Bab 9 Pembezaan

Posted on June 2, 2016 by user

9.7 Pembezaan, SPM Praktis (Kertas 1)

Soalan 11:

Diberi fungsi graf

f (x) = h x^{3} + \frac{k}{x^{2}}

mempunyai fungsi kecerunan

f' (x) = 12 x^{2} - \frac{258}{x^{3}}

dengan hdan k ialah pemalar. Cari nilai h dan k.

Penyelesaian:

\begin{array}{l} f (x) = h x^{3} + \frac{k}{x^{2}} = h x^{3} + k x^{- 2} \\ f' (x) = 3 h x^{2} - 2 k x^{- 3} \\ f' (x) = 3 h x^{2} - \frac{2 k}{x^{3}} \end{array}

Tetapi, diberi

f' (x) = 12 x^{2} - \frac{258}{x^{3}}

dengan perbandingan,

3h = 12 atau 2k = 258

h = 4 k = 129

Soalan 12:

\begin{array}{l} Diberi y = \frac{x^{2}}{x + 3}, tunjukkan \frac{d y}{d x} = \frac{x^{2} + 6 x}{{(x + 3)}^{2}} \\ Cari \frac{d^{2} y}{d x^{2}} dalam bentuk paling ringkas . \end{array}

Penyelesaian

\begin{array}{l} y = \frac{x^{2}}{x + 3} \\ \frac{d y}{d x} = \frac{(x + 3) (2 x) - x^{2} .1}{{(x + 3)}^{2}} = \frac{2 x^{2} + 6 x - x^{2}}{{(x + 3)}^{2}} \\ \frac{d y}{d x} = \frac{x^{2} + 6 x}{{(x + 3)}^{2}} (tertunjuk) \end{array}

\begin{array}{l} \frac{d^{2} y}{d x^{2}} = \frac{{(x + 3)}^{2} (2 x + 6) - (x^{2} + 6 x) .2 (x + 3)}{{(x + 3)}^{4}} \\ \frac{d^{2} y}{d x^{2}} = \frac{(x + 3) [(x + 3) (2 x + 6) - 2 (x^{2} + 6 x)]}{{(x + 3)}^{4}} \\ \frac{d^{2} y}{d x^{2}} = \frac{[2 x^{2} + 6 x + 6 x + 18 - 2 x^{2} - 12 x]}{{(x + 3)}^{3}} \\ \frac{d^{2} y}{d x^{2}} = \frac{18}{{(x + 3)}^{3}} \end{array}

Soalan 13:

Jika y = x² + 4x, tunjukkan

x^{2} \frac{d^{2} y}{d x^{2}} - 2 x \frac{d y}{d x} + 2 y = 0.

Penyelesaian

\begin{array}{l} y = x^{2} + 4 x \\ \frac{d y}{d x} = 2 x + 4 \\ \frac{d^{2} y}{d x^{2}} = 2 \\ x^{2} \frac{d^{2} y}{d x^{2}} - 2 x \frac{d y}{d x} + 2 y \\ = x^{2} (2) - 2 x (2 x + 4) + 2 (x^{2} + 4 x) \\ = 2 x^{2} - 4 x^{2} - 8 x + 2 x^{2} + 8 x \\ = 0 (tertunjuk) \end{array}

Soalan 14:

Diberi y = x (6 – x), ungkapkan

y \frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + 18

dalam sebutan x yang paling ringkas.

Seterusnya, cari nilai xyang memuaskan persamaan

y \frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + 18 = 0

Penyelesaian:

\begin{array}{l} y = x (6 - x) = 6 x - x^{2} \\ \frac{d y}{d x} = 6 - 2 x \\ \frac{d^{2} y}{d x^{2}} = - 2 \\ ∴ y \frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + 18 = (6 x - x^{2}) (- 2) + x (6 - 2 x) + 18 \\ = - 12 x + 2 x^{2} + 6 x - 2 x^{2} + 18 \\ = - 6 x + 18 \\ y \frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + 18 = 0 \\ - 6 x + 18 = 0 \\ x = 3 \end{array}