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Cell["Ellen\[ODoubleAcute]rz\[ODoubleAcute] k\[EAcute]rd\[EAcute]sek \
\[EAcute]s feladatok", "Title",
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Cell["1. h\[EAcute]t, elemi feladatok, \[AAcute]br\[AAcute]zol\[AAcute]s", \
"Section",
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Cell[TextData[{
 "A Wolfram Alpha kapcsolat seg\[IAcute]ts\[EAcute]g\[EAcute]vel  \
v\[AAcute]laszoljon a k\[ODoubleDot]vetkez\[ODoubleAcute] k\[EAcute]rd\
\[EAcute]sekre:\nMennyi a C vitamin \
s\[UDoubleAcute]r\[UDoubleAcute]s\[EAcute]ge?\nAz \[ODoubleDot]t legmagasabbb \
hegycs\[UAcute]cs\nAz \[ODoubleDot]t legnagyobb DGP-t el\[EAcute]r\
\[ODoubleAcute] orsz\[AAcute]g\nAz ",
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  FormBox[
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 " f\[UDoubleDot]ggv\[EAcute]ny x0=1 helyen vett \
\[EAcute]rint\[ODoubleAcute]j\[EAcute]nk \[AAcute]br\[AAcute]zol\[AAcute]sa\n\
A 3x-2y+ z =1 s\[IAcute]k norm\[AAcute]lvektora\nA 3x-2y+ z =1 s\[IAcute]k \
\[EAcute]s x-2y+z s\[IAcute]kok metsz\[EAcute]svonala\n..."
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Cell[TextData[{
 "K\[EAcute]sz\[IAcute]tsen lok\[AAcute]lis \[EAcute]s glob\[AAcute]lis \
\[AAcute]br\[AAcute]kat az  ",
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 " f\[UDoubleDot]ggv\[EAcute]nyr\[ODoubleAcute]l. Mit tudunk mondani a \
szingularit\[AAcute]sokr\[OAcute]l \[EAcute]s az aszimptotikus viselked\
\[EAcute]sr\[ODoubleAcute]l."
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Cell["\<\
Mit l\[AAcute]tunk, ha az |sin(x)| f\[UDoubleDot]ggv\[EAcute]nyt az orig\
\[OAcute] eg\[EAcute]sz kis k\[ODoubleDot]rnyezet\[EAcute]ben \[AAcute]br\
\[AAcute]zoljuk?\
\>", "Text",
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Cell[TextData[{
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 " polinomot. \[OpenCurlyDoubleQuote]Nagyon glob\[AAcute]lis\
\[CloseCurlyDoubleQuote] \[AAcute]bra eset\[EAcute]n melyik f\[UDoubleDot]ggv\
\[EAcute]nyt\[ODoubleAcute]l nem tudn\[AAcute]nk megk\[UDoubleDot]l\
\[ODoubleDot]nb\[ODoubleDot]ztetni?"
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Cell[TextData[{
 "Vegy\[UDoubleDot]nk az ",
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 " \[EAcute]s ",
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 " f\[UDoubleDot]ggv\[EAcute]nyeket. Mit l\[AAcute]tunk \
\[OpenCurlyDoubleQuote]nagyon glob\[AAcute]lis\[CloseCurlyDoubleQuote] \
\[AAcute]bra eset\[EAcute]n ([0,X] szakaszon \[AAcute]br\[AAcute]zolva, ahol \
X \[OpenCurlyDoubleQuote]kell\[ODoubleAcute]en\[CloseCurlyDoubleQuote] nagy)."
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Cell[TextData[{
 "Tekints\[UDoubleDot]k az ",
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     RowBox[{
      RowBox[{"-", "2"}], "x"}]], "-", 
    SuperscriptBox["\[ExponentialE]", 
     RowBox[{
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 " f\[UDoubleDot]ggv\[EAcute]nyt. Melyik line\[AAcute]ris f\[UDoubleDot]ggv\
\[EAcute]nnyel k\[ODoubleDot]zel\[IAcute]thet\[ODoubleAcute] az ",
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  FormBox[
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 "k\[ODoubleDot]rnyezet\[EAcute]ben, \[EAcute]s milyen f\[UDoubleDot]ggv\
\[EAcute]ny \[IAcute]rja le az aszimptotikus viselked\[EAcute]st (ha x nagyon \
nagy)."
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Cell["\<\
Hogyan hasonl\[IAcute]tan\[AAcute]nk \[ODoubleDot]ssze k\[EAcute]t f\
\[UDoubleDot]ggv\[EAcute]ny aszimptotikus trendj\[EAcute]t? \
\>", "Text",
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Cell["\<\
Mit l\[AAcute]tunk, ha a cos(x) f\[UDoubleDot]ggv\[EAcute]nyt az \
orig\[OAcute] kell\[ODoubleAcute]en kis k\[ODoubleDot]rnezet\[EAcute]ben \
\[AAcute]br\[AAcute]zoljuk?\
\>", "Text",
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Cell["2. h\[EAcute]t, v\[AAcute]ltoz\[AAcute]s, deriv\[AAcute]lt, integr\
\[AAcute]l", "Section",
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Cell[TextData[{
 "\[CapitalAAcute]br\[AAcute]zolja dinamikusan (Manipulate)  az ",
 Cell[BoxData[
  FormBox[
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 " f\[UDoubleDot]ggv\[EAcute]nyeket ({n,1,10,1})"
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Cell["\<\
A f\[UDoubleDot]ggv\[EAcute]ny v\[AAcute]ltoz\[AAcute]s\[AAcute]t milyen m\
\[OAcute]don hangs\[UAcute]lyozhatjuk \[AAcute]gr\[AAcute]zol\[AAcute]s sor\
\[AAcute]n?\
\>", "Text",
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Cell["\<\
A stroboszk\[OAcute]pikus \[AAcute]br\[AAcute]n k\[EAcute]t \
szomsz\[EAcute]dos pont ismeret\[EAcute]ben a f\[UDoubleDot]ggv\[EAcute]ny \
milyen tulajdons\[AAcute]g\[AAcute]ra \
k\[ODoubleDot]vetkeztehet\[UDoubleDot]nk?\
\>", "Text",
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Cell["\<\
Olvassa le a deriv\[AAcute]lt \[EAcute]rt\[EAcute]k\[EAcute]t a \
f\[UDoubleDot]ggv\[EAcute]ny grafikonj\[AAcute]r\[OAcute]l egy adott helyen.\
\>", "Text",
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  3.694428586612278*^9}}],

Cell["A grafikon ismeret\[EAcute]ben \[AAcute]br\[AAcute]zolja a f\
\[UDoubleDot]ggv\[EAcute]ny deriv\[AAcute]ltj\[AAcute]t.", "Text",
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Cell["\<\
A deriv\[AAcute]lt illetve m\[AAcute]sodik deriv\[AAcute]lt grafikonja \
ismeret\[EAcute]ben k\[ODoubleDot]vetkeztessen a f\[UDoubleDot]ggv\[EAcute]ny \
tulajdons\[AAcute]gaira.\
\>", "Text",
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Cell["\<\
A sebes\[EAcute]gf\[UDoubleDot]ggv\[EAcute]ny grafikonja ismeret\[EAcute]ben \
adja meg a test \[AAcute]ltal megtett utat,... (l\[AAcute]sd p\[EAcute]lda...)\
\>", "Text",
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Cell[TextData[{
 "Sz\[AAcute]m\[IAcute]tsa ki k\[EAcute]t f\[UDoubleDot]ggv\[EAcute]ny \
grafikonja \[AAcute]ltal bez\[AAcute]rt tarom\[AAcute]ny ter\[UDoubleDot]let\
\[EAcute]t, pl.\nsin(x) \[EAcute]s cos(x); x+1 \[EAcute]s ",
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Cell[TextData[{
 "K\[ODoubleDot]zel\[IAcute]tse az ",
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     RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]]],
 " integr\[AAcute]lt baloldali (jobboldali, trap\[EAcute]z) k\[ODoubleDot]zel\
\[IAcute]t\[EAcute]ssel ",
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Cell["\<\
\[CapitalIAcute]rja fel az al\[AAcute]bbi grafikonnal megadott \
f\[UDoubleDot]ggv\[EAcute]ny k\[EAcute]plet\[EAcute]t. Indokolja a megold\
\[AAcute]st.\
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