Page 49 - Hybrid PBD 2022 Tg 5 - Matematik Tambahan

Matematik Tambahan Tingkatan 5 Bab 2 Pembezaan

2.3 Pembezaan Peringkat Kedua Buku Teks
PBD The Second Derivative ms. 49 – 50
PBD
PBD
15. Cari terbitan kedua bagi fungsi berikut terhadap terbitan x. SP 2.3.1 TP3 Terbitan pertama dan
kedua bagi fungsi algebra
Find the first and the second derivatives for the following functions with respect to x. The first and second derivative
NOTA of algebraic function

(a) y = x –5x + 8
2
2 1
1
2
4
y = 4x – 2x + x = x – 5x + 8
2
2 Tip Penting d y 1 – 1
1
2
d y = 12x – 4x +   2 dy dy dx = x – 10x
2
dy
2
dx 2 dx = dx × dx 2
3
d y d y = – x – 10
1 –
2
2
= 24x – 4 d y =   dx 2 4
d dy
2
dx 2 dx 2 dx dx 1
d y dy 2 = –  3 – 10
2
≠
2   4 x
dx dx
(3x) 2
(b) y = (x – 3)(2 – x) 2 (c) y =
d y 2x – 1
dx = (x – 3)(–2)(2 – x) + (2 – x) 2 d y = (2x – 1)(18x) – 9x (2)
2
= (2 – x)[–2(x – 3) + 2 – x] dx (2x – 1) 2
= (2 – x)[–3x + 8] = 18x [2x – 1 – x]
(2x – 1)
= (2 – x)[8 – 3x] 18x [x – 1] 2
2
d y 2 = (2 – x)(–3) + (8 – 3x)(–1) = (2x – 1) 2
dx
= –6 + 3x – 8 + 3x
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(2x – 1) (36x – 18) – 18x(x – 1)(4)(2x – 1)
d y
2
2
= 6x – 14 = 2(3x – 7) dx 2 = (2x – 1) 4
18(2x – 1)[(2x – 1)(2x – 1) – 4x (x – 1)]
=
(2x – 1) 4
2
2
= 18[4x – 4x + 1 – 4x + 4x]
(2x – 1) 3
18
=
(2x – 1) 3
16. Selesaikan. SP 2.3.1 TP4
Solve.
2
1 3
Diberi y = x + 2x, tunjukkan bahawa (a) Diberi/Given y = x – x – 3x + 1,
2
3
Given y = x + 2x, show that d y
2
2
2 d y 2 d y (i) cari nilai x jika = 0.
2x 2 2   + 4 = 4. dx 2
–
dx dx dx d y
2
y = x + 2x find the value of x if dx 2 = 0.
2
2
d y d y d y d y
2
= 2x + 2, = 2 (ii) cari nilai-nilai 2 jika = 0.
dx dx 2 dx dx
d y d y
2
find the values of if = 0.
d y d y d y dx 2 dx
2
2
Maka 2x 2 2   + 4
–
dx dx dx d y d y
2
2
= 2x (2) – (2x + 2) + 4(2x + 2) (i) dx = x – 2x – 3 (ii) dx = 0 = x – 2x – 3 = 0
2
2
= 4x – (4x + 8x + 4) + 8x + 8 d y (x + 2)(x – 3) = 0
2
2
2
= 4 = Sebelah kanan / Right hand side dx 2 = 2x – 2 = 0 x = –1 atau 3
∴ x = 1 d y
2
∴ = –4 atau 4
dx 2
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