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Diberikan fungsi trigonometri f(x)=sinx2+cosx​ dengan daerah asal Df​={x∣x∈R, sin x=0}. Carilah persamaan garis singgung kurva fungsi f di titik yang berabsis 21​π.

Pertanyaan

Diberikan fungsi trigonometri f left parenthesis x right parenthesis equals fraction numerator 2 plus cos x over denominator sin x end fraction dengan daerah asal D subscript f equals open curly brackets x left enclose x element of R comma space sin space x not equal to 0 end enclose close curly brackets. Carilah persamaan garis singgung kurva fungsi f di titik yang berabsis 1 half straight pi.

G. Albiah

Master Teacher

Mahasiswa/Alumni Universitas Galuh Ciamis

Jawaban terverifikasi

Pembahasan

Tentukan titik singgung, karena x equals 1 half straight pi maka

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator 2 plus cos x over denominator sin x end fraction end cell row y equals cell fraction numerator 2 plus cos open parentheses begin display style 1 half end style straight pi close parentheses over denominator sin open parentheses 1 half straight pi close parentheses end fraction end cell row y equals cell fraction numerator 2 plus cos 90 over denominator sin 90 end fraction end cell row blank equals cell fraction numerator 2 plus 0 over denominator 1 end fraction end cell row blank equals 2 end table

Di dapatkan titik singgung open parentheses x subscript 1 comma y subscript 1 close parentheses equals open parentheses 1 half straight pi comma 2 close parentheses.

Tentukan turunan pertama fungsi f left parenthesis x right parenthesis equals fraction numerator 2 plus cos x over denominator sin x end fraction

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 fraction numerator 2 plus cos open parentheses x close parentheses over denominator sin open parentheses x close parentheses end fraction close parentheses end cell row blank equals cell fraction numerator fraction numerator d over denominator d x end fraction open parentheses 2 plus cos open parentheses x close parentheses close parentheses sin open parentheses x close parentheses minus fraction numerator d over denominator d x end fraction open parentheses sin open parentheses x close parentheses close parentheses open parentheses 2 plus cos open parentheses x close parentheses close parentheses over denominator open parentheses sin open parentheses x close parentheses close parentheses squared end fraction end cell row blank equals cell fraction numerator fraction numerator d over denominator d x end fraction open parentheses 2 close parentheses plus fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses over denominator open parentheses sin open parentheses x close parentheses close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 0 minus sin open parentheses x close parentheses close parentheses minus cos open parentheses x close parentheses open parentheses 2 plus cos open parentheses x close parentheses close parentheses over denominator open parentheses sin open parentheses x close parentheses close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses negative sin open parentheses x close parentheses close parentheses sin open parentheses x close parentheses minus cos open parentheses x close parentheses open parentheses 2 plus cos open parentheses x close parentheses close parentheses over denominator open parentheses sin open parentheses x close parentheses close parentheses squared end fraction end cell end table

Tentukan gradien

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 fraction numerator negative sin squared open parentheses begin display style 1 half end style straight pi close parentheses minus cos open parentheses 1 half straight pi close parentheses open parentheses 2 plus cos open parentheses 1 half straight pi close parentheses close parentheses over denominator sin squared open parentheses 1 half straight pi close parentheses end fraction end cell row blank equals cell fraction numerator negative 1 squared minus 0 times open parentheses 2 plus 0 close parentheses over denominator 1 squared end fraction end cell row blank equals cell equals fraction numerator negative 1 over denominator 1 end fraction end cell row blank equals cell negative 1 end cell end table

Tentukan 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 2 end cell equals cell negative 1 open parentheses x minus 1 half straight pi close parentheses end cell row y equals cell negative x plus 1 half straight pi plus 2 end cell end table

 

Jadi, persamaan garis singgungnya adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank 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 minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x 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 cell 1 half end cell 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 plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table.

 

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