Roboguru

Tentukan persamaan garis singgung dan persamaan garis normal di titik dengan absis x=1 pada setiap fungsi berikut. Petunjuk: carilah gradien persamaan garis singgung dengan menggunakan limit fungsi. e. f(x)=x22​

Pertanyaan

Tentukan persamaan garis singgung dan persamaan garis normal di titik dengan absis begin mathsize 14px style x equals 1 end style pada setiap fungsi berikut. Petunjuk: carilah gradien persamaan garis singgung dengan menggunakan limit fungsi.

e. begin mathsize 14px style f left parenthesis x right parenthesis equals 2 over x squared end style

Pembahasan Soal:

Gradien garis singgung suatu kurva begin mathsize 14px style f left parenthesis x right parenthesis end style didefinisikan oleh

begin mathsize 14px style m subscript g s end subscript equals f apostrophe open parentheses a close parentheses equals limit as h rightwards arrow 0 of fraction numerator f open parentheses a plus h close parentheses minus f open parentheses a close parentheses over denominator h end fraction end style.

dengan begin mathsize 14px style open parentheses a comma space b close parentheses end style adalah koordinat titik singgungnya.

Maka gradien garis singgung kurva begin mathsize 14px style f left parenthesis x right parenthesis equals 2 over x squared end style di titik dengan absis begin mathsize 14px style x equals 1 end style adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript g s end subscript end cell equals cell f apostrophe open parentheses 1 close parentheses equals limit as h rightwards arrow 0 of fraction numerator f open parentheses 1 plus h close parentheses minus f open parentheses 1 close parentheses over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style 2 over open parentheses 1 plus h close parentheses squared end style minus begin display style 2 over 1 squared end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style fraction numerator 2 over denominator 1 plus 2 h plus h squared end fraction end style minus begin display style 2 over 1 end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style fraction numerator 2 minus 2 open parentheses 1 plus 2 h plus h squared close parentheses over denominator 1 plus 2 h plus h squared end fraction end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator open parentheses 2 minus 2 minus 4 h minus 2 h squared close parentheses over denominator 1 plus 2 h plus h squared end fraction times 1 over h end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator left parenthesis negative 4 h minus 2 h squared right parenthesis over denominator 1 plus 2 h plus h squared end fraction times 1 over h end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator left parenthesis negative 4 minus 2 h right parenthesis up diagonal strike h to the power of 1 over denominator 1 plus 2 h plus h squared end fraction times 1 over up diagonal strike h to the power of 1 end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator negative 4 minus 2 h over denominator 1 plus 2 h plus h squared end fraction end cell row blank equals cell fraction numerator negative 4 minus 0 over denominator 1 plus 0 plus 0 end fraction end cell row blank equals cell negative 4 end cell end table end style

Selanjutnya, kita akan menghitung ordinat titik singgungnya. Substitusikan begin mathsize 14px style x equals 1 end style ke persamaan kurva begin mathsize 14px style f left parenthesis x right parenthesis equals 2 over x squared end style sehingga diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell f open parentheses 1 close parentheses end cell row blank equals cell 2 over 1 squared end cell row blank equals 2 end table end style

Dengan demikian, persamaan garis singgungnya adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus b end cell equals cell m subscript g s end subscript open parentheses x minus a close parentheses end cell row cell y minus 2 end cell equals cell negative 4 open parentheses x minus 1 close parentheses end cell row cell y minus 2 end cell equals cell negative 4 x plus 4 end cell row cell 4 x plus y minus 6 end cell equals 0 end table end style

Selanjutnya, karena garis singgung dan garis normal kurva begin mathsize 14px style f left parenthesis x right parenthesis end style saling berpotongan tegak lurus di titik singgung begin mathsize 14px style open parentheses a comma space b close parentheses end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript g s end subscript times m subscript g n end subscript end cell equals cell negative 1 end cell row cell negative 4 times m subscript g n end subscript end cell equals cell negative 1 end cell row cell m subscript g n end subscript end cell equals cell 1 fourth end cell end table end style

dan persamaan garis normalnya adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus b end cell equals cell m subscript g n end subscript left parenthesis x minus a right parenthesis end cell row cell y minus 2 end cell equals cell 1 fourth left parenthesis x minus 1 right parenthesis end cell row cell y minus 2 end cell equals cell 1 fourth x minus 1 fourth end cell row cell 4 y minus 8 end cell equals cell x minus 1 end cell row cell 4 y minus x minus 7 end cell equals 0 end table end style

Jadi, persamaan garis singgung dan garis normal kurva begin mathsize 14px style f left parenthesis x right parenthesis equals 2 over x squared end style pada titik dengan absis begin mathsize 14px style x equals 1 end style dan ordinat begin mathsize 14px style y equals 2 end style berturut-turut adalah begin mathsize 14px style 4 x plus y minus 6 equals 0 end style dan begin mathsize 14px style 4 y minus x minus 7 equals 0 end style.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Eka

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 07 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

Membantu

Kurang Membantu

Apakah pembahasan ini membantu?

Belum menemukan yang kamu cari?

Post pertanyaanmu ke Tanya Jawab, yuk

Mau Bertanya

Pertanyaan yang serupa

Tentukan persamaan garis singgung dan persamaan garis normal di titik dengan absis x=1 pada setiap fungsi berikut. Petunjuk: carilah gradien persamaan garis singgung dengan menggunakan limit fungsi. ...

Pembahasan Soal:

Gradien garis singgung suatu kurva begin mathsize 14px style f left parenthesis x right parenthesis end style didefinisikan sebagai berikut.

begin mathsize 14px style m subscript g s end subscript equals f apostrophe open parentheses a close parentheses equals limit as h rightwards arrow 0 of fraction numerator f open parentheses a plus h close parentheses minus f open parentheses a close parentheses over denominator h end fraction end style.

 begin mathsize 14px style open parentheses a comma space b close parentheses end style adalah koordinat titik singgungnya.

Berdasarkan rumus di atas, diperoleh gradien garis singgung kurva begin mathsize 14px style f left parenthesis x right parenthesis equals open parentheses 2 x minus 1 close parentheses cubed end style di titik dengan absis begin mathsize 14px style x equals 1 end style sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript g s end subscript end cell equals cell f apostrophe open parentheses 1 close parentheses end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator f open parentheses 1 plus h close parentheses minus f open parentheses 1 close parentheses over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style open parentheses 2 open parentheses 1 plus h close parentheses minus 1 close parentheses cubed minus open parentheses 2 open parentheses 1 close parentheses minus 1 close parentheses cubed end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style open parentheses 2 plus 2 h minus 1 close parentheses cubed minus 1 end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style open parentheses 2 h plus 1 close parentheses cubed minus 1 end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style 8 h cubed plus 12 h squared plus 6 h plus 1 minus 1 end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator 8 h cubed plus 12 h squared plus 6 h over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator left parenthesis 8 h squared plus 12 h plus 6 right parenthesis up diagonal strike h to the power of 1 over denominator up diagonal strike h to the power of 1 end fraction end cell row blank equals cell limit as h rightwards arrow 0 of left parenthesis 8 h squared plus 12 h plus 6 right parenthesis end cell row blank equals cell 0 minus 0 plus 6 end cell row blank equals 6 end table

Selanjutnya, kita akan menghitung ordinat titik singgungnya. Substitusikan begin mathsize 14px style x equals 1 end style ke persamaan kurva begin mathsize 14px style f left parenthesis x right parenthesis equals open parentheses 2 x minus 1 close parentheses cubed end style sehingga diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell f left parenthesis 1 right parenthesis end cell row blank equals cell open parentheses 2 open parentheses 1 close parentheses minus 1 close parentheses cubed end cell row blank equals 1 end table end style

Dengan demikian, persamaan garis singgungnya adalah:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus b end cell equals cell m subscript g s end subscript left parenthesis x minus a right parenthesis end cell row cell y minus 1 end cell equals cell 6 left parenthesis x minus 1 right parenthesis end cell row cell y minus 1 end cell equals cell 6 x minus 6 end cell row cell y minus 6 x plus 5 end cell equals 0 end table end style

Selanjutnya, dikarenakan garis singgung dan garis normal kurva begin mathsize 14px style f left parenthesis x right parenthesis end style saling berpotongan tegak lurus di titik singgung begin mathsize 14px style open parentheses a comma space b close parentheses end style, maka diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript g s end subscript times m subscript g n end subscript end cell equals cell negative 1 end cell row cell 6 times m subscript g n end subscript end cell equals cell negative 1 end cell row cell m subscript g n end subscript end cell equals cell negative 1 over 6 end cell end table end style

Kemudian, diperoleh persamaan garis normalnya sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus b end cell equals cell m subscript g n end subscript left parenthesis x minus a right parenthesis end cell row cell y minus 1 end cell equals cell negative 1 over 6 left parenthesis x minus 1 right parenthesis end cell row cell 6 y minus 6 end cell equals cell negative open parentheses x minus 1 close parentheses end cell row cell 6 y minus 6 plus open parentheses x minus 1 close parentheses end cell equals 0 row cell 6 y plus x minus 7 end cell equals 0 end table end style

Dengan demikian, persamaan garis singgung dan garis normal kurva begin mathsize 14px style f left parenthesis x right parenthesis equals open parentheses 2 x minus 1 close parentheses cubed end style pada titik dengan absis begin mathsize 14px style x equals 1 end style dan ordinat undefined berturut-turut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 6 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 5 end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 0 end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 6 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 7 end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 0 end table.

1

Roboguru

Tentukan persamaan garis singgung dan persamaan garis normal di titik dengan absis x=1 pada setiap fungsi berikut. Petunjuk: carilah gradien persamaan garis singgung dengan menggunakan limit fungsi. ...

Pembahasan Soal:

Gradien garis singgung suatu kurva begin mathsize 14px style f left parenthesis x right parenthesis end style didefinisikan oleh

begin mathsize 14px style m subscript g s end subscript equals f apostrophe open parentheses a close parentheses equals limit as h rightwards arrow 0 of fraction numerator f open parentheses a plus h close parentheses minus f open parentheses a close parentheses over denominator h end fraction end style.

dengan begin mathsize 14px style open parentheses a comma space b close parentheses end style adalah koordinat titik singgungnya.

Maka gradien garis singgung kurva begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator 2 over denominator x plus 1 end fraction end style di titik dengan absis begin mathsize 14px style x equals 1 end style adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript g s end subscript end cell equals cell f apostrophe left parenthesis 1 right parenthesis equals limit as h rightwards arrow 0 of fraction numerator f left parenthesis 1 plus h right parenthesis minus f left parenthesis 1 right parenthesis over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style 2 over open parentheses 2 plus h close parentheses squared minus 2 over open parentheses 2 close parentheses squared end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style fraction numerator 2 over denominator 4 plus 4 h plus h squared end fraction minus 2 over 4 end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style fraction numerator 2 over denominator 4 plus 4 h plus h squared end fraction minus 1 half end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style fraction numerator 4 minus open parentheses 4 plus 4 h plus h squared close parentheses over denominator 8 plus 8 h plus 2 h squared end fraction end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator left parenthesis 4 minus 4 minus 4 h minus h squared right parenthesis over denominator 8 plus 8 h plus 2 h squared end fraction times 1 over h end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator left parenthesis negative 4 h minus h squared right parenthesis over denominator 8 plus 8 h plus 2 h squared end fraction times 1 over h end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator left parenthesis negative 4 minus h right parenthesis up diagonal strike h to the power of 1 over denominator 8 plus 8 h plus 2 h squared end fraction times 1 over up diagonal strike h to the power of 1 end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator negative 4 minus 2 h over denominator 8 plus 8 h plus 2 h squared end fraction end cell row blank equals cell fraction numerator negative 4 minus 0 over denominator 8 plus 0 plus 0 end fraction end cell row blank equals cell fraction numerator negative 4 over denominator 8 end fraction end cell row blank equals cell negative 1 half end cell end table end style

Selanjutnya, kita akan menghitung ordinat titik singgungnya. Substitusikan begin mathsize 14px style x equals 1 end style ke persamaan kurva begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator 2 over denominator x plus 1 end fraction end style sehingga diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell f left parenthesis 1 right parenthesis end cell row blank equals cell fraction numerator 2 over denominator 1 plus 1 end fraction end cell row blank equals cell 2 over 2 end cell row blank equals 1 end table end style

Dengan demikian, persamaan garis singgungnya adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus b end cell equals cell m subscript g s end subscript left parenthesis x minus a right parenthesis end cell row cell y minus 1 end cell equals cell negative 1 half left parenthesis x minus 1 right parenthesis end cell row cell 2 y minus 2 end cell equals cell negative open parentheses x minus 1 close parentheses end cell row cell 2 y minus 2 plus open parentheses x minus 1 close parentheses end cell equals 0 row cell 2 y plus x minus 3 end cell equals 0 end table end style

Selanjutnya, karena garis singgung dan garis normal kurva begin mathsize 14px style f left parenthesis x right parenthesis end style saling berpotongan tegak lurus di titik singgung begin mathsize 14px style open parentheses a comma space b close parentheses end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript g s end subscript times m subscript g n end subscript end cell equals cell negative 1 end cell row cell negative 1 half times m subscript g n end subscript end cell equals cell negative 1 end cell row cell m subscript g n end subscript end cell equals 2 end table end style

dan persamaan garis normalnya adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus b end cell equals cell m subscript g n end subscript left parenthesis x minus a right parenthesis end cell row cell y minus 1 end cell equals cell 2 left parenthesis x minus 1 right parenthesis end cell row cell y minus 1 end cell equals cell 2 x minus 2 end cell row cell y minus 2 x minus 1 plus 2 end cell equals 0 row cell y minus 2 x plus 1 end cell equals 0 end table end style

Jadi, persamaan garis singgung dan garis normal kurva begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator 2 over denominator x plus 1 end fraction end style pada titik dengan absis begin mathsize 14px style x equals 1 end style dan ordinat size 14px y size 14px equals size 14px 1berturut-turut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 0 end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 0 end table.

0

Roboguru

Tentukan persamaan garis singgung dan persamaan garis normal di titik dengan absis x=1 pada setiap fungsi berikut. Petunjuk: carilah gradien persamaan garis singgung dengan menggunakan limit fungsi. ...

Pembahasan Soal:

Gradien garis singgung suatu kurva begin mathsize 14px style f left parenthesis x right parenthesis end style didefinisikan oleh

begin mathsize 14px style m subscript g s end subscript equals f apostrophe open parentheses a close parentheses equals limit as h rightwards arrow 0 of fraction numerator f open parentheses a plus h close parentheses minus f open parentheses a close parentheses over denominator h end fraction end style.

dengan begin mathsize 14px style open parentheses a comma space b close parentheses end style adalah koordinat titik singgungnya.

Maka gradien garis singgung kurva begin mathsize 14px style f left parenthesis x right parenthesis equals 2 x squared end style di titik dengan absis begin mathsize 14px style x equals 1 end style adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript g s end subscript end cell equals cell f apostrophe left parenthesis 1 right parenthesis end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator f left parenthesis 1 plus h right parenthesis minus f left parenthesis 1 right parenthesis over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style 2 open parentheses 1 plus h close parentheses squared minus 2 open parentheses 1 close parentheses squared end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style 2 open parentheses 1 plus 2 h plus h squared close parentheses minus 2 end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style 2 plus 4 h plus 2 h squared minus 2 end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator 4 h plus 2 h squared over denominator h end fraction times end cell row blank equals cell limit as h rightwards arrow 0 of open parentheses 4 plus 2 h close parentheses end cell row blank equals cell 4 plus 0 end cell row blank equals 4 end table end style

Selanjutnya, kita akan menghitung ordinat titik singgungnya. Substitusikan begin mathsize 14px style x equals 1 end style ke persamaan kurva begin mathsize 14px style f left parenthesis x right parenthesis equals 2 x squared end style sehingga diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell f left parenthesis 1 right parenthesis end cell row blank equals cell 2 open parentheses 1 close parentheses squared end cell row blank equals 2 end table end style

Dengan demikian, persamaan garis singgungnya adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus b end cell equals cell m subscript g s end subscript left parenthesis x minus a right parenthesis end cell row cell y minus 2 end cell equals cell 4 left parenthesis x minus 1 right parenthesis end cell row cell y minus 2 end cell equals cell 4 x plus 4 end cell row cell y minus 4 x minus 6 end cell equals 0 end table end style

Selanjutnya, karena garis singgung dan garis normal kurva begin mathsize 14px style f left parenthesis x right parenthesis end style saling berpotongan tegak lurus di titik singgung begin mathsize 14px style open parentheses a comma space b close parentheses end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript g s end subscript times m subscript g n end subscript end cell equals cell negative 1 end cell row cell 4 times m subscript g n end subscript end cell equals cell negative 1 end cell row cell m subscript g n end subscript end cell equals cell negative 1 fourth end cell end table end style

dan persamaan garis normalnya adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus b end cell equals cell m subscript g n end subscript left parenthesis x minus a right parenthesis end cell row cell y minus 2 end cell equals cell negative 1 fourth left parenthesis x minus 1 right parenthesis end cell row cell y minus 2 end cell equals cell negative 1 fourth x plus 1 fourth end cell row cell 4 y minus 8 end cell equals cell negative x plus 1 end cell row cell 4 y plus x minus 9 end cell equals 0 end table end style

Jadi, persamaan garis singgung dan garis normal kurva begin mathsize 14px style f left parenthesis x right parenthesis equals 2 x squared end style pada titik dengan absis begin mathsize 14px style x equals 1 end style dan ordinat begin mathsize 14px style y equals 2 end style berturut-turut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 6 end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 0 end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 9 end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 0 end table.

1

Roboguru

Tentukan persamaan garis singgung dan persamaan garis normal di titik dengan absis x=1 pada setiap fungsi berikut. Petunjuk: carilah gradien persamaan garis singgung dengan menggunakan limit fungsi. ...

Pembahasan Soal:

Gradien garis singgung suatu kurva begin mathsize 14px style f left parenthesis x right parenthesis end style didefinisikan oleh

begin mathsize 14px style m subscript g s end subscript equals f apostrophe open parentheses a close parentheses equals limit as h rightwards arrow 0 of fraction numerator f open parentheses a plus h close parentheses minus f open parentheses a close parentheses over denominator h end fraction end style.

dengan begin mathsize 14px style open parentheses a comma space b close parentheses end style adalah koordinat titik singgungnya.

Maka gradien garis singgung kurva begin mathsize 14px style f open parentheses x close parentheses equals 2 x end style di titik dengan absis begin mathsize 14px style x equals 1 end style adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript g s end subscript end cell equals cell f apostrophe left parenthesis 1 right parenthesis end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator f left parenthesis 1 plus h right parenthesis minus f left parenthesis 1 right parenthesis over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style 2 open parentheses 1 plus h close parentheses minus 2 open parentheses 1 close parentheses end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style 2 h end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of 2 end cell row blank equals 2 end table end style

Selanjutnya, kita akan menghitung ordinat titik singgungnya. Substitusikan begin mathsize 14px style x equals 1 end style ke persamaan kurva begin mathsize 14px style f open parentheses x close parentheses equals 2 x end style sehingga diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell f left parenthesis 1 right parenthesis end cell row blank equals cell 2 open parentheses 1 close parentheses end cell row blank equals 2 end table end style

Dengan demikian, persamaan garis singgungnya adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus b end cell equals cell m subscript g s end subscript left parenthesis x minus a right parenthesis end cell row cell y minus 2 end cell equals cell 2 left parenthesis x minus 1 right parenthesis end cell row cell y minus 2 end cell equals cell 2 x minus 2 end cell row y equals cell 2 x end cell end table end style

Selanjutnya, karena garis singgung dan garis normal kurva begin mathsize 14px style f left parenthesis x right parenthesis end style saling berpotongan tegak lurus di titik singgung begin mathsize 14px style open parentheses a comma space b close parentheses end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript g s end subscript times m subscript g n end subscript end cell equals cell negative 1 end cell row cell 2 times m subscript g n end subscript end cell equals cell negative 1 end cell row cell m subscript g n end subscript end cell equals cell negative 1 half end cell end table end style

dan persamaan garis normalnya adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus b end cell equals cell m subscript g n end subscript left parenthesis x minus a right parenthesis end cell row cell y minus 2 end cell equals cell negative 1 half left parenthesis x minus 1 right parenthesis end cell row cell y minus 2 end cell equals cell negative 1 half x plus 1 half end cell row y equals cell negative 1 half x plus 5 over 2 end cell end table end style

Jadi, persamaan garis singgung dan garis normal kurva begin mathsize 14px style f open parentheses x close parentheses equals 2 x end style pada titik dengan absis begin mathsize 14px style x equals 1 end style dan ordinat begin mathsize 14px style y equals 2 end style berturut-turut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px y end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table (kurva itu sendiri) dan begin mathsize 14px style y equals negative 1 half x plus 5 over 2 end style.

0

Roboguru

Tentukan persamaan garis singgung dan persamaan garis normal di titik dengan absis x=1 pada setiap fungsi berikut. f(x)=2x2

Pembahasan Soal:

Absis begin mathsize 14px style x equals 1 end style, maka ordinat atau begin mathsize 14px style y end style dapat dicari dengan mensubstusikan nilai begin mathsize 14px style x equals 1 end style ke fungsi,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell 2 x squared end cell row cell f left parenthesis 1 right parenthesis end cell equals cell 2 left parenthesis 1 right parenthesis squared space end cell row cell f left parenthesis 1 right parenthesis end cell equals 2 end table end style

maka ordinat begin mathsize 14px style y equals 2 end style

Untuk menentukan gradien garis singgung kita dapat menggunakan turunan pertama atau begin mathsize 14px style f apostrophe left parenthesis x right parenthesis end style sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell 2 x squared end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell 4 x end cell row cell absis space x end cell equals cell 1 comma space maka colon end cell row cell m subscript s end cell equals cell 4 left parenthesis 1 right parenthesis end cell row cell m subscript s end cell equals 4 end table end style

Maka persamaan garis singgungnya:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus y subscript 1 end cell equals cell m subscript s left parenthesis x minus x subscript 1 right parenthesis end cell row cell y minus 2 end cell equals cell 4 left parenthesis x minus 1 right parenthesis end cell row cell y minus 2 end cell equals cell 4 x minus 4 end cell row cell 4 x minus y plus 2 end cell equals 0 end table end style 

Hubungan garis singgung dan garis normal adalah saling tegak lurus, maka berlaku:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript n end cell equals cell negative 1 over m subscript s end cell row blank equals cell negative 1 fourth end cell end table end style 

Maka persamaan garis normalnya adalah:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus y subscript 1 end cell equals cell m subscript n left parenthesis x minus x subscript 1 right parenthesis end cell row cell y minus 2 end cell equals cell negative 1 fourth left parenthesis x minus 1 right parenthesis end cell row cell y minus 2 end cell equals cell negative 1 fourth x plus 1 fourth end cell row cell 4 y minus 8 end cell equals cell negative x plus 1 end cell row cell x plus 4 y minus 8 minus 1 end cell equals 0 row cell x plus 4 y minus 9 end cell equals 0 end table end style 

Dengan demikian, persamaan garis singgung adalah begin mathsize 14px style 4 x minus y plus 2 equals 0 end style dan persamaan garis normalnya adalah begin mathsize 14px style x plus 4 y minus 9 equals 0 end style

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

Membantu

Kurang Membantu

Apakah pembahasan ini membantu?

Belum menemukan yang kamu cari?

Post pertanyaanmu ke Tanya Jawab, yuk

Mau Bertanya

RUANGGURU HQ

Jl. Dr. Saharjo No.161, Manggarai Selatan, Tebet, Kota Jakarta Selatan, Daerah Khusus Ibukota Jakarta 12860

Coba GRATIS Aplikasi Ruangguru

Produk Ruangguru

Produk Lainnya

Hubungi Kami

Ikuti Kami

©2021 Ruangguru. All Rights Reserved