tag:blogger.com,1999:blog-6473348776046755982024-03-08T11:41:25.194-08:00Rumus Lengkap MatematikaUswati Khoiriahhttp://www.blogger.com/profile/00146480953413302763noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-647334877604675598.post-66245889627669751672010-10-22T01:22:00.000-07:002010-10-22T01:26:14.310-07:00Rumus Matematika<table border="0" cellpadding="9" cellspacing="0"><tbody>
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<tr bgcolor="#cccccc"> <td width="410"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><!-- #BeginEditable "Judul" --><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><b><span style="font-size: small;">Beberapa Benda Ruang Yang Beraturan</span><br />
</b><span style="font-size: xx-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Matematika%203.htm">Matematika Kelas 3</a> > Geometri Ruang Dimensi Tiga</span></span><!-- #EndEditable --></span></td> <td bgcolor="#000000" width="50"> <div align="center"><b><!-- #BeginEditable "halaman" --><b><span style="color: white; font-family: Arial,Helvetica,sans-serif;">442</span></b><!-- #EndEditable --></b></div></td> </tr>
<tr bgcolor="#ffffff"> <td colspan="2"> <div align="center"><span style="font-family: Arial,Helvetica,sans-serif; font-size: xx-small;"><!-- #BeginEditable "Sb%20atas" --><span style="font-family: Arial,Helvetica,sans-serif; font-size: xx-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/0441%20Mat%203-4c3.htm">< Sebelum</a></span><!-- #EndEditable --> <img height="10" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/image/putih%203x3.gif" width="370" /><!-- #BeginEditable "Ss%20Atas" --><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/0443%20Mat%203-5b.htm"><span style="font-family: Arial,Helvetica,sans-serif; font-size: xx-small;">Sesudah ></span></a><!-- #EndEditable --></span></div></td> </tr>
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<tr valign="top"> <td width="50%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">Kubus</span></td> <td width="50%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">Tabung</span></td> </tr>
<tr align="center" valign="top"> <td width="50%"><span style="font-size: x-small;"><img height="90" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Image/3-5a-1.jpg" width="103" /></span></td> <td width="50%"><span style="font-size: x-small;"><img height="98" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Image/3-5a-2.jpg" width="89" /></span></td> </tr>
<tr valign="top"> <td width="50%"> <div align="center"><i><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: xx-small;">rusuk kubus = a</span></i><br />
<i><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: xx-small;">volume = a³ panjang diagonal bidang = a<span style="font-family: Symbol;">Ö</span>2<br />
luas = 6a² panjang diagonal ruang = a<span style="font-family: Symbol;">Ö</span>3</span></i></div></td> <td align="left" width="50%"> <div align="center"><i><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: xx-small;">r = jari-jari<br />
t = tinggi<br />
volume = <span style="font-family: Symbol;">p</span> r² t luas = 2<span style="font-family: Symbol;">p</span>rt</span></i></div></td> </tr>
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<tr valign="top"> <td width="50%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">Prisma</span></td> <td width="50%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">Kerucut</span></td> </tr>
<tr align="center" valign="top"> <td width="50%"><span style="font-size: x-small;"><img height="106" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Image/3-5a-3.jpg" width="105" /></span></td> <td width="50%"><span style="font-size: x-small;"><img height="102" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Image/3-5a-4.jpg" width="86" /></span></td> </tr>
<tr valign="top"> <td width="50%"> <div align="center"><i><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: xx-small;">LA = luas alas<br />
t = tinggi<br />
volume = LA.t</span></i></div></td> <td align="left" width="50%"> <div align="center"><i><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: xx-small;">r = jari-jari<br />
t= tinggi<br />
g = garis pelukis<br />
volume = 1/3 <span style="font-family: Symbol;">p</span>r²t luas = <span style="font-family: Symbol;">p</span>rs</span></i></div></td> </tr>
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<tr valign="top"> <td width="50%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">Limas</span></td> <td width="50%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">Bola</span></td> </tr>
<tr align="center" valign="top"> <td width="50%"><span style="font-size: x-small;"><img height="95" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Image/3-5a-5.jpg" width="115" /></span></td> <td width="50%"><span style="font-size: x-small;"><img height="91" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Image/3-5a-6.jpg" width="97" /></span></td> </tr>
<tr valign="top"> <td width="50%"> <div align="center"><span style="font-size: xx-small;"><i><span style="font-family: Verdana,Arial,Helvetica,sans-serif;">LA = luas alas<br />
t = tinggi<br />
volume = 1/3 LA t</span></i></span></div></td> <td align="left" width="50%"> <div align="center"><span style="font-size: xx-small;"><i><span style="font-family: Verdana,Arial,Helvetica,sans-serif;">r =jari-jari<br />
volume = 4/3 <span style="font-family: Symbol;">p</span>r³</span></i></span></div></td></tr>
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</tbody></table></td> </tr>
<tr bgcolor="#ffffff"> <td colspan="2"><div align="left"><table border="0"><tbody>
<tr valign="top"> <td width="48%"><table border="0" cellpadding="9" cellspacing="0"><tbody>
<tr bgcolor="#cccccc"> <td width="410"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><b><span style="font-size: small;">Limit Fungsi Trigonometri</span><br />
</b><span style="font-size: xx-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Matematika%203.htm">Matematika Kelas 3 </a>> Limit</span></span></span></td> <td bgcolor="#000000" width="50"><div align="center"><b><b><span style="color: white; font-family: Arial,Helvetica,sans-serif;">433</span></b></b></div></td> </tr>
<tr bgcolor="#ffffff"> <td colspan="2"><div align="center"><span style="font-family: Arial,Helvetica,sans-serif; font-size: xx-small;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: xx-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/0432%20Mat%203-2a.htm">< Sebelum</a></span> <img height="10" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/image/putih%203x3.gif" width="370" /><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/0434%20Mat%203-3a.htm"><span style="font-family: Arial,Helvetica,sans-serif; font-size: xx-small;">Sesudah ></span></a></span></div></td> </tr>
<tr bgcolor="#ffffff"> <td colspan="2"><div align="left"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>KETENTUAN</b></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<br />
Untuk x <<< ( x </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol;">®</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> 0 ) maka sin x </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol;">» </span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">x</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><br />
(<span style="font-size: x-small;">x <<< kecil sekali ;</span> </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol;">»</span></span> <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">setara </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;">)<br />
<br />
<span style="font-size: x-small;"> l i m <u>sin x</u> = 1 l i m <u>tg x</u> = 1<br />
x <span style="font-family: symbol;">®</span> 0 x </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: x-small;"> x <span style="font-family: symbol;">®</span> 0 x</span></span></span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: x-small;"> l i m <u> x </u> = 1 l i m <u> x </u> = 1<br />
x <span style="font-family: symbol;">®</span> 0 sin x </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: x-small;"> x <span style="font-family: symbol;">®</span> 0 tg x</span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b><br />
</b></span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>PERLUASAN</b></span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: x-small;"> l i m <u>sin ax</u> = a/b l i m <u>tg ax</u> = a/b</span><span style="font-size: x-small;"><br />
x <span style="font-family: symbol;">®</span> 0 bx </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: x-small;"> x <span style="font-family: symbol;">®</span> 0 bx</span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b><br />
</b></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: x-small;"><br />
l i m <u> ax </u> = a/b l i m <u> ax </u> = a/b</span><span style="font-size: x-small;"><br />
x <span style="font-family: symbol;">®</span> 0 sin bx </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: x-small;"> x <span style="font-family: symbol;">®</span> 0 tg bx</span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b><br />
</b></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><br />
<span style="font-size: x-small;"> l i m <u>sin ax</u> = a/b l i m <u>tg ax</u> = a/b</span><span style="font-size: x-small;"><br />
x <span style="font-family: symbol;">®</span> 0 sin bx </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: x-small;"> x <span style="font-family: symbol;">®</span> 0 tg bx</span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b><br />
</b></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"> <br />
<br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: x-small;"> l i m <u>sin ax</u> = a/b l i m <u>tg ax</u> = a/b</span><span style="font-size: x-small;"><br />
x <span style="font-family: symbol;">®</span> 0 tg bx </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: x-small;"> x <span style="font-family: symbol;">®</span> 0 sin bx</span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b><br />
</b></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"> </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"> </span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">Rumus-rumus trigonometri yang sering digunakan untuk merubah fungsi</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;">:<br />
<br />
<span style="font-size: x-small;">cos x = sin (90° - x)<br />
ctg x = tg (90° - x)<br />
sin ax = 2 sin ½ax cos ½ax</span><br />
<span style="font-size: x-small;">cos ax = 1- 2 sin² ½ax<br />
cos²x = 1 - sin²x</span><br />
<b><br />
<br />
</b><span style="color: black; font-size: x-small;"><b>HAL-HAL KHUSUS</b><br />
<br />
</span></span><br />
<table border="0"><tbody>
<tr valign="top"> <td width="67%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> l i m <u> ax<sup>m</sup> + bx<sup>m-1</sup> + .... </u> =<u><br />
</u>x <span style="font-family: symbol;">® <span style="font-size: small;">¥</span></span> px<sup>n</sup> + qx<sup>n-1</sup> + ...<u><br />
</u></span></td> <td width="33%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: small;"> <span style="font-family: symbol;">¥</span></span><span style="color: black; font-size: x-small;"> untuk m > n ; </span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="color: black; font-size: x-small;">a/p untuk m =n ;</span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="color: black; font-size: x-small;">0 untuk m < n</span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span></td> </tr>
<tr valign="top"> <td width="67%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <u> </u> <u> </u><br />
l i m <span style="font-family: symbol;">Ö</span>ax<sup>2</sup> + bx + c - <span style="font-family: symbol;"> Ö</span>dx<sup>2</sup> + ex + f<u><br />
</u>x <span style="font-family: symbol;">® <span style="font-size: small;">¥</span></span> </span></td> <td width="33%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: symbol;">¥</span></span><span style="color: black; font-size: x-small;"> untuk a > d ; </span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="color: black; font-size: x-small;"><u>b-e</u> untuk m =n ;</span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">2<span style="font-family: symbol;">Ö</span>a</span> <br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="color: black; font-size: x-small;">-</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: symbol;">¥</span></span></span><span style="color: black; font-size: x-small;"> untuk a < d</span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span></td> </tr>
</tbody></table><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="color: black; font-size: x-small;">Bila salah satu suku belum berbentuk tanda akar maka dibentuk dengan cara mengkuadratkan kemudian menarik tanda akar.<br />
<br />
<br />
</span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><span style="color: black; font-size: x-small;"> <b>DALIL L'HOSPITAL</b><br />
<br />
</span></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> Jika fungsi f dan g masing-masing terdifferensir pada titik x= a<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> dan f(a) = g(a) = 0 atau f(a) = g(a) = </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol; font-size: small;">¥</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> maka<br />
<br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> l i m <u> f(x) </u> = l i m <u> f(x)<br />
</u>x <span style="font-family: symbol;">® <span style="font-size: small;">¥</span></span> g(x) x <span style="font-family: symbol;">® <span style="font-family: Verdana,Arial,Helvetica,sans-serif;"> a</span></span> g(x) <u><br />
</u><b> <br />
<br />
CONTOH LIMIT FUNGSI ALJABAR</b><br />
<br />
1. l i m x<sup>2</sup> - 5x + 6 = (3)<sup>2</sup> - 5(3) + 6 = 0 <u><br />
</u> x <span style="font-family: symbol;">® <span style="font-family: Verdana,Arial,Helvetica,sans-serif;">3</span></span> <br />
<br />
2. l i m <u> 3x - 2 </u> = <u> <span style="font-family: symbol;"><span style="font-size: small;">¥</span></span> </u> (*) Uraikan <u><br />
</u> x <span style="font-family: symbol;">® <span style="font-size: small;">¥</span></span> 2x + 1 <span style="font-family: symbol;"><span style="font-size: small;">¥</span></span> <u><br />
</u><b> <br />
</b><u>x(3 - 2/x)</u> = <u>3 - 2/x</u> = <u>3 - 0</u> = <u>3</u><b><br />
</b>x(2 - 1/x) 2 + 1/x 2 - 0 2<br />
<b> <br />
</b><b> </b>atau langsung gunakan hal khusus<br />
<br />
3. l i m <u> x<sup>2</sup> - x - 1 </u> = <u> <span style="font-family: symbol;"><span style="font-size: small;">¥</span></span> </u> (*) Uraikan <u><br />
</u> x <span style="font-family: symbol;">® <span style="font-size: small;">¥</span></span> 10x + 9 <span style="font-family: symbol;"><span style="font-size: small;">¥</span></span> <u><br />
</u><b> <br />
</b><u>x(x - 1 - 1/x)</u> = <u>x - 1 - 1/x</u> = <u><span style="font-family: symbol;"><span style="font-size: small;">¥</span></span> - 1 - 0</u> = <u><span style="font-family: symbol;"><span style="font-size: small;">¥</span></span></u><b> =</b><span style="font-family: symbol;"><span style="font-size: small;">¥</span></span><b> <br />
</b>x(10 - 9/x) 10 + 9/x 10 + 0 10<br />
<br />
<b> </b>atau langsung gunakan hal khusus<br />
<br />
<br />
4. l i m <u> x<sup>2</sup> - 3x + 2 </u> = <u> 0 </u> (*) Uraikan <u><br />
</u> x <span style="font-family: symbol;">® </span> 2 x<sup>2</sup> - 5x + 6 0 <u><br />
</u><b> <br />
</b><u>(x - 1)(x - 2)</u> = <u>(x - 1)</u> = <u>2 - 1</u> = -1 <b> <br />
</b><b> </b><b> </b>(x - 3)(x - 2) = (x - 3) = 2 - 3 <br />
<br />
<b> </b>atau langsung gunakan hal khusus <span style="font-family: symbol;">®</span> Differensial<br />
<br />
<br />
5. l i m <u> x<sup>3</sup> - 3x<sup>2</sup> + 3x - 1 </u> = <u> 0 </u> (*) Uraikan <u><br />
</u> x <span style="font-family: symbol;">® </span> 1 x<sup>2</sup> - 5x + 6 0 <u><br />
</u><b> <br />
</b><u> (x - 1)<sup>3 </sup></u> = <u>(x - 1)<sup>2</sup></u> = <u>(1 - 1)<sup>2</sup></u> = <u>0</u> <b> <br />
</b>(x - 1) (x - 5) (x + 5) (1 + 5) 6<br />
<br />
<b> </b>atau langsung gunakan hal khusus <span style="font-family: symbol;">®</span> Differensial<br />
<br />
<br />
<u> </u> <u> </u> <br />
6. l i m <u> </u><span style="font-family: symbol;">Ö</span><u><sup>2</sup> + x - </u><span style="font-family: symbol;"><u>Ö</u></span><u>2x </u> = <u> 0 </u> (*) Hilangkan tanda akar dengan <u><br />
</u> x <span style="font-family: symbol;">® </span> 2 x - 2 0 mengalikan bentuk sekawan<u><br />
</u><b> <br />
</b><u> (x - 1)<sup>3 </sup></u> = <u>(x - 1)<sup>2</sup></u> = <u>(1 - 1)<sup>2</sup></u> = <u>0</u> = 0 <b> <br />
</b>(x - 1) (x - 5) (x + 5) (1 + 5) 6<br />
<br />
<b> </b>atau langsung gunakan hal khusus <span style="font-family: symbol;">®</span> Differensial<br />
<br />
<br />
<u> </u> <br />
7. l i m (<sup>3x</sup> - <span style="font-family: symbol;">Ö</span>9x<sup>2</sup> + 4x) = <span style="font-family: Symbol; font-size: small;">¥</span> - <span style="font-family: Symbol; font-size: small;">¥</span> (*) Hilangkan tanda akar<u><br />
</u> x <span style="font-family: symbol;">® </span> <span style="font-family: Symbol; font-size: small;">¥</span> <b><br />
</b> <u> </u> <u> </u><br />
<b> </b> l i m (<sup>3x</sup> - <span style="font-family: symbol;">Ö</span>9x<sup>2</sup> + 4x ) = <span style="font-family: Symbol;">é</span> <sup><u>3x</u></sup><u> - <span style="font-family: symbol;">Ö</span>9x<sup>2</sup> + 4x</u> <span style="font-family: symbol;">ù</span> = (*) Hilangkan tanda<u><br />
</u> x <span style="font-family: symbol;">® </span> <span style="font-family: Symbol;">¥ </span> <img height="8" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/image/putih%203x3.gif" width="142" /><span style="font-family: Symbol;">ë</span><b> </b><sup>3x</sup> - <span style="font-family: symbol;">Ö</span>9x<sup>2</sup> + 4x <span style="font-family: symbol;"> û</span><b> </b>akar</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> l i m (<sup>9x<sup>2</sup></sup> - (9x<sup>2</sup> + 4x) = l i m <u> -4x </u> =<u><br />
</u> x <span style="font-family: symbol;">® </span> <span style="font-family: Symbol; font-size: small;">¥</span><span style="font-family: Symbol;"> </span> 3x + <span style="font-family: symbol;">Ö</span>(9x<sup>2</sup> + 4x)<b> </b>x <span style="font-family: symbol;">® </span> <span style="font-family: Symbol; font-size: small;">¥</span> 3x + 3x <span style="font-family: symbol;">Ö</span>[1+(a/9x)]<b><br />
<br />
</b> l i m <u> -4 </u> = <u>-4</u> = <u>-2</u><br />
x <span style="font-family: symbol;">® </span> <span style="font-family: Symbol; font-size: small;">¥</span><span style="font-family: Symbol;"> </span> 3 + 3<span style="font-family: symbol;">Ö</span><sup>(1</sup> + 0)<b> </b>6 3<b><br />
</b><br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b> </b>atau langsung gunakan hal khusus<br />
<br />
<b>CONTOH LIMIT FUNGSI TRIGONOMETRI</b></span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">1. l i m <u>sin 2x</u> = <u> 0</u> (*)<br />
x <span style="font-family: symbol;">® </span>0 tg 3x 0</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <u>sin 2x</u> = <u> 3x</u> <u>2</u> = 1 . 1 . <u>2</u> = <u>2</u><br />
2x tg 3x 3 3 3</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">2. l i m <u>1 - cos 2x</u> = <u> 0</u><br />
x <span style="font-family: symbol;">® </span>0 sin 2x 0</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><u>1 - (1 - 2 sin² 2x)</u> = <u> 2 sin² x </u> = <u> sin x</u> = tg x = 0<br />
2 sin x cos x 2 sin x cos cos x</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">3. l i m <u>1 - cos x</u> = <u> 0</u><br />
x <span style="font-family: symbol;">® </span>0 3x² 0</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><u>2 sin² (½x)</u> = <u> sin (½x) . sin (½x)</u> = <u>1</u> . 1 . 1 = <u>1</u><br />
3 . 4 . (½x) 6 (½x) (½x) 6 6</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> atau langsung gunakan hal khusus <span style="font-family: symbol;">®</span> Differensial</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">4. l i m <u>sin x - sin a</u> = <u> 0</u> (*) <br />
x <span style="font-family: symbol;">® </span>0 x - a 0</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><u>2 cos ½(x+a) sin ½(x-a)</u> = <u> cos ½(x+a) . sin ½(x-a)</u> = <br />
x - a ½ (x - a )</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> cos ½(x+a) . 1 = cos ½(a+a) . 1 = cos a</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> atau langsung gunakan hal khusus <span style="font-family: symbol;">®</span> Differensial</span></div></td></tr>
</tbody></table></td> <td width="52%"><br />
</td> </tr>
<tr align="center" valign="top"> <td width="48%"> <table border="0" cellpadding="9" cellspacing="0"><tbody>
<tr bgcolor="#cccccc"> <td width="410"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><!-- #BeginEditable "Judul" --> <span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><b><span style="font-size: small;">Penggunaan</span><br />
</b><span style="font-size: xx-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Matematika%203.htm">Matematika Kelas 3</a> > Differensial</span></span><!-- #EndEditable --></span></td> <td bgcolor="#000000" width="50"> <div align="center"><b><!-- #BeginEditable "halaman" --><b><span style="color: white; font-family: Arial,Helvetica,sans-serif;">435</span></b><!-- #EndEditable --></b></div></td> </tr>
<tr bgcolor="#ffffff"> <td colspan="2"> <div align="center"><span style="font-family: Arial,Helvetica,sans-serif; font-size: xx-small;"><!-- #BeginEditable "Sb%20atas" --><span style="font-family: Arial,Helvetica,sans-serif; font-size: xx-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/0434%20Mat%203-3a.htm">< Sebelum</a></span><!-- #EndEditable --> <img height="10" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/image/putih%203x3.gif" width="370" /><!-- #BeginEditable "Ss%20Atas" --><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/0436%20Mat%203-4a.htm"><span style="font-family: Arial,Helvetica,sans-serif; font-size: xx-small;">Sesudah ></span></a><!-- #EndEditable --></span></div></td> </tr>
<tr bgcolor="#ffffff"> <td colspan="2"> <div align="left"><!-- #BeginEditable "Isi" --> <span style="color: black; font-family: Garamond; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"><b>1. MENENTUKAN KOEFISIEN ARAN GARIS SINGGUNG </b><br />
<b>(Gradien) di titik </b></span> </span> <b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> (x</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: xx-small;">1</span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">y</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-size: xx-small;">1</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">) pada kurva y = f(x)<br />
</span></b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span><br />
<table border="0"><tbody>
<tr valign="top"> <td width="22%"> <div align="center"><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>m = f`(x1)</b></span></div></td> <td align="center" width="78%"><img height="105" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Image/3-3b-1.jpg" width="150" /></td> </tr>
</tbody></table><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> f`(x1) berarti nilai turunan f(x) pada titik dengan absis x = x1, <br />
<br />
<b> Ket : <br />
</b>Khusus untuk jenis fungsi kuadrat. </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> Jika titik tidak terletak pada grafik, </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> maka gradien garis singgungnya dimisalkan dengan m yang dicari dengan menggunakan persamaan garis y - y1 = m (x - x1) disinggungkan dengan persamaan kurva </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> y = f(x) </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> dengan syarat D = 0 (D = diskriminan dari hasil eliminasi kedua persamaan)<br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="color: black;"><b>2. MENENTUKAN MONOTON FUNGSI<br />
</b></span></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> • Fungsi y = f(x) monoton </span> <b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> naik</span></b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> pada suatu interval, <br />
jika pada interval itu berlaku <b>f'(x) > 0 </b><br />
<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> • Fungsi y = f(x) monoton </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <b>turun</b> </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> pada suatu interval, <br />
jika pada interval itu berlaku</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> f'(x) < 0<br />
<br />
</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="color: black;"><b>3. MENENTUKAN TITIK STASIONER<br />
</b></span></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> Fungsi <b>y = f(x)</b> </span> <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <span style="font-family: Symbol;">®</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> Syarat stasioner</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <b>f'(x) = 0</b><br />
<b><br />
<span style="color: black;">JENIS - JENISNYA</span></b><br />
<br />
<span style="color: black;"><b>STASIONER</b></span><b> :<br />
</b><b><br />
<span style="color: black;">MAKSIMUM</span></b><span style="color: black;"><br />
</span></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> Syarat : f`(x) = 0 </span> <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <span style="font-family: Symbol;">®</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> x = x0</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">; f'' (x0) < 0 <span style="font-family: Symbol;">®</span> <span style="color: black;">Titik maksimum </span></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> (xo, f(xo)) </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<span style="color: black;"> <br />
<b>MINIMUM</b><br />
</span></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> Syarat : f '(x) = 0 </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol;">®</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> x = x0</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">; f'' (x0) > 0 <span style="font-family: Symbol;">® </span><span style="color: black;"> Titik Minimum </span></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> (xo, f(xo)) </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <span style="color: black;"> </span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<br />
<b>BELOK</b><br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> Syarat : f '(x) = 0 </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol;">®</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> x = x0</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">; </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">f'' (x0) = 0 </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol;">® </span><span style="color: black;"> </span></span> Titik belok<span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> (xo, f(xo)) <br />
<br />
<b>Nilai Stasioner </b></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> adalah nilai fungsi di absis titik stasioner</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<b><br />
<span style="color: black;">Keterangan :</span></b><span style="color: black;"><br />
</span></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> 1. Untuk </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> menentukan jenis jenis titik stasioner </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> dapat juga dicari dengan </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> melihat perubahan tanda </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> disekitar titik stasioner.<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> Langkah : <br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> a. Tentukan </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> absis </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> titik stasioner dengan syarat f '(x) = 0 </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol;">® </span><span style="color: black;"> </span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> x = xo<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> b. Buat garis bilangan f '(x)<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> c. Tentukan tanda-tanda disekitar titik stasioner dengan mensubstitusi sembarang titik pada f '(x)<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> d. Jenis titik stasioner ditentukan oleh perubahan tanda di sekitar <br />
titik stasioner.</span><br />
<div align="center"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <img height="74" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Image/3-3b-2.jpg" width="450" /></span></div><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">ket : f`(x) > 0 grafik naik<br />
f`(x) > 0 grafik turun <br />
<br />
<span style="color: black;">2. Nilai maksimum/minimum suatu fungsi dalam </span></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <b>interval tertutup </b></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> didapat dari </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <b>nilai stasioner fungsi </b></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> dalam interval </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> itu </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> atau dari </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <b>nilai fungsi pada ujung - ujung interval </b></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<br />
</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b><span style="color: black;">4. MASALAH FISIKA<br />
</span></b><span style="color: black;"><br />
</span></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> Jika S(t) = Jarak (fungsi waktu)<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> V(t) = Kecepatan (fungsi waktu)<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> a(t) = Percepatan (fungsi waktu) <br />
t = waktu</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<br />
maka <b>V = dS/dt</b> dan <b>a = dV/dt</b><br />
<br />
<span style="color: black;"><b>5. </b></span></span><b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">MENYELESAIKAN MASALAH LIMIT<br />
</span> </b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> DALIL L'Hospital<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> Jika fungsi-fungsi <b>f</b> dan <b>g</b> masing-masing terdifferensir pada </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> x = a dan f(a) = g(a) = 0 </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> atau </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> f(a) = g(a) = </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol; font-size: small;">¥</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> sehingga :<br />
<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> lim <u> f(x) </u> = <u> 0 </u> atau lim <u>f(x)</u> = </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol; font-size: small;"><u>¥</u></span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">, maka <br />
x</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">®</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">a g(x) 0 x</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">®</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">a g(x) </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol; font-size: small;">¥</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<br />
</span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> lim <u> f(x) </u> = lim <u>f`(x)</u> = </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol; font-size: small;"><u>¥</u></span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">, maka <br />
x</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">®</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">a g(x) x</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">®</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">a g`(x) </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Symbol; font-size: small;">¥</span></span></div></td></tr>
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</td> </tr>
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</td> <td width="52%"><br />
</td> </tr>
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<tr><td valign="top" width="496"><table border="0" cellpadding="9" cellspacing="0"><tbody>
<tr bgcolor="#cccccc"><td width="410"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><b><span style="font-size: small;">Proyeksi</span><br />
</b><span style="font-size: xx-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Matematika%203.htm">Matematika Kelas 3</a> > Geometri Ruang Dimensi Tiga</span></span><!-- #EndEditable --></span></td> <td bgcolor="#000000" width="50"> <div align="center"><b><!-- #BeginEditable "halaman" --><b><span style="color: white; font-family: Arial,Helvetica,sans-serif;">445</span></b><!-- #EndEditable --></b></div></td> </tr>
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<tr bgcolor="#ffffff"> <td colspan="2"> <div align="left"><!-- #BeginEditable "Isi" --><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>PROYEKSI TITIK PADA GARIS</b></span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">Proyeksi sebuah titik P pada sebuah garis g dapat diperoleh dengan menarik garis tegak lurus dari titik P terhadap garis g. Perpotongan garis tegak lurus dari titik P dengan dengan garis g yaitu titik P' , disebut proyeksi titik P pada garis g. </span><br />
<table border="0"><tbody>
<tr> <td align="center" width="41%"><img height="77" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Image/3-5d-1.jpg" width="122" /></td> <td width="59%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">P = titik yang diproyeksikan (proyektum)<br />
P' = titik hasil proyeksi<br />
PP' = garis yang memproyeksikan<br />
g = garis yang menerima proyeksi (garis<br />
proyeksi) dan PP' g</span></td> </tr>
</tbody></table><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>PROYEKSI TITIK PADA BIDANG<br />
</b>Proyeksi sebuah titik P pada bidang V dapat diperoleh dengan menarik garis tegak lurus dari P ke bidang V. Perpotongan garis lurus dari P dengan bidang V, yaitu titik P' disebut sebagai proyeksi titik P pada bidang V.<b> </b></span><br />
<table border="0"><tbody>
<tr> <td align="center" width="32%"><img height="88" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Image/3-5d-2.jpg" width="136" /></td> <td width="68%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">P = titik yang diproyeksikan (proyektum)<br />
P' = titik hasil proyeksi<br />
PP' = garis yang memproyeksikan (proyektor)<br />
V = bidang yang menerima proyeksi (bidang <br />
proyeksikan) dan PP' V)</span></td> </tr>
</tbody></table><span style="color: black; font-family: Times Roman; font-size: x-small;"> </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>PROYEKSI GARIS PADA BIDANG</b></span><b><span style="color: black; font-family: Times Roman; font-size: x-small;"> </span></b><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">Proyeksi sebuah garis g pada bidang V dapat diperoleh dengan membuat proyeksi titik-titik yang terletak pada garis g ke bidang V. Selanjutnya titik-titik proyeksi ini kita hubungkan, maka diperoleh proyeksi dari garis g, yaitu g'</span><span style="color: black; font-family: Times Roman; font-size: x-small;"> </span><br />
<table border="0"><tbody>
<tr> <td align="center" width="42%"><img height="137" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Image/3-5d-3.jpg" width="198" /></td> <td width="58%"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">Garis g = garis yang diproyeksikan<br />
(proyektum)<br />
<br />
Bidang v = bidang yang menerima <br />
proyeksi (bidang proyeksi)<br />
<br />
AA', BB', CC' = garis yang memproyeksi-<br />
kan (proyektor)<br />
<br />
Garis g' = proyeksi garis g pada bidang V<br />
<br />
Bidang yang dibentuk oleh garis-garis proyektor yaitu bidang a disebut bidang proyektor.</span></td> </tr>
</tbody></table><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>GARIS TEGAK LURUS PADA SEBUAH BIDANG</b></span><span style="color: black; font-family: Times Roman; font-size: x-small;"> </span><br />
<br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">• Sebuah garis tegak lurus bidang, jika garis tersebut tegak lurus dua <br />
garis yang berpotongan pada bidang tersebut.<br />
• Garis g tegak lurus bidang V, berarti garis g tegak lurus pada setiap <br />
garis yang terletak pada bidang V.</span> <span style="color: black; font-family: Times Roman; font-size: small;"> </span> <span style="color: black; font-family: Arial Narrow; font-size: small;"> </span><br />
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<tr bgcolor="#cccccc"><td width="410"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><b><span style="font-size: small;">Integral Tak Tentu</span><br />
</b><span style="font-size: xx-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Matematika%203.htm">Matematika Kelas 3</a> > Integral</span></span><!-- #EndEditable --></span></td> <td bgcolor="#000000" width="50"> <div align="center"><b><!-- #BeginEditable "halaman" --><b><span style="color: white; font-family: Arial,Helvetica,sans-serif;">437</span></b><!-- #EndEditable --></b></div></td> </tr>
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<tr bgcolor="#ffffff"> <td colspan="2"> <div align="left"><!-- #BeginEditable "Isi" --> <span style="color: black; font-family: Times Roman; font-size: small;"> </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>1. RUMUS<br />
</b><br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <b>FUNGSI ALJABAR <br />
</b></span><b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ò</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> x<sup>n</sup> dx = <sup>1</sup>/<sub>n+1</sub> x<sup>n+1</sup> + c ; n </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">¹</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> -1</span></b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<b><br />
FUNGSI TRIGONOMETRI<br />
</b> <b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ò</span></span> sin x dx = - cos x + c<br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ò </span></span>cos x dx = sin x + c <br />
</b><br />
sifat-sifat:<br />
a. <b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ò </span></span>c f(x) dx = c <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ò </span></span>f(x) dx<br />
</b> b. <b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ò </span></span>( f(x) ± g(x) )</b> <b>dx = <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ò </span></span>f(x) dx</b> ± <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span><b>g(x) dx</b><br />
c. jika <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span><b>f(x) dx = F(x) + c <br />
</b>maka <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>f(ax) dx</b></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b> </b></span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"> <span style="font-family: Verdana,Arial,Helvetica,sans-serif;"> <b>=</b></span> </span></span></span><b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">1/a F(ax) + c</span></b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>f(ax+b) dx</b></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b> </b></span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"> <span style="font-family: Verdana,Arial,Helvetica,sans-serif;"> <b>=</b></span> </span></span></span><b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">1/a F(ax+b) + c</span></b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<br />
<b> Perluasan :<br />
</b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>(ax + b)<sup>n</sup> dx = 1/a 1/(n+1) (ax + b)<sup>n+1</sup> + c</b></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<b> </b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>sin (ax + b) dx = -1/a cos (ax + b) + c</b></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<b> </b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>cos (ax + b) dx = 1/a sin (ax + b) + c</b></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span><br />
<b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="color: black;">CARA MENGINTEGRIR</span></span></b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="color: black;"><b>a. SUBSTITUSI</b></span><b><br />
</b><br />
<b>I = </b></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">f(x) dx<br />
substitusi : <b>x = Q(u) ; dx = Q`(u) du</b> <br />
<b>I = </b></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>f(Q(u)) Q`(u) du</b><br />
jika ruas kanan telah diintegrir, subtitusi kembali dengan fungsi invers dari x = Q(u)<br />
(ket : Prinsipnya adalah <b><i>merubah variabel </i></b>sehingga rumus dapat digunakan)<br />
<br />
<span style="color: black;"><b>b. SUBSTITUSI TRIGONOMETRI</b></span><b><br />
</b> <u> </u><br />
<span style="color: black;">1. Bentuk </span><span style="font-family: symbol;">Ö</span><span style="color: black;"> a<sup>2</sup> - x<sup>2</sup><br />
misalkan <b>x = a sin </b></span><b><span style="font-family: symbol;">q ® q</span><span style="color: black;"> = arc sin x/a<br />
dx = a cos </span><span style="font-family: symbol;">q</span><span style="color: black;"> d</span><span style="font-family: symbol;">q</span></b></span><br />
<div align="center"><img height="96" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Image/3-4b.jpg" width="96" /></div><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="color: black;"> <u> </u> <u> </u><br />
</span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span></span><span style="font-family: symbol;">Ö</span> <span style="color: black;">a2 - x2 dx = a </span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span></span><span style="font-family: symbol;">Ö</span> 1 - sin<sup>2</sup><span style="font-family: symbol;">q</span> (a cos <span style="font-family: symbol;">q</span> d<span style="font-family: symbol;">q</span>)<br />
<img height="8" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/image/putih%203x3.gif" width="100" />= a<sup>2</sup> </span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span></span>cos<sup>2</sup><span style="font-family: symbol; font-size: x-small;">q</span> d<span style="font-family: symbol; font-size: x-small;">q</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <img height="8" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/image/putih%203x3.gif" width="100" />= ½a<sup>2</sup> </span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">(1 + cos<sup>2</sup><span style="font-family: symbol;">q</span>) d<span style="font-family: symbol;">q</span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <img height="8" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/image/putih%203x3.gif" width="100" />= ½a<sup>2</sup> (<span style="font-family: symbol;">q</span> + sin<span style="font-family: symbol;">q</span> cos<span style="font-family: symbol;">q</span>) + c</span> <span style="color: black;"><br />
<u> </u> <br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <img height="8" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/image/putih%203x3.gif" width="100" />= ½a<sup>2</sup> </span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><b>ò </b></span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">[arc sin <u>x</u> + <u>x</u> </span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><u>Ö</u></span></span><u><span style="color: black;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">a<sup>2</sup> - x<sup>2</sup></span></span></u><span style="color: black;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> ] + c<br />
a </span> a a<br />
<u> </u> <u> </u><br />
</span> <b><span style="font-family: symbol;">ò</span></b> <span style="font-family: symbol;">Ö</span><b><span style="color: black;"> a<sup>2</sup> - x<sup>2</sup> dx = ½ a<sup>2</sup> arc sin x/a + ½ x </span></b><span style="font-family: symbol;">Ö</span><b><span style="color: black;"> a<sup>2</sup> - x<sup>2</sup> + c</span></b></span><br />
<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="color: black;"> <u> </u><br />
2. Bentuk </span><span style="font-family: symbol;">ò Ö</span><span style="color: black;">a<sup>2</sup> + b<sup>2</sup>x<sup>2</sup><br />
Gunakan substitusi : x = a/b tg</span><span style="font-family: symbol;">q</span><span style="color: black;"><br />
dx = a/b sec<sup>2</sup></span><span style="font-family: symbol;">q</span><span style="color: black;"> d</span><span style="font-family: symbol;">q</span><span style="color: black;"><br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="color: black;"> <u> </u><br />
</span></span><span style="color: black;"> 3. </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="color: black;">Bentuk </span><span style="font-family: symbol;">ò Ö</span><span style="color: black;">b<sup>2x<sup>2</sup></sup> - a<sup>2</sup><br />
Gunakan substitusi : x = a/b sec</span><span style="font-family: symbol;">q</span><span style="color: black;"><br />
dx = a/b tg</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">q</span></span></span><span style="color: black;"> sec<sup>2</sup></span><span style="font-family: symbol;">q</span><span style="color: black;"> </span><span style="color: black;"><br />
</span></span><span style="color: black;"> <br />
</span></span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <span style="color: black;"><b>c. PARSIIL</b></span><b><br />
</b><br />
<span style="color: black;">Yaitu mengenai integral dari suatu bentuk yang merupakan hasil </span></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> perkalian antara suatu fungsi x </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> dengan turunan </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> dari suatu fungsi x </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> yang </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> lain.</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<b><br />
I = </b></span><b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ò </span></span></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">f(x) g(x) dx</span></b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
Misalkan : <b>u = f(x) ; dv = g(x) dx</b> <br />
<b> </b></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>du = ..... dx ; v = <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ò </span></span></span>g(x) dx</b> <b>= .....</b> maka : <br />
<br />
<b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ò </span></span></span>u du</b> <b>= u v - <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ò </span></span></span>v du</b><br />
<br />
<span style="color: black;">Pemisalan dibuat sedemikian sehingga bentuk<span style="font-family: Symbol;"> </span></span></span><span style="font-family: Symbol; font-size: x-small;"> <b>ò </b></span><b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">v du</span></b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> jadi lebih mudah</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
<span style="color: black;">Untuk </span></span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> hal-hal khusus </span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> dapat digunakan cara <b>TABULA</b></span><b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">SI</span></b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <br />
<br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span><br />
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<tr bgcolor="#cccccc"> <td width="410"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><!-- #BeginEditable "Judul" --><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><b><span style="font-size: small;">Integral Tertentu</span><br />
</b><span style="font-size: xx-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/Matematika%203.htm">Matematika Kelas 3</a> > Integral</span></span><!-- #EndEditable --></span></td> <td bgcolor="#000000" width="50"> <div align="center"><b><!-- #BeginEditable "halaman" --><b><span style="color: white; font-family: Arial,Helvetica,sans-serif;">438</span></b><!-- #EndEditable --></b></div></td> </tr>
<tr bgcolor="#ffffff"> <td colspan="2"> <div align="center"><span style="font-family: Arial,Helvetica,sans-serif; font-size: xx-small;"><!-- #BeginEditable "Sb%20atas" --><span style="font-family: Arial,Helvetica,sans-serif; font-size: xx-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/0437%20Mat%203-4b.htm">< Sebelum</a></span><!-- #EndEditable --> <img height="10" src="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/image/putih%203x3.gif" width="370" /><!-- #BeginEditable "Ss%20Atas" --><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Matematika/0439%20Mat%203-4c1.htm"><span style="font-family: Arial,Helvetica,sans-serif; font-size: xx-small;">Sesudah ></span></a><!-- #EndEditable --></span></div></td> </tr>
<tr bgcolor="#ffffff"> <td colspan="2"> <div align="left"><!-- #BeginEditable "Isi" --> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>1. Pengertian<br />
</b><br />
Bila suatu fungsi F(x) mempunyai turunan f(x), maka bila f(x) diintegrasikan pada selang (a, b) menjadi <br />
<span style="font-size: xx-small;">a</span> <span style="font-size: xx-small;">a</span><br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <b><span style="font-family: symbol; font-size: small;">ò</span></b></span><b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> c dx = c(x) </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ï<span style="font-family: Verdana,Arial,Helvetica,sans-serif;">= F(b) - F(a)</span></span></span></b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <br />
<span style="font-size: xx-small;">b</span> <span style="font-size: xx-small;">b</span><br />
</span><br />
<span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b>2. Sifat<br />
</b></span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><br />
</span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-size: xx-small;"> b</span> <span style="font-size: xx-small;">b</span><br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> a. <b><span style="font-family: symbol; font-size: small;">ò</span></b></span><b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> c dx = c(x) </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;">ï <span style="font-family: Verdana,Arial,Helvetica,sans-serif;">= c(b - c) </span></span></span></b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><i><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> c = konstanta</span></i></span></span><b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"> </span></span></span></b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <br />
<span style="font-size: xx-small;"> a</span> <span style="font-size: xx-small;">a</span><br />
<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-size: xx-small;"> b</span> <span style="font-size: xx-small;">a</span><br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> b. <b><span style="font-family: symbol; font-size: small;">ò</span></b></span><b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> f(x) dx = - </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b><span style="font-family: symbol; font-size: small;">ò</span></b></span> </b> <b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">f</span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">(x) </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;">dx </span></span></span></b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><i><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> c = <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">batas ditukar</span></span></i></span></span><b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"> </span></span></span></b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <br />
<span style="font-size: xx-small;"> a</span> <span style="font-size: xx-small;">b</span><br />
<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-size: xx-small;"> a</span><br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> c. <b><span style="font-family: symbol; font-size: small;">ò</span></b></span><b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> f(x) dx = </span></b><b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"> 0 </span></span></span></b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><i><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> c = <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">batas sama </span></span></i></span></span><b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"> </span></span></span></b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> <br />
<span style="font-size: xx-small;"> a</span><br />
<br />
</span> <span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-size: xx-small;"> b</span> <span style="font-size: xx-small;">a</span> <span style="font-size: xx-small;">b</span><br />
</span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> d. <b><span style="font-family: symbol; font-size: small;">ò</span></b></span><b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> f(x) dx = </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><b><span style="font-family: symbol; font-size: small;">ò</span></b></span> </b> <b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">f</span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">(x) </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;">dx + </span></span><b><span style="font-family: symbol; font-size: small;">ò</span></b></span> </b> <b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">f</span><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">(x) </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;">dx</span></span></span></b><b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"> </span></span></span></b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><i><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">c = </span><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;">( a < c < b)</span></i></span></span><b><span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-family: symbol;"><span style="font-family: Verdana,Arial,Helvetica,sans-serif;"> </span></span></span></b><span style="color: black; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: x-small;"> </span></div></td></tr>
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<tr bgcolor="#cccccc"> <td> <div align="center"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;">Bio</span></div></td> <td width="25"> <div align="center"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Biologi/Biologi%201.htm">1</a></span></div></td> <td width="25"> <div align="center"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Biologi/Biologi%202.htm">2</a></span></div></td> <td width="25"> <div align="center"><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><a href="http://kambing.ui.ac.id/bebas/v12/sponsor/Sponsor-Pendamping/Praweda/Biologi/Biologi%203.htm">3</a></span></div></td> </tr>
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</tbody></table>Uswati Khoiriahhttp://www.blogger.com/profile/00146480953413302763noreply@blogger.com0