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 21π.

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 $D_f=\left\{x\left|x\in R,\;\sin\;x\neq0\right.\right\}$ . Carilah persamaan garis singgung kurva fungsi $f$ di titik yang berabsis $\frac12\mathrm\pi$ .

G. Albiah

Master Teacher

Mahasiswa/Alumni Universitas Galuh Ciamis

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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 .

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