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Pertanyaan

Diketahui f left parenthesis x right parenthesis equals sin space open parentheses x over 2 plus straight pi close parentheses space cos space open parentheses x minus straight pi close parentheses dengan 0 less or equal than x less or equal than 2 straight pi. Grafik fungsi f melalui titik straight A. Jika titik A berabsis x equals fraction numerator 3 straight pi over denominator 2 end fraction, tentukan: 

a. gradien garis singgung grafik fungsi f di titik straight A

b. persamaan garis singgung grafik fungsi f di titik straight A

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

 gradien dan persamaan garis singgung grafik fungsi f di titik straight A berturut-turut adalah negative 1 half square root of 2 dan y equals negative 1 half square root of 2 x plus 3 over 4 square root of 2 straight pi

Pembahasan

Untuk menjawab bagian (a) gunakan aturan turunan perkalian yaitu: 

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell u times v end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell u apostrophe v plus u v apostrophe end cell end table 

Pada soal diberikan f left parenthesis x right parenthesis equals sin space open parentheses x over 2 plus straight pi close parentheses space cos space open parentheses x minus straight pi close parentheses maka kita misalkan terlebih dahulu menjadi: 

u equals sin open parentheses x over 2 plus straight pi close parentheses v equals cos open parentheses x minus straight pi close parentheses 

Selanjutnya kita turunkan masing-masing, menjadi: 

table attributes columnalign right center left columnspacing 0px end attributes row u equals cell sin open parentheses x over 2 plus straight pi close parentheses end cell row cell u apostrophe end cell equals cell 1 half space cos open parentheses x over 2 plus straight pi close parentheses end cell row blank blank blank row v equals cell cos left parenthesis x minus straight pi right parenthesis end cell row cell straight v apostrophe end cell equals cell negative sin left parenthesis straight x minus straight pi right parenthesis end cell end table  

Setelah kita turunkan, masukkan pada rumus aturan pembagian turunan yaitu: 

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis x right parenthesis end cell equals cell u apostrophe v plus u v apostrophe end cell row blank equals cell 1 half space cos open parentheses x over 2 plus straight pi close parentheses times cos left parenthesis x minus straight pi right parenthesis plus sin open parentheses straight x over 2 plus straight pi close parentheses times negative sin open parentheses straight x minus straight pi close parentheses end cell row blank equals cell 1 half space cos open parentheses straight x over 2 plus straight pi close parentheses space cos left parenthesis straight x minus straight pi right parenthesis minus sin open parentheses straight x over 2 plus straight pi close parentheses space sin open parentheses straight x minus straight pi close parentheses end cell end table end style 

Untuk mencari gradien, kita substitusikan absis x equals fraction numerator 3 straight pi over denominator 2 end fraction pada turunan menjadi; 

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe open parentheses fraction numerator 3 straight pi over denominator 2 end fraction close parentheses end cell equals cell 1 half space cos open parentheses fraction numerator begin display style fraction numerator 3 straight pi over denominator 2 end fraction end style over denominator 2 end fraction plus straight pi close parentheses space cos open parentheses fraction numerator 3 straight pi over denominator 2 end fraction minus straight pi close parentheses minus sin open parentheses fraction numerator begin display style fraction numerator 3 straight pi over denominator 2 end fraction end style over denominator 2 end fraction plus straight pi close parentheses space sin open parentheses fraction numerator 3 straight pi over denominator 2 end fraction minus straight pi close parentheses end cell row blank equals cell 1 half space cos open parentheses fraction numerator 3 straight pi over denominator 4 end fraction plus straight pi close parentheses space cos open parentheses straight pi over 2 close parentheses minus sin open parentheses fraction numerator 3 straight pi over denominator 4 end fraction plus straight pi close parentheses space sin open parentheses straight pi over 2 close parentheses end cell row blank equals cell 1 half space cos open parentheses fraction numerator 7 straight pi over denominator 4 end fraction close parentheses space cos open parentheses straight pi over 2 close parentheses minus sin open parentheses fraction numerator 7 straight pi over denominator 4 end fraction close parentheses space sin open parentheses straight pi over 2 close parentheses end cell row blank equals cell 1 half times 1 half square root of 2 times 0 minus 1 half square root of 2 times 1 end cell row blank equals cell 0 minus 1 half square root of 2 end cell row m equals cell negative 1 half square root of 2 end cell end table end style 

 

Untuk menjawab bagian (b) gunakan rumus berikut ini: 

y minus y subscript 1 equals m left parenthesis x minus x subscript 1 right parenthesis 

Pada bagian ini kita baru memperoleh m equals negative 1 half square root of 2 dan  x subscript 1 equals fraction numerator 3 straight pi over denominator 2 end fraction, maka kita cari terlebih dahulu y subscript 1 dengan cara mensubstitusi x subscript 1 pada fungsi f 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 x over 2 plus straight pi close parentheses space cos space open parentheses x minus straight pi close parentheses end cell row cell f open parentheses fraction numerator 3 straight pi over denominator 2 end fraction close parentheses end cell equals cell sin space open parentheses fraction numerator begin display style fraction numerator 3 straight pi over denominator 2 end fraction end style over denominator 2 end fraction plus straight pi close parentheses space cos space open parentheses fraction numerator 3 straight pi over denominator 2 end fraction minus straight pi close parentheses end cell row blank equals cell sin open parentheses fraction numerator 3 straight pi over denominator 4 end fraction plus straight pi close parentheses space cos open parentheses straight pi over 2 close parentheses end cell row blank equals cell sin open parentheses fraction numerator 7 straight pi over denominator 4 end fraction close parentheses space cos open parentheses straight pi over 2 close parentheses end cell row blank equals cell 1 half square root of 2 times 0 end cell row cell y subscript 1 end cell equals 0 end table 

Setelah itu, kita substitusikan pada rumus: 

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 0 end cell equals cell negative 1 half square root of 2 open parentheses x minus fraction numerator 3 straight pi over denominator 2 end fraction close parentheses end cell row y equals cell negative 1 half square root of 2 x plus 3 over 4 square root of 2 straight pi end cell end table 

 

Dengan demikian, gradien dan persamaan garis singgung grafik fungsi f di titik straight A berturut-turut adalah negative 1 half square root of 2 dan y equals negative 1 half square root of 2 x plus 3 over 4 square root of 2 straight pi

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