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Diketahui kurva f(x)=cos2x−2cosx−1 melalui titik berabsis 2π​. Persamaan garis singgung kurva pada titik tersebut adalah...

Pertanyaan

Diketahui kurva f left parenthesis x right parenthesis equals cos squared x minus 2 cos x minus 1 melalui titik berabsis straight pi over 2. Persamaan garis singgung kurva pada titik tersebut adalah...

  1. y equals 2 x plus straight pi minus 1

  2. y equals 2 x plus straight pi minus 2

  3. y equals 2 x minus straight pi minus 1

  4. y equals negative 2 x plus straight pi minus 1

  5. y equals negative 2 x plus straight pi minus 2

Pembahasan Soal:

Tentukan titik singgung, absis straight pi over 2. Ingatlah!

cos open parentheses pi over 2 close parentheses equals 0

Maka,

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell cos squared x minus 2 cos x minus 1 end cell row y equals cell cos squared x minus 2 cos x minus 1 end cell row blank equals cell cos squared open parentheses straight pi over 2 close parentheses minus 2 cos open parentheses straight pi over 2 close parentheses minus 1 end cell row blank equals cell 0 squared minus 2 times thin space 0 minus 1 end cell row blank equals cell negative 1 end cell end table

Di dapatkan titik open parentheses x subscript 1 comma y subscript 1 close parentheses equals open parentheses straight pi over 2 comma negative 1 close parentheses

Tentukan turunan pertama dari f left parenthesis x right parenthesis equals cos squared x minus 2 cos x minus 1. Ingatlah!

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f left parenthesis u right parenthesis over denominator d x end fraction end cell equals cell fraction numerator d f over denominator d u end fraction times fraction numerator d u over denominator d x end fraction end cell row f equals cell u squared end cell row u equals cell cos left parenthesis negative 2 cos left parenthesis x right parenthesis minus 1 right parenthesis end cell end table

Maka,

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator d over denominator d x end fraction open parentheses cos squared open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses close parentheses end cell row blank equals cell fraction numerator d over denominator d u end fraction open parentheses u squared close parentheses fraction numerator d over denominator d x end fraction open parentheses cos open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses close parentheses end cell row blank equals cell 2 u to the power of 2 minus 1 end exponent fraction numerator d over denominator d x end fraction open parentheses cos open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses close parentheses end cell row blank equals cell 2 u fraction numerator d over denominator d x end fraction open parentheses cos open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses close parentheses end cell row blank equals cell 2 cos open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses fraction numerator d over denominator d x end fraction open parentheses cos open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses close parentheses end cell end table

Kemudian cari turunan fraction numerator d over denominator d x end fraction open parentheses cos open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f left parenthesis u right parenthesis over denominator d x end fraction end cell equals cell fraction numerator d f over denominator d u end fraction times fraction numerator d u over denominator d x end fraction end cell row f equals cell cos space u end cell row u equals cell negative 2 cos left parenthesis x right parenthesis minus 1 end cell end table

Maka,

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator d over denominator d x end fraction open parentheses cos open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses close parentheses end cell row blank equals cell negative sin open parentheses u close parentheses fraction numerator d over denominator d x end fraction open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses end cell row blank equals cell negative sin open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses times fraction numerator d over denominator d x end fraction open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses end cell row blank equals cell negative sin open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses times negative fraction numerator d over denominator d x end fraction open parentheses 2 cos open parentheses x close parentheses close parentheses minus fraction numerator d over denominator d x end fraction open parentheses 1 close parentheses end cell row blank equals cell negative sin open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses times negative 2 fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses end cell row blank equals cell negative sin open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses times negative 2 open parentheses negative sin open parentheses x close parentheses close parentheses end cell row blank equals cell negative sin open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses times 2 sin open parentheses x close parentheses end cell end table

Di dapatkan,

fraction numerator d over denominator d x end fraction open parentheses cos squared open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses close parentheses equals 2 cos open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses open parentheses negative sin open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses times thin space 2 sin open parentheses x close parentheses close parentheses equals negative 4 cos open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses sin open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses sin open parentheses x close parentheses

Tentukan gradien garis singgung melalui titik berabsis straight pi over 2

Ingatlah!

table attributes columnalign right center left columnspacing 0px end attributes row cell sin open parentheses pi over 2 close parentheses end cell equals 1 row cell cos open parentheses pi over 2 close parentheses end cell equals 0 row cell cos open parentheses x close parentheses sin open parentheses x close parentheses end cell equals cell fraction numerator sin open parentheses 2 x close parentheses over denominator 2 end fraction end cell end table

Maka,

table attributes columnalign right center left columnspacing 0px end attributes row m equals cell fraction numerator d y over denominator d x end fraction end cell row m equals cell negative 4 cos open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses sin open parentheses negative 2 cos open parentheses x close parentheses minus 1 close parentheses sin open parentheses x close parentheses end cell row blank equals cell negative 4 cos open parentheses negative 2 cos open parentheses straight pi over 2 close parentheses minus 1 close parentheses sin open parentheses negative 2 cos open parentheses straight pi over 2 close parentheses minus 1 close parentheses sin open parentheses straight pi over 2 close parentheses end cell row blank equals cell negative 4 cos open parentheses 2 cos open parentheses pi over 2 close parentheses plus 1 close parentheses sin open parentheses negative 2 cos open parentheses pi over 2 close parentheses minus 1 close parentheses sin open parentheses pi over 2 close parentheses end cell row blank equals cell negative 4 cos open parentheses 2 cos open parentheses pi over 2 close parentheses plus 1 close parentheses open parentheses negative sin open parentheses 2 cos open parentheses pi over 2 close parentheses plus 1 close parentheses close parentheses sin open parentheses pi over 2 close parentheses end cell row blank equals cell 4 sin open parentheses pi over 2 close parentheses cos open parentheses 2 cos open parentheses pi over 2 close parentheses plus 1 close parentheses sin open parentheses 2 cos open parentheses pi over 2 close parentheses plus 1 close parentheses end cell row blank equals cell 4 times thin space 1 times cos open parentheses 2 times thin space 0 plus 1 close parentheses sin open parentheses 2 times thin space 0 plus 1 close parentheses end cell row blank equals cell 4 cos open parentheses 1 close parentheses sin open parentheses 1 close parentheses end cell row blank equals cell 4 times fraction numerator sin open parentheses 2 times thin space 1 close parentheses over denominator 2 end fraction end cell row blank equals cell 4 times fraction numerator sin open parentheses 2 close parentheses over denominator 2 end fraction end cell row blank equals cell 2 sin open parentheses 2 close parentheses end cell row blank equals cell 1 comma 8 end cell row blank almost equal to 2 end table

Di dapatkan gradien garis singgung m equals 2 dan titik open parentheses x subscript 1 comma y subscript 1 close parentheses equals open parentheses straight pi over 2 comma negative 1 close parentheses. Persamaan garis singgung, 

table attributes columnalign right center left columnspacing 0px end attributes row cell y minus y subscript 1 end cell equals cell m open parentheses x minus x subscript 1 close parentheses end cell row cell y minus left parenthesis negative 1 right parenthesis end cell equals cell 2 open parentheses x minus straight pi over 2 close parentheses end cell row cell y plus 1 end cell equals cell 2 open parentheses x minus straight pi over 2 close parentheses end cell row y equals cell 2 open parentheses x minus straight pi over 2 close parentheses minus 1 end cell row y equals cell 2 x minus fraction numerator 2 straight pi over denominator 2 end fraction minus 1 end cell row y equals cell 2 x minus straight pi minus 1 end cell end table

Jadi, persamaan garis singgung kurva pada titik tersebut adalah y equals 2 x minus straight pi minus 1.


Oleh karena itu, jawaban yang benar adalah C.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

G. Albiah

Mahasiswa/Alumni Universitas Galuh Ciamis

Terakhir diupdate 07 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Tentukan persamaan garis singgung pada kurva f(x)=sin(2x−3π​) di titik (0,21​3​)!

Pembahasan Soal:

Gradien garis singgung kurva dapat diketahui dengan m equals f apostrophe left parenthesis x right parenthesis. Sedangkan apabila garis singgung memiliki titik open parentheses a comma b close parentheses maka persamaan garis singgungnya adalah :

y minus b equals m left parenthesis x minus a right parenthesis

Diketahui f left parenthesis x right parenthesis equals sin space open parentheses 2 x minus straight pi over 3 close parentheses, maka m equals f apostrophe left parenthesis x right parenthesis :

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell sin space open parentheses 2 x minus straight pi over 3 close parentheses end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell cos space open parentheses 2 x minus straight pi over 3 close parentheses end cell end table

Titik open parentheses 0 comma space 1 half square root of 3 close parentheses, maka :

table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe open parentheses 0 close parentheses end cell equals cell cos space open parentheses 2 open parentheses 0 close parentheses minus straight pi over 3 close parentheses end cell row blank equals cell cos space open parentheses negative straight pi over 3 close parentheses end cell row blank equals cell cos space minus 60 degree end cell row blank equals cell negative 1 half end cell end table

Persamaan garis singgungnya :

table attributes columnalign right center left columnspacing 0px end attributes row cell y minus b end cell equals cell m left parenthesis x minus a right parenthesis end cell row cell y minus open parentheses negative 1 half square root of 3 close parentheses end cell equals cell negative 1 half open parentheses straight x minus 0 close parentheses end cell row cell y plus 1 half square root of 3 end cell equals cell negative 1 half x end cell row y equals cell negative 1 half x minus 1 half square root of 3 end cell end table

Maka, persamaan garis singgungnya adalah table attributes columnalign right center left columnspacing 0px end attributes row y equals cell negative 1 half x minus 1 half square root of 3 end cell end table.

2

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Selesaikanlah permasalahan berikut dengan tepat! Diketahui sebuah fungsi trigonometri yaitu f(x)=2xcos3x dengan −π<x<π. Tentukan beberapa hal berikut. 3. Persamaan garis singgung kurva di titik...

Pembahasan Soal:

Turunan pertama,

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses 2 x cos open parentheses 3 x close parentheses close parentheses end cell equals cell 2 fraction numerator d over denominator d x end fraction open parentheses x cos open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 2 open parentheses fraction numerator d over denominator d x end fraction open parentheses x close parentheses cos open parentheses 3 x close parentheses plus fraction numerator d over denominator d x end fraction open parentheses cos open parentheses 3 x close parentheses close parentheses x close parentheses end cell row blank equals cell 2 open square brackets open parentheses 1 cos open parentheses 3 x close parentheses close parentheses plus open parentheses negative sin open parentheses 3 x close parentheses fraction numerator d over denominator d x end fraction open parentheses 3 x close parentheses close parentheses x close square brackets end cell row blank equals cell 2 open square brackets cos open parentheses 3 x close parentheses close square brackets plus open parentheses negative sin open parentheses 3 x close parentheses times thin space 3 close parentheses x end cell row blank equals cell 2 open parentheses cos open parentheses 3 x close parentheses minus 3 x sin open parentheses 3 x close parentheses close parentheses end cell end table end style

Kemudian menentukan gradien garis singgung.

table attributes columnalign right center left columnspacing 0px end attributes row m equals cell fraction numerator d y over denominator d x end fraction end cell row m equals cell 2 open parentheses cos open parentheses 3 x close parentheses minus 3 x sin open parentheses 3 x close parentheses close parentheses end cell row m equals cell 2 open parentheses cos open parentheses 3 times 1 half straight pi close parentheses minus 3 times 1 half πsin open parentheses 3 times 1 half straight pi close parentheses close parentheses end cell row m equals cell 2 open parentheses cos open parentheses 3 over 2 straight pi close parentheses minus 3 over 2 πsin open parentheses 3 over 2 straight pi close parentheses close parentheses end cell row m equals cell 2 open parentheses cos open parentheses 270 degree close parentheses minus 3 over 2 πsin open parentheses 270 degree close parentheses close parentheses end cell row m equals cell 2 open parentheses 0 minus 3 over 2 pi open parentheses negative 1 close parentheses close parentheses end cell row m equals cell 2 open parentheses 0 plus 3 over 2 pi close parentheses end cell row m equals cell 2 open parentheses 3 over 2 pi close parentheses end cell row m equals cell 3 straight pi end cell end table

Tentukan garis singgung,

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 2 x cos 3 x end cell row y equals cell 2 open parentheses 1 half straight pi close parentheses cos 3 open parentheses 1 half straight pi close parentheses end cell row y equals cell πcos 3 over 2 straight pi end cell row straight y equals cell πcos 270 degree end cell row straight y equals cell straight pi times 0 end cell row straight y equals 0 end table

Di dapatkan titik open parentheses x subscript 1 comma y subscript 1 close parentheses equals open parentheses 1 half straight pi comma 0 close parentheses.

Persamaan garis singgung,

table attributes columnalign right center left columnspacing 0px end attributes row cell y minus y subscript 1 end cell equals cell m open parentheses x minus x subscript 1 close parentheses end cell row cell y minus 0 end cell equals cell 3 straight pi open parentheses straight x minus 1 half straight pi close parentheses end cell row straight y equals cell 3 straight pi open parentheses straight x minus 1 half straight pi close parentheses end cell end table

 

Jadi, persamaan garis singgungnya adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight pi end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses straight x minus 1 half straight pi close parentheses end cell end table.

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Roboguru

Tentukan gradien garis singgung pada kurva f(x)=4cos(x−6π​) di titik (3π​,23​)!

Pembahasan Soal:

Gradien garis singgung kurva dapat diketahui dengan m equals f apostrophe left parenthesis x right parenthesis. Sedangkan apabila garis singgung memiliki titik open parentheses a comma b close parentheses maka persamaan garis singgungnya adalah :

y minus b equals m left parenthesis x minus a right parenthesis

Diketahui f left parenthesis x right parenthesis equals 4 space cos space open parentheses x minus straight pi over 6 close parentheses, maka m equals f apostrophe left parenthesis x right parenthesis :

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell 4 space cos space open parentheses x minus straight pi over 6 close parentheses end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell negative 4 space sin space open parentheses x minus straight pi over 6 close parentheses end cell end table

Titik open parentheses straight pi over 3 comma space 2 square root of 3 close parentheses, maka :

table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe open parentheses straight pi over 3 close parentheses end cell equals cell negative 4 space sin space open parentheses straight pi over 3 minus straight pi over 6 close parentheses end cell row blank equals cell negative 4 space sin space open parentheses straight pi over 6 close parentheses end cell row blank equals cell negative 4 space sin space 30 degree end cell row blank equals cell negative 4 cross times 1 half end cell row blank equals cell negative 2 end cell end table

Maka, gradien garis singgung f left parenthesis x right parenthesis equals 4 space cos space open parentheses x minus straight pi over 6 close parentheses adalah negative 2.

1

Roboguru

Persamaan garis singgung dari fungsi f(x)=cos4x pada x=4π​ adalah...

Pembahasan Soal:

Tentukan titik singgung, karena x equals straight pi over 4 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell cos to the power of 4 x end cell row y equals cell cos to the power of 4 x end cell row blank equals cell cos to the power of 4 open parentheses straight pi over 4 close parentheses end cell row blank equals cell cos to the power of 4 open parentheses 180 over 4 close parentheses end cell row blank equals cell cos to the power of 4 open parentheses 45 close parentheses end cell row blank equals cell open parentheses 1 half square root of 2 close parentheses to the power of 4 end cell row blank equals cell 1 over 16 times 2 squared end cell row blank equals cell 4 over 16 end cell row blank equals cell 1 fourth end cell end table

Di dapatkan titik singgungnya open parentheses x subscript 1 comma y subscript 1 close parentheses equals open parentheses straight pi over 4 comma 1 fourth close parentheses. Kemudian tentukan turunan pertama dari f left parenthesis x right parenthesis equals cos to the power of 4 x

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses cos to the power of 4 open parentheses x close parentheses close parentheses end cell equals cell 4 open parentheses cos open parentheses x close parentheses close parentheses cubed fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses end cell row blank equals cell 4 open parentheses cos open parentheses x close parentheses close parentheses cubed open parentheses negative sin open parentheses x close parentheses close parentheses end cell row blank equals cell negative 4 cos cubed open parentheses x close parentheses sin open parentheses x close parentheses end cell end table

Tentukan gradiennya,

table attributes columnalign right center left columnspacing 0px end attributes row m equals cell fraction numerator d y over denominator d x end fraction end cell row blank equals cell negative 4 cos cubed open parentheses x close parentheses sin open parentheses x close parentheses end cell row blank equals cell negative 4 cos cubed open parentheses straight pi over 4 close parentheses sin open parentheses straight pi over 4 close parentheses end cell row blank equals cell negative 4 cos cubed open parentheses 45 degree close parentheses sin open parentheses 45 degree close parentheses end cell row blank equals cell negative 4 open parentheses 1 half square root of 2 close parentheses cubed open parentheses 1 half square root of 2 close parentheses end cell row blank equals cell negative 4 open parentheses 3 over 2 square root of 2 close parentheses open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses end cell row blank equals cell negative 4 open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses to the power of 4 end cell row blank equals cell negative 4 times 1 over 2 squared end cell row blank equals cell fraction numerator negative 4 over denominator 4 end fraction end cell row blank equals cell negative 1 end cell end table

Persamaan garis singgung,

table attributes columnalign right center left columnspacing 0px end attributes row cell y minus y subscript 1 end cell equals cell m left parenthesis x minus x subscript 1 right parenthesis end cell row cell y minus 1 fourth end cell equals cell negative 1 open parentheses x minus straight pi over 4 close parentheses end cell row y equals cell negative straight x plus straight pi over 4 plus 1 fourth end cell row cell 4 straight y end cell equals cell negative 4 straight x plus straight pi plus 1 end cell row cell 4 straight y plus 4 straight x minus straight pi minus 1 end cell equals 0 end table

Jadi, persamaan garis singgungnya adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight pi end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 0 end table.

 

Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

Persamaan garis singgung kurva y=−2tanx pada titik (4π​,−2) adalah ....

Pembahasan Soal:

Terlebih dahulu, tentukan gradien garis singgungnya, yaitu dengan cara mencari turunan pertama dari kurva tersebut.

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell negative 2 tan x end cell row cell y apostrophe end cell equals cell negative 2 sec squared x end cell end table

Kemudian, subtitusikan nilai x equals pi over 4 ke y apostrophe sehingga didapat gradien garis singgung sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row m equals cell negative 2 sec squared open parentheses pi over 4 close parentheses end cell row blank equals cell negative 2 left parenthesis 2 right parenthesis end cell row blank equals cell negative 4 end cell end table

Dengan demikian, persamaan garis singgung di titik open parentheses pi over 4 comma negative 2 close parentheses adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell y minus y subscript 1 end cell equals cell m open parentheses x minus x subscript 1 close parentheses end cell row cell y minus left parenthesis negative 2 right parenthesis end cell equals cell negative 4 open parentheses x minus pi over 4 close parentheses end cell row cell y plus 2 end cell equals cell negative 4 x plus pi end cell row y equals cell negative 4 x plus pi minus 2 end cell end table

Jadi, jawaban yang tepat adalah B.

0

Roboguru

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