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Pertanyaan

Jika garis kurva y equals 2 x space cos cubed x spacedi titik open parentheses straight pi comma negative 2 straight pi close parentheses tegak lurus dengan garis g, maka persamaan garis g adalah ....

  1. y equals x minus 3 over 2 straight pi

  2. y equals 1 half x minus 3 over 2 straight pi

  3. y equals x minus 5 over 2 straight pi

  4. y equals 1 half x minus 5 over 2 straight pi

  5. y equals x minus 3 over 2 straight pi

G. Albiah

Master Teacher

Mahasiswa/Alumni Universitas Galuh Ciamis

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Karena garis saling tegak lurus maka m equals negative 1. Cari dulu turunan dari y equals 2 x space cos cubed x space

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 cubed open parentheses x close parentheses close parentheses end cell equals cell 2 fraction numerator d over denominator d x end fraction open parentheses x cos cubed open parentheses x close parentheses close parentheses end cell row cell open parentheses f times g close parentheses apostrophe end cell equals cell f thin space apostrophe times g plus f times g apostrophe end cell row f equals x row g equals cell cos cubed open parentheses x close parentheses end cell row cell 2 fraction numerator d over denominator d x end fraction open parentheses x cos cubed open parentheses x close parentheses close parentheses end cell equals cell 2 open parentheses fraction numerator d over denominator d x end fraction open parentheses x close parentheses cos cubed open parentheses x close parentheses plus fraction numerator d over denominator d x end fraction open parentheses cos cubed open parentheses x close parentheses close parentheses x close parentheses end cell row blank equals cell 2 open parentheses 1 cos cubed open parentheses x close parentheses plus 3 open parentheses cos open parentheses x close parentheses close parentheses squared fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses close parentheses end cell row blank equals cell 2 open parentheses 1 cos cubed open parentheses x close parentheses plus 3 open parentheses cos open parentheses x close parentheses close parentheses squared close parentheses open parentheses negative sin open parentheses x close parentheses close parentheses end cell row blank equals cell 2 open parentheses 1 times cos cubed open parentheses x close parentheses plus open parentheses negative 3 cos squared open parentheses x close parentheses sin open parentheses x close parentheses close parentheses x close parentheses end cell row blank equals cell 2 open parentheses cos cubed open parentheses x close parentheses minus 3 x cos squared open parentheses x close parentheses sin open parentheses x close parentheses close parentheses end cell end table end style

Maka,

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator y equals 2 x cos cubed x y apostrophe equals 2 cos cubed x plus 2 x times 3 cos squared x left parenthesis negative sin x right parenthesis y apostrophe equals 2 cos cubed x minus 2 x times 3 cos squared x sin x over denominator m subscript x equals straight pi end subscript equals 2 cos cubed straight pi minus 2 straight pi times 3 cos squared πsinπ end fraction end cell row cell straight m subscript straight x equals straight pi end subscript end cell equals cell 2 cos cubed straight pi minus 2 straight pi times 3 cos squared πsinπ end cell row blank equals cell 2 left parenthesis negative 1 right parenthesis cubed minus 2 straight pi times 3 times open parentheses negative 1 close parentheses squared times 0 end cell row blank equals cell 2 open parentheses negative 1 close parentheses minus 0 end cell row blank equals cell negative 2 end cell end table

Karena garis saling tegak lurus maka gradiennya negative 1 sehingga gradien garis yang tegak lurus dengan garis bergradien m subscript g equals negative 2 adalah 

table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript l times m subscript g end cell equals cell negative 1 end cell row cell m subscript l times negative 2 end cell equals cell negative 1 end cell row cell m subscript l end cell equals cell fraction numerator negative 1 over denominator negative 2 end fraction end cell row cell m subscript l end cell equals cell 1 half end cell end table

Persamaan garis di titik open parentheses straight pi comma negative 2 straight pi close parentheses yang tegak lurus dengan garis g yaitu,

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 left parenthesis negative 2 straight pi right parenthesis end cell equals cell 1 half open parentheses straight x minus straight pi close parentheses end cell row cell straight y plus 2 straight pi end cell equals cell 1 half open parentheses straight x minus straight pi close parentheses end cell row straight y equals cell 1 half straight x minus 1 half straight pi minus 2 straight pi end cell row straight y equals cell 1 half straight x minus 1 half straight pi minus 4 over 2 straight pi end cell row straight y equals cell 1 half straight x minus 5 over 2 straight pi end cell end table

Di dapatkan y equals 1 half x minus 5 over 2 straight pi
 

Oleh karena itu, jawaban yang benar adalah D.

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