Grupa II
1. U zavisnosti od realnog parametra m rijeiti i diskutovati
sistem :
2. Nai vrijednost x u izrazu iji je trei lan razvoja
1000000.
3. Data su tjemena paralelograma ABCD.Sastaviti jednainu
dijagonale BD ovog paralelograma.
4. Nacrtati grafik funkcije .
5. .
6. Rijeiti diferencijalnu jednainu : .
1.
neodreen
sistem je nemogu.2.
3.
Jednaina prave kroz dvije take
4.
1.) .
2.) Nije ni parna ni neparna funkcija.
3.) Nije periodina.
4.) 5.) Znak funkcije
6.) Prvi izvod funkcije
7.) Znak prvog izvoda
8.)Grafik
5.
6.
VI Ispitni rok
Grupa I
1. Odrediti x u izrazu u kome je odnos sedmog lana od poetka
razvoja binoma prema sedmom lanu od kraja .
2. Izraunati .
3. Sastaviti jednainu normale povuene iz take (2,3,1) na pravu
:
.
4. Nacrtati grafik funkcije .
5. Odrediti integral .
6. Rijeiti diferencijalnu jednainu : .
1.
Sedmi lan razvoja binoma je:
a sedmi lan od kraja je :
Binomni koeficijenti su im jednaki, a to proizilazi iz njihove
osobine simetrinosti.
Prema uslovu zadatka je:
2.
3.
zadovoljava jednainu prave
uvrstimo u
normala :
Traena jednaina normale : .
4.
3.Nije periodina
Znak y
Prvi izvod :
Znak
5.
6.
Grupa II1. Rijeiti matrinu jednainu :
.
2. Dokazati jednakost : .
3. Nai presjeke ravni sa koordinatnim osama i koordinatnim
ravnima.
4. Nacrtati grafik funkcije .
5. .
6. .
1.
2.
1.)
2.) , Pretpostavimo da je (F) tano za .
tj. (T)
3.) Treba dokazati da je (F) tano i za
(T)3.
Datu ravan svedemo na segmentni oblik:
Vidi se da su odsjeci na koordinatnim osama:
Presjek ravni sa xy ravni
4.
1.) .
2.) parna funkcija
3.)
Funkcija je periodina sa periodom
4.)
5.) Znak funkcije
6.) Prvi izvod funkcije
(keZ)
7.) Znak prvog izvoda
8.) Grafik5.
6.
VII Ispitni rok
Grupa I
1. Dato je , gdje je . Pokazati da je .
2. Rijeiti matrinu jednainu :
3. Odrediti rastojanje paralelnih pravih : .4. Nacrtati grafik
funkcije .
5. Odrediti integral .
6. Odrediti opte rjeenje diferencijalne jednaine : .
1.
Dakle, vrijedi :
2.
3.
4.
Domen
Parnost funkcija nije ni parna ni neparna.
Nije periodina Nule funkcije
Znak funkcije
Prvi izvod funkcije
Asimptote
kosa asimptota funkcije .
Grafik
5.
6.
Grupa II
1. Nai vrijednost x u izrazu iji je etvrti lan razvoja binoma
200.
2. Rijeiti matrini jednainu AX=B.
3.Napisati jednainu ravni koja prolazi kroz taku A(3,-5,1) i
paralelna je ravni
.4. Nacrtati grafik funkcije .
5.
6.
1.
IV lan je
2.
3.
Jednaina traene ravni
4.
1.)
2.) Funkcija nije parna ni neparna.3.) Period .
4.)
5.) Znak funkcije
8.) Grafik
5.
6.
Sadraj Stranica
I Ispitni rok
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II Ispitni rok
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III Ispitni rok
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IV Ispitni rok
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V Ispitni rok
.....................................................................................
VI Ispitni rok
....................................................................................
VII Ispitni rok
..................................................................................
Sadraj
.............................................................................................
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