Pertanyaan

Jika f ( x ) = x 3 + 2 2 x 3 − 1 , maka nilai h → 0 lim h f ( 2 + h ) − f ( 2 ) adalah ....

Jika $f (x) = \frac{2 x ^{3} - 1}{x ^{3} + 2}$ , maka nilai $h \to 0 lim \frac{f ( 2 + h ) - f ( 2 )}{h}$ adalah ....

M. Nasrullah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Makassar

Jawaban terverifikasi

Jawaban

nilai adalah

nilai $limit as h rightwards arrow 0 of fraction numerator f left parenthesis 2 plus h right parenthesis minus f left parenthesis 2 right parenthesis over denominator h end fraction$ adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 5 end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 5 end cell end table

Pembahasan

Petama kita tentukan fungsi : Kemudian, kita tentukan fungsi : Sehingga diperoleh perhitungan: Jadi, nilai adalah

Petama kita tentukan fungsi f open parentheses 2 plus h close parentheses :

$table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell fraction numerator 2 x cubed minus 1 over denominator x cubed plus 2 end fraction end cell row cell f open parentheses 2 plus h close parentheses end cell equals cell fraction numerator 2 open parentheses 2 plus h close parentheses cubed minus 1 over denominator open parentheses 2 plus h close parentheses cubed plus 2 end fraction end cell row blank equals cell fraction numerator 2 open parentheses 8 plus 12 h plus 6 h squared plus h cubed close parentheses minus 1 over denominator 8 plus 12 h plus 6 h squared plus h cubed plus 2 end fraction end cell row blank equals cell fraction numerator 16 plus 24 h plus 12 h squared plus 2 h cubed minus 1 over denominator 10 plus 12 h plus 6 h squared plus h cubed end fraction end cell row blank equals cell fraction numerator 15 plus 24 h plus 12 h squared plus 2 h cubed over denominator 10 plus 12 h plus 6 h squared plus h cubed end fraction end cell end table$

Kemudian, kita tentukan fungsi f open parentheses 2 close parentheses :

Sehingga diperoleh perhitungan:

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as h rightwards arrow 0 of fraction numerator f left parenthesis 2 plus h right parenthesis minus f left parenthesis 2 right parenthesis over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator fraction numerator 15 plus 24 h plus 12 h squared plus 2 h cubed over denominator 10 plus 12 h plus 6 h squared plus h cubed end fraction minus begin display style 3 over 2 end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator fraction numerator 2 open parentheses 15 plus 24 h plus 12 h squared plus 2 h cubed close parentheses minus 3 open parentheses 10 plus 12 h plus 6 h squared plus h cubed close parentheses over denominator 2 open parentheses 10 plus 12 h plus 6 h squared plus h cubed close parentheses end fraction over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style fraction numerator up diagonal strike 30 plus 48 h plus 24 h squared plus 4 h cubed up diagonal strike negative 30 end strike minus 36 h minus 18 h squared minus 3 h cubed over denominator 20 plus 24 h plus 12 h plus 2 h cubed end fraction end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style fraction numerator 12 h plus 6 h squared plus h cubed over denominator 20 plus 24 h plus 12 h plus 2 h cubed end fraction end style over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator begin display style fraction numerator up diagonal strike h open parentheses 12 plus 6 h plus h squared close parentheses over denominator 20 plus 24 h plus 12 h plus 2 h cubed end fraction end style over denominator up diagonal strike h end fraction end cell row blank equals cell fraction numerator 12 plus 0 plus 0 over denominator 20 plus 0 plus 0 plus 0 end fraction end cell row blank equals cell 12 over 20 end cell row blank equals cell 3 over 5 end cell end table$

Jadi, nilai $limit as h rightwards arrow 0 of fraction numerator f left parenthesis 2 plus h right parenthesis minus f left parenthesis 2 right parenthesis over denominator h end fraction$ adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 5 end cell end table