Aula 01 noções de funções
- 1. ƪƝƟƪƙƫ ƜƝ ƜƝƪơƮƙDŽƶƧƪƝƟƪƙƫ ƜƝ ƜƝƪơƮƙDŽƶƧƪƝƟƪƙƫ ƜƝ ƜƝƪơƮƙDŽƶƧƪƝƟƪƙƫ ƜƝ ƜƝƪơƮƙDŽƶƧ
̰˷̛ ( ) 1−
= nn
nxx
dx
d
̱˷̛
( ) x
dx
xd
cos
sen
=
̲˷̛
( ) x
dx
xd
sen
cos
−=
̳˷̛
( ) x
dx
xtgd 2
sec=
̴˷̛
( ) =
dx
xd cotg
̓ ȱȽɁɁȳȱ
̱
Ɇ
̵˷̛
( ) xtgx
dx
xd
sec
sec
=
̶˷̛
( ) =
dx
xd cossec
̓ ȱȽɁɁȳȱ Ɇ˷ȱȽɂȵ Ɇ
̷˷̛
( )
2
1
1sen
xdx
xarcd
−
=
̸˷̛
( )
2
1
1cos
xdx
xarcd
−
−=
̰̯˷̛
( )
2
1
1
xdx
xtgarcd
+
=
̰̰˷̛
( )
2
1
1cotgarc
xdx
xd
+
−=
̰̱˷̛
( )
1
1secarc
2
−
=
xxdx
xd
̰̲˷̛
( )
1
1cossecarc
2
−
−=
xxdx
xd
̰̳˷̛
( ) e
xdx
xd
a
a log
1log
=
̴̰˷̛
( )
xdx
xd 1ln
=
̵̰˷̛
( ) aa
dx
ad x
x
ln=
̶̰˷̛
( ) x
x
e
dx
ed
=
̷̰˷̛
( )
dx
dv
uu
dx
du
vu
dx
ud vv
v
ln1
+= −
ɂȯȷɁ ȿɃȳ Ʉ ͆ Ʉ̚Ɇ̛ ȳ Ƀ ͆ Ƀ ̚Ɇ̛
ƜȳɀȷɄȯȲȯ Ȳȯ ȴɃȼɚɌȽ ȱȽȻȾȽɁɂȯƜȳɀȷɄȯȲȯ Ȳȯ ȴɃȼɚɌȽ ȱȽȻȾȽɁɂȯƜȳɀȷɄȯȲȯ Ȳȯ ȴɃȼɚɌȽ ȱȽȻȾȽɁɂȯƜȳɀȷɄȯȲȯ Ȳȯ ȴɃȼɚɌȽ ȱȽȻȾȽɁɂȯ
dx
du
du
dy
dx
dy
du
dx
du
dy
.==
ƜȳɀȷɄȯȲȯ Ȳȯ ȴɃȼɚɌȽ ȷȼɄȳɀɁȯƜȳɀȷɄȯȲȯ Ȳȯ ȴɃȼɚɌȽ ȷȼɄȳɀɁȯƜȳɀȷɄȯȲȯ Ȳȯ ȴɃȼɚɌȽ ȷȼɄȳɀɁȯƜȳɀȷɄȯȲȯ Ȳȯ ȴɃȼɚɌȽ ȷȼɄȳɀɁȯ
dx
dydy
dx 1
=
’‘•–‹Žƒ ͳ ʹͲͳ͵
- 2. Ͳͳ ͳ
ͳǤ ‘…‡‹–‘• —†ƒ‡–ƒ‹•ͳǤ ‘…‡‹–‘• —†ƒ‡–ƒ‹•ͳǤ ‘…‡‹–‘• —†ƒ‡–ƒ‹•ͳǤ ‘…‡‹–‘• —†ƒ‡–ƒ‹•
ͳǤͳ ‘• ‡ƒ‹•ͳǤͳ ‘• ‡ƒ‹•ͳǤͳ ‘• ‡ƒ‹•ͳǤͳ ‘• ‡ƒ‹•
‘†‘• ‘• …‘…‡‹–‘• †‘ …žŽ…—Ž‘ „ƒ•‡‹ƒǦ•‡ ‡ ’”‘’”‹‡†ƒ†‡• †‘ …‘Œ—–‘
- 4. ‡ ‘• ’‘–‘• †‡ —ƒ ”‡–ƒ ”‡ƒŽ ሺ”‡–ƒ
…‘‘”†‡ƒ†ƒሻ ‘†‡ Ͳ ± ƒ ‘”‹‰‡ǡ …‘ˆ‘”‡ ˜‡‘• ƒ ˆ‹‰—”ƒ ƒ„ƒ‹š‘ǣ
- 5. ͳǤͳǤͳͳǤͳǤͳͳǤͳǤͳͳǤͳǤͳ ‡•‹‰—ƒŽ†ƒ†‡•‡•‹‰—ƒŽ†ƒ†‡•‡•‹‰—ƒŽ†ƒ†‡•‡•‹‰—ƒŽ†ƒ†‡•
•‹‰‹ˆ‹…ƒ†‘ ‰‡‘±–”‹…‘ †ƒ †‡•‹‰—ƒŽ†ƒ†‡ ba ± “—‡ ƒ ‡•–ž ƒ ‡•“—‡”†ƒ †‡ „Ǣ ƒ
†‡•‹‰—ƒŽ†ƒ†‡ ‡“—‹˜ƒŽ‡–‡ ba •‹‰‹ˆ‹…ƒ “—‡ „ ‡•–ž †‹”‡‹–ƒ †‡ ƒ Ǥ ‘ ± ’‘•‹–‹˜‘ ‘—
‡‰ƒ–‹˜‘ …‘ˆ‘”‡ 0a ‘— 0a Ǥ
ͳǤͳǤʹ ƒŽ‘” „•‘Ž—–‘ͳǤͳǤʹ ƒŽ‘” „•‘Ž—–‘ͳǤͳǤʹ ƒŽ‘” „•‘Ž—–‘ͳǤͳǤʹ ƒŽ‘” „•‘Ž—–‘
˜ƒŽ‘” ƒ„•‘Ž—–‘ †‡ — ‘ šǡ ‹†‹…ƒ†‘ ’‘” ȁšȁǡ ± †‡ˆ‹‹†‘ ’‘”ǣ
¯
®
≥
=
0xse,-
0xse,
||
x
x
x
‡‘‡–”‹…ƒ‡–‡ǡ ”‡’”‡•‡–ƒ ƒ †‹•–Ÿ…‹ƒ †‘ ’‘–‘ š ƒ ‘”‹‰‡ ͲǤ
š‡’Ž‘•š‡’Ž‘•š‡’Ž‘•š‡’Ž‘•
ȁ͵ȁ ൌ ͵ ȁͲȁ ൌ Ͳ
ȁǦ͵ȁ ൌ ǦሺǦ͵ሻ ൌ ͵ |3| π− ൌ ( ) 33 −=−− ππ
‡‘•–”ƒǦ•‡ “—‡ ǣ 0,
ouy|x|)
ouy|x|)
|x|)
°
¯
°
®
==⇔=
⇔
−⇔
y
-yxyxiii
-yxyxii
yxyyi
ͳǤͳǤ͵
- 9. –‡”˜ƒŽ‘•
ƒ†‘• †‘‹• ‘• ”‡ƒ‹• ƒ ‡ „ǡ …‘ ƒ ൏ „ǡ †‡•‹‰ƒ‘• ’‘” ሿƒǡ„ሾ ‘ ‹–‡”˜ƒŽ‘ ƒ„‡”–‘ǡ ’‘”
ሾƒǡ„ሿ ‘ ‹–‡”˜ƒŽ‘ ˆ‡…Šƒ†‘ǡ ’‘” ሾƒǡ„ሾ ‡ ሿƒǡ„ሿ ‘• ‹–‡”˜ƒŽ‘• •‡‹Ǧƒ„‡”–‘•ǡ ‡ ‹–‡”˜ƒŽ‘• †‡ˆ‹‹†‘• …‘ ‘
•À„‘Ž‘ ∞+ ‘— ∞− ǡ ’‘” ‹–‡”˜ƒŽ‘• ‹ˆ‹‹–‘•Ǥ
765432101-2-3-4-5-6-7-8-
π2ba2,332,- 51
- 12. ȁ bxa ሽ ሻ ൌ
ƒ „
ሾƒǡ„ሿ ൌ ሼš ∈
- 16. ȁ ax ሽ ൌ
ƒ
ሾƒǡ ∞ ሾ ൌ ሼš ∈
- 17. ȁ ax ≥ ሽ ൌ
ƒ
ሿ ∞− ǡ„ሾ ൌ ሼš ∈
- 18. ȁ bx ሽ ൌ
„
ሿ ∞− ǡ„ሿ ൌ ሼš ∈
- 20. ൌ
„•ǣ
ƒ ± †‡‘‹ƒ†‘ ‡š–”‡‹†ƒ†‡ ‹ˆ‡”‹‘” †‘ ‹–‡”˜ƒŽ‘Ǣ
„ ± †‡‘‹ƒ†‘ ‡š–”‡‹†ƒ†‡ •—’‡”‹‘” †‘ ‹–‡”˜ƒŽ‘Ǣ
Š ൌ ȁ„Ǧƒȁ ± ƒ ƒ’Ž‹–—†‡ †‘ ‹–‡”˜ƒŽ‘Ǥ
š‡”…À…‹‘•š‡”…À…‹‘•š‡”…À…‹‘•š‡”…À…‹‘•
ͳǤሻ ² ƒ• ”‡’”‡•‡–ƒÙ‡• †‘• …‘Œ—–‘• ƒ„ƒ‹š‘ǣ
ƒǤሻ ሿͲǡ͵ሾ †Ǥሻ ሿ ∞,a ሾ
„ǤሻሾǦͳǡʹሾ ‡Ǥሻ ሺ ∞∞− , ሻ
…Ǥሻሺ ∞− ǡͷሻ
ʹǤሻ •–—†‡ ƒ• ‡š’”‡••Ù‡• ƒ„ƒ‹š‘ǣ
ƒǤሻȁš Ǧ ʹȁ ൌ ͷ „Ǥሻ ȁš Ǧ ͳȁ 3≥ …Ǥሻ ȁš Ǧ ͵ȁ 4≤
’‘•–‹Žƒ ͳ ʹͲͳ͵
- 21. Ͳͳ ͵
ͳǤʹ —Ù‡•ͳǤʹ —Ù‡•ͳǤʹ —Ù‡•ͳǤʹ —Ù‡•
ƒ ˆ— ‘ ˆ †‡ — …‘Œ—–‘ ‡ — …‘Œ—–‘ ± —ƒ ”‡Žƒ ‘ “—‡ ƒ••‘…‹ƒ ƒ –‘†‘
‡Ž‡‡–‘ š †‡ — ï‹…‘ ‡Ž‡‡–‘ › ൌ ˆሺšሻ †‡ Ǥ
‘–ƒ ‘
ˆ ǣ →
š › ൌ ˆሺšሻ
‘À‹‘ †‡ ˆ ‘–”ƒǦ ‘À‹‘ †‡ ˆ
š ± …Šƒƒ†‘ †‡ ˜ƒ”‹ž˜‡Ž ‹†‡’‡†‡–‡ ሺƒ”‰—‡–‘ሻ ‡ › ± ƒ ˜ƒ”‹ž˜‡Ž †‡’‡†‡–‡ ሺ‹ƒ‰‡ ‘—
ˆ— ‘ሻǤ
ͳǤʹǤͳ ‹’‘• †‡ ˆ—Ù‡• ”‡ƒ‹•ͳǤʹǤͳ ‹’‘• †‡ ˆ—Ù‡• ”‡ƒ‹•ͳǤʹǤͳ ‹’‘• †‡ ˆ—Ù‡• ”‡ƒ‹•ͳǤʹǤͳ ‹’‘• †‡ ˆ—Ù‡• ”‡ƒ‹•
ƒ ˆ— ‘ ˆ ± —ƒ ˆ— ‘ ’‘Ž‹‘‹ƒŽ •‡ ˆሺšሻ ± — ’‘Ž‹Ø‹‘ǡ ‹•–‘ ±ǡ •‡
ˆሺšሻ ൌ ƒš ƒǦͳšǦͳ ǤǤǤ ƒͳš ƒͲǡ
‘†‡ ‘• …‘‡ˆ‹…‹‡–‡• ƒ ǡ ƒǦͳ ǡǤǤǤǡ ƒͳ ‡ ƒͲǡ • ‘ ‘• ”‡ƒ‹• ‡ ‘• ‡š’‘‡–‡• • ‘ ‹–‡‹”‘• ‘Ǧ
‡‰ƒ–‹˜‘• Ǥ
‡ ƒ ≠ Ͳǡ ‡– ‘ ˆ ± †‡ ‰”ƒ— Ǥ
ƒ ˆ— ‘ ”ƒ…‹‘ƒŽ ± ‘ “—‘…‹‡–‡ †‡ †—ƒ• ˆ—Ù‡• ’‘Ž‹‘‹ƒ‹•Ǥ
ƒ ˆ— ‘ ƒŽ‰±„”‹…ƒ ± —ƒ ˆ— ‘ “—‡ ’‘†‡ •‡” ‡š’”‡••ƒ ‡ –‡”‘• †‡ •‘ƒ•ǡ
†‹ˆ‡”‡ƒ•ǡ ’”‘†—–‘•ǡ “—‘…‹‡–‡• ‘— ’‘–²…‹ƒ• ”ƒ…‹‘ƒ‹• †‡ ’‘Ž‹Ø‹‘•Ǥ ‘” ‡š‡’Ž‘ǡ
ˆሺšሻൌ
( )
xx
xx
xx
+
+
+−
3
2
32 5
25
• ˆ—Ù‡• “—‡ ‘ • ‘ ƒŽ‰±„”‹…ƒ• • ‘ †‹–ƒ• –”ƒ•…‡†‡–‡•ǡ ’‘” ‡š‡’Ž‘ǡ ƒ• ˆ—Ù‡•
–”‹‰‘‘±–”‹…ƒ•ǡ ‡š’‘‡…‹ƒ‹• ‡ Ž‘‰ƒ”À–‹…ƒ•Ǥ
š‡’Ž‘š‡’Ž‘š‡’Ž‘š‡’Ž‘
ƒǤሻ ˆሺšሻ ൌ …‘• ʹš Ȃ ƒ”… –‰ ሺš Ȃ ͵ሻ
- 23. Ͳͳ Ͷ
„Ǥሻ ‘•‹†‡”‡ ƒ• •‡‰—‹–‡• ˆ—Ù‡• –”ƒ•…‡†‡–‡• ˆሺšሻ ൌ ‡š ( )*
+∈IRx ‡ ‰ሺšሻ ൌ Ž šǡ ‡– ‘ǣ
ሺˆ‘‰ሻሺšሻ ൌ ˆሺ‰ሺšሻሻ ൌ ‡Ž š ൌ š
ሺ‰‘ˆሻሺšሻ ൌ ‰ሺˆሺšሻሻ ൌ Ž ‡š ൌ š
‡•–‡ …ƒ•‘ –‡‘• “—‡ ˆ ± ƒ ˆ— ‘ ‹˜‡”•ƒ †‡ ‰ ‡
- 24. Ǥ
š‡”…À…‹‘•š‡”…À…‹‘•š‡”…À…‹‘•š‡”…À…‹‘•
͵Ǥሻ‡†‘ ˆሺšሻ ൌ •‡ ʹš …‘• šǡ ‡…‘–”‡ ˆሺͲሻǤ
ͶǤሻ‡†‘ ( )=Φ z Ͷœǡ ‘•–”‡ “—‡ ( ) ( ) ( )zzz Φ=Φ−+Φ 31 Ǥ
ͷǤሻ ‡–‡”‹‡ ‘ †‘À‹‘ ሺ…ƒ’‘ †‡ †‡ˆ‹‹ ‘ሻ †ƒ• ˆ—Ù‡• ƒ„ƒ‹š‘ǣ
ƒǤሻ› ൌ 42
−x „Ǥሻ › ൌ
x−1
1
…Ǥሻ › ൌ
9
1
2
−x
†Ǥሻ › ൌ
( ) 1xcos
1
−
‡Ǥሻ › ൌ ( )2
1log x−
ͳǤʹǤʹ
”žˆ‹…‘ †‡ —ƒ ˆ— ‘ ”‡ƒŽͳǤʹǤʹ
”žˆ‹…‘ †‡ —ƒ ˆ— ‘ ”‡ƒŽͳǤʹǤʹ
”žˆ‹…‘ †‡ —ƒ ˆ— ‘ ”‡ƒŽͳǤʹǤʹ
”žˆ‹…‘ †‡ —ƒ ˆ— ‘ ”‡ƒŽ
‡Œƒ ˆ ǣ → Ǥ ‰”žˆ‹…‘ †ƒ ˆ— ‘ ˆ ± ‘ …‘Œ—–‘ǣ
‰”ƒˆሺˆሻ ൌ ሼሺšǡ›ሻ ∈
- 25. ȁ › ൌ ˆሺšሻሽǤ
š‡’Ž‘•ǣš‡’Ž‘•ǣš‡’Ž‘•ǣš‡’Ž‘•ǣ
ƒǤሻ ˆሺšሻ ൌ šʹ „Ǥሻ ˆሺšሻ ൌ
x
1
’‘•–‹Žƒ ͳ ʹͲͳ͵
- 26. Ͳͳ ͷ
1
1
…Ǥሻ ˆሺšሻ ൌ Ž š †Ǥሻ ˆሺšሻ ൌ ‡ š
‡Ǥሻ ˆሺšሻ ൌ •‡ š ˆǤሻ ˆሺšሻ ൌ …‘• š
‰Ǥሻ ˆሺšሻ ൌ –‰ š ŠǤሻ ˆሺšሻ ൌ …‘–‰ š
- 28. Ͳͳ
‹Ǥሻ ˆሺšሻ ൌ …‘••‡… š ŒǤሻ ˆሺšሻ ൌ •‡… š
Ǥሻ ˆሺšሻ ൌ ƒ”… •‡ š ŽǤሻ ˆሺšሻ ൌ ƒ”… …‘• š
’‘•–‹Žƒ ͳ ʹͲͳ͵
- 29. Ͳͳ
Ǥሻ ˆሺšሻ ൌ ƒ”… •‡… š Ǥሻ ˆሺšሻ ൌ ƒ”… …‘••‡… š
‘Ǥሻ ˆሺšሻ ൌ ƒ”…–‰ š ’Ǥሻ ˆሺšሻ ൌ ƒ”… …‘–‰ š
