Termodinamika

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Sebuah sistem termodinamika

Termodinamika (bahasa Yunani: thermos = 'panas' and dynamic = 'perubahan') adalah fisika energi , panas, kerja, entropi dan kespontanan proses. Termodinamika berhubungan dekat dengan mekanika statistik di mana banyak hubungan termodinamika berasal.
Pada sistem di mana terjadi proses perubahan wujud atau pertukaran energi, termodinamika klasik tidak berhubungan dengan kinetika reaksi (kecepatan suatu proses reaksi berlangsung). Karena alasan ini, penggunaan istilah "termodinamika" biasanya merujuk pada termodinamika setimbang. Dengan hubungan ini, konsep utama dalam termodinamika adalah proses kuasistatik, yang diidealkan, proses "super pelan". Proses termodinamika bergantung-waktu dipelajari dalam termodinamika tak-setimbang.
Karena termodinamika tidak berhubungan dengan konsep waktu, telah diusulkan bahwa termodinamika setimbang seharusnya dinamakan termostatik.
Hukum termodinamika kebenarannya sangat umum, dan hukum-hukum ini tidak bergantung kepada rincian dari interaksi atau sistem yang diteliti. Ini berarti mereka dapat diterapkan ke sistem di mana seseorang tidak tahu apa pun kecual perimbangan transfer energi dan wujud di antara mereka dan lingkungan. Contohnya termasuk perkiraan Einstein tentang emisi spontan dalam abad ke-20 dan riset sekarang ini tentang termodinamika benda hitam.
Konsep dasar dalam termodinamika Pengabstrakkan dasar atas termodinamika adalah pembagian dunia menjadi sistem dibatasi oleh kenyataan atau ideal dari batasan. Sistem yang tidak termasuk dalam pertimbangan digolongkan sebagai lingkungan. Dan pembagian sistem menjadi subsistem masih mungkin terjadi, atau membentuk beberapa sistem menjadi sistem yang lebih besar. Biasanya sistem dapat diberikan keadaan yang dirinci dengan jelas yang dapat diuraikan menjadi beberapa parameter !

Sistem termodinamika

Sistem termodinamika adalah bagian dari jagat raya yang diperhitungkan. Sebuah batasan yang nyata atau imajinasi memisahkan sistem dengan jagat raya, yang disebut lingkungan. Klasifikasi sistem termodinamika berdasarkan pada sifat batas sistem-lingkungan dan perpindahan materi, kalor dan entropi antara sistem dan lingkungan.
Ada tiga jenis sistem berdasarkan jenis pertukaran yang terjadi antara sistem dan lingkungan:

• sistem terisolasi: tak terjadi pertukaran panas, benda atau kerja dengan lingkungan. Contoh dari sistem terisolasi adalah wadah terisolasi, seperti tabung gas terisolasi.
• sistem tertutup: terjadi pertukaran energi (panas dan kerja) tetapi tidak terjadi pertukaran benda dengan lingkungan. Rumah hijau adalah contoh dari sistem tertutup di mana terjadi pertukaran panas tetapi tidak terjadi pertukaran kerja dengan lingkungan. Apakah suatu sistem terjadi pertukaran panas, kerja atau keduanya biasanya dipertimbangkan sebagai sifat pembatasnya:
  • pembatas adiabatik: tidak memperbolehkan pertukaran panas.
  • pembatas rigid: tidak memperbolehkan pertukaran kerja.
• sistem terbuka: terjadi pertukaran energi (panas dan kerja) dan benda dengan lingkungannya. Sebuah pembatas memperbolehkan pertukaran benda disebut permeabel. Samudra merupakan contoh dari sistem terbuka.

Dalam kenyataan, sebuah sistem tidak dapat terisolasi sepenuhnya dari lingkungan, karena pasti ada terjadi sedikit pencampuran, meskipun hanya penerimaan sedikit penarikan gravitasi. Dalam analisis sistem terisolasi, energi yang masuk ke sistem sama dengan energi yang keluar dari sistem.

Keadaan termodinamika

Ketika sistem dalam keadaan seimbang dalam kondisi yang ditentukan, ini disebut dalam keadaan pasti (atau keadaan sistem).
Untuk keadaan termodinamika tertentu, banyak sifat dari sistem dispesifikasikan. Properti yang tidak tergantung dengan jalur di mana sistem itu membentuk keadaan tersebut, disebut fungsi keadaan dari sistem. Bagian selanjutnya dalam seksi ini hanya mempertimbangkan properti, yang merupakan fungsi keadaan.
Jumlah properti minimal yang harus dispesifikasikan untuk menjelaskan keadaan dari sistem tertentu ditentukan oleh Hukum fase Gibbs. Biasanya seseorang berhadapan dengan properti sistem yang lebih besar, dari jumlah minimal tersebut.
Pengembangan hubungan antara properti dari keadaan yang berlainan dimungkinkan. Persamaan keadaan adalah contoh dari hubungan tersebut.

 Hukum-hukum Dasar Termodinamika

Terdapat empat Hukum Dasar yang berlaku di dalam sistem termodinamika, yaitu:

• Hukum Awal (Zeroth Law) Termodinamika

Hukum ini menyatakan bahwa dua sistem dalam keadaan setimbang dengan sistem ketiga, maka ketiganya dalam saling setimbang satu dengan lainnya.

• Hukum Pertama Termodinamika

Hukum ini terkait dengan kekekalan energi. Hukum ini menyatakan perubahan energi dalam dari suatu sistem termodinamika tertutup sama dengan total dari jumlah energi kalor yang disuplai ke dalam sistem dan kerja yang dilakukan terhadap sistem.

• Hukum kedua Termodinamika

Hukum kedua termodinamika terkait dengan entropi. Hukum ini menyatakan bahwa total entropi dari suatu sistem termodinamika terisolasi cenderung untuk meningkat seiring dengan meningkatnya waktu, mendekati nilai maksimumnya.

• Hukum ketiga Termodinamika

Hukum ketiga termodinamika terkait dengan temperatur nol absolut. Hukum ini menyatakan bahwa pada saat suatu sistem mencapai temperatur nol absolut, semua proses akan berhenti dan entropi sistem akan mendekati nilai minimum. Hukum ini juga menyatakan bahwa entropi benda berstruktur kristal sempurna pada temperatur nol absolut bernilai nol.

Momentum dan Impuls

Momentum dan Impuls

Momentum dan Impuls dalam pembahasan fisika adalah sebagai satu kesatuan karena momentum dan Impuls dua besaran yang setara. Dua besaran dikatakan setara seperti momentum dan Impuls bila memiliki satuan Sistim Internasional(SI) sama atau juga dimensi sama seperti yang sudah dibahas dalam besaran dansatuan.  Posting kali ini akan sedikit membahas mengenai pengertian momentum dan impuls.

 

Pengertian Momentum

Momentum adalah hasil kali antara massa dan kecepatan. Secara matematis dapat dituliskan sebagai berikut

  P = m.v

Keterangan

• P = momentum(kg.m/s)
• M=massa(kg)
• V=kecepatan(m/s)

Jadi momentum adalah besaran yang dimiliki oleh sebuah benda atau partikel yang bergerak.

Contoh

Sebuah bus bermassa 5 ton bergerak dengan kecepatan tetap 10 m/s. Berapa momentum yang dimiliki bus tersebut?

Penyelesaian:

Dengan menggunakan persamaan diatas maka kita mendapatkan besar momentum bus sebesar P = mv

P = 5000 kg x 20 m/s

P= 100000 kg m/s

(catatan 1 ton = 1000 kg)

Pengertian Impuls

Impuls adalah peristiwa gaya yang bekerja pada benda dalam waktu hanya sesaat. Atau Impuls adalah peristiwa bekerjanya gaya dalam waktu yang sangat singkat. Contoh dari kejadian impuls adalah: peristiwa seperti bola ditendang, bola tenis dipukul karena pada saat tendangan dan pukulan, gaya yang bekerja sangat singkat.

  I=F.Δt

Keterangan

• I= impuls
• F=gaya(N)
• Δt=selang waktu(s)

Contoh:

Sebuah bola dipukul dengan gaya 50 Newton dengan waktu 0,01 sekon. Berapa besar Impus pada bola tersebut?

Penyelesaian

Dengan menggunakan persamaan diatas maka

I=F.Δt

I=50 N. 0,01s

I=0,5 Ns

Impuls sama dengan perubahan momentum

Suatu partikel yang bermassa m bekerja gaya F yang konstan, maka setelah waktu  Δt partikel tersebut bergerak dengan kecepatan

V_t=V₀+ a Δt seperti yang sudah dibahas pada post glbb(gerak lurus berubah beraturan)

Menurut hukum ke-2 Newton:

  F=m.a,

Dengan subtitusi kedua persamaan tersebut maka diperoleh

 I=F.Δt = mv_t – mv₀

Keterangan

• mv_t = mementum benda pada saat kecepatan v_t
• mv₀ = mementum benda pada saat kecepatan v₀

Suku Banyak

Suku banyak (polinomial) adalah sebuah ungkapan aljabar yang variabel (peubahnya) berpangkat Bilangan bulat non negative.

Bentuk umum :

y = F(x) = a0xn + a1xn-1 + a2xn-2 + … + an-1x + an

Dengan n Є bilangan bulat
an ≠ 0
Pengertian-pengertian:
a0, a1, a2 ,…, an-1 , an
Disebut koefisien masing-masing bilangan real (walaupun boleh juga bilangan kompleks)

Derajat Suku Banyak adalah pangkat tertinggi dari pangkat-pangkat pada tiap-tiap suku, disebut n.Untuk suku banyak nol dikatakan tidak memiliki derajat.

Suku : a0xn , a1xn-1 , a2xn-2 , … , an-1x , an
Masing-masing merupakan suku dari suku banyak


Suku Tetap (konstanta)
A0 adalah suku tetap atau konstanta, tidak mengandung variabel/peubah. Sedangkan anxn adalah suku berderajat tinggi.

Soal
1. Diketahui suku banyak: f(x) = 2x5+3x4-5x2+x-7
Tentukan suku tetapnya.
Jawab :
Suku tetap adalah konstanta.
Maka, suku tetapnya adalah -7
2. Diketehui suku banyak: f(x) = 2x5+3x4-5x2+x-7
tentukan derajat suku banyaknya
Jawab:
Derajat suku banyak adalah pangkat tertinggi dari suku-suku yang ada.
x5 adalah pangkat tertinggi. Jadi f(x) berderajat 5

NILAI SUKU BANYAK

Jika f(x) = axn + bxn-1+CXN-2+…+f maka nilai suku banyak dapat dicari dengan cara subtitusi dan skematik.

Soal
1. Diketahui fungsi polinom f(x) = 2x5+3x4-5x2+x-7
Maka nilai fungsi tersebut untuk x=-2 adalah
a. -90 d. 45
b. -45 e. 90
c. 0
Pembahasan
f(x) = 2x5+3x4-5x2+x-7

Cara 1 (subtitusi): x = -2
f(-2)= 2(-2)5+3(-2)4+5(-2)2+(-2)-7
f(-2)= -45
Cara 2 (skematik)
f(x) = 2x5+3x4-5x2+x-7, x=-2
Ambil koefisiennya:
-2 2 3 0 -5 1 -7
-4 2 -4 18 -38 +
2 -1 2 -9 19 -45
Jadi nilai suku banyaknya -45

2. Diketahui fungsi kuadrat : f (x) = 1 x2 + 3 x - 5
2 4
untuk x=2 maka nilai suku banyak tersebut adalah:
Pembahasan:
Cara Substitusi: f(2) = 1 (2)2 + 3 (2) - 5
2 4
= 2 + 3 - 5
2
= - 3
2
Cara skematik:
2 1 3 - 5
2 4
1 7
2
1 7 -3
2 4 2
Jadi nilai suku banyaknya -3/2

OPERASI PADA SUKU BANYAK
Penjumlahan, pengurangn dan perkalian Suku Banyak

1. Penjumlahan
contohnya: f (x) = 3x4 – 2x3 + 5x2 – 4x + 3 , g(x) = 4x3 – 6x2 + 7x - 1
Tentukan : f (x) + g(x)
Jawab : f (x) + g(x) = (3x4 – 2x3 + 5x2 – 4x + 3) + (4x3 – 6x2 + 7x – 1)
= 3x4 + (-2 +4)x3 + (5-6)x2 + (-4+7)x + (3-1)
= 3x4 + 2 x3 – 1x2 + 3x + 2
2. Pengurangan
contoh: : f (x) = 3x4 – 2x3 + 5x2 – 4x + 3 , g(x) = 4x3 – 6x2 + 7x - 1
Tentukan : f (x) - g(x)
Jawab : f (x) - g(x) = (3x4 – 2x3 + 5x2 – 4x + 3) - (4x3 – 6x2 + 7x – 1)
= 3x4 + (-2 -4)x3 + (5+6)x2 + (-4-7)x + (3+1)
= 3x4 - 6x3 +11x2 - 11x + 4
3. Perkalian
Contohnya: f (x) = 2x3 + 5x2 – 4x + 3 , g(x) = 6x2 + 7x - 1
Tentukan : f (x) x g(x)
Jawab : f (x) x g(x) = (2x3 + 5x2 – 4x + 3) x (6x2 + 7x – 1)
= 2x3 (6x2 + 7x – 1) + 5x2 (6x2 + 7x – 1)
– 4x (6x2 + 7x – 1) + 3 (6x2 + 7x – 1)
= 12x5 + 14x4 – 2x3 + 30x4 + 35x3 – 5x2
- 24x3 – 28x2 + 4x + 18x2 +21x - 3
= 12x5 + 34x4 – 26x3 – 15x2 + 25x – 3

PEMBAGIAN PADA SUKU BANYAK

Pembagian sukubanyak P(x) oleh (x – a) dapat ditulis dengan

P(x) = (x – a)H(x) + S

Keterangan:
P(x) sukubanyak yang dibagi,
(x – a) adalah pembagi,
H(x) adalah hasil pembagian,
dan S adalah sisa pembagian

TOREMA SISA

Jika sukubanyak P(x) dibagi (x – a), sisanya P(a) dibagi (x + a) sisanya P(-a)
dibagi (ax – b) sisanya P(b/a)

Contoh 1:
Tentukan sisanya jika 2x3 – x2 + 7x + 6 dibagi x + 1 atau dibagi x – (-1)

Jawab: sisanya adalah
P(-1) = 2.(-1)3 – (-1)2 + 7(-1) + 6
= - 2 – 1 – 7 + 6
= -4

Contoh 2:
Tentukan sisa dan hasil baginya jika x3 + 4x2 - 5x – 8 dibagi x - 2
Jawab:
Dengan teorema sisa, dengan mudah kita dapatkan sisanya,
yaitu P(2) = 8 + 16 - 10 - 8
= 6
tapi untuk menentukan hasil baginya kita gunakan: Pembagian Horner:
dengan menggunakan bagan seperti berikut:
x3 + 4x2 - 5x – 8 dibagi x - 2

2 1 4 -5 -8 koefisien
2 12 14 Polinum

1 6 7 6

Koefisien hasil bagi 1 6 7
Jadi hasil baginya: x2 + 6x + 7

Contoh 3:
Tentukan sisa dan hasil baginya jika 2x3 - 7x2 + 11x + 5 dibagi 2x - 1
Jawab:
(2x3 - 7x2 + 11x + 5) : (2x – 1)
Sisa:
P(½) = 2(½)3 – 7(½)2 + 11.½ + 5
= 2.⅛ - 7.¼ + 5½ + 5
= ¼ - 1¾ + 5½ + 5
= 9
2x3 - 7x2 + 11x + 5 dibagi 2x – 1
Kita gunakan pembagian horner
2x3 - 7x2 + 11x + 5 dibagi 2x – 1 →x = 1
2
2 -7 11 5
1
2 1 -3 4

2 -6 8 9

Koefisien hasil bagi 2 -6 8 9

Sehingga 2x3 - 7x2 + 11x + 5 dibagi 2x – 1
Dapat ditulis: 2x3 – 7x2 + 11x + 5 = (x - ½)(2x2 – 6x + 8) + 9
= (2x – 1)(x2 – 3x + 4) + 9
Pembagi : 2x - 1
Hasil bagi : x2 – 3x + 4
Sisa : 9

Contoh 4:
Nilai m supaya 4x4 – 12x3 + mx2 + 2 habis dibagi 2x – 1 adalah….
Jawab: habis dibagi → S = 0
P(½) = 0
4(½)4 – 12(½)3 + m(½)2 + 2 = 0
¼ - 1½ + ¼m + 2 = 0
¼m = -¼ + 1½ - 2 (dikali 4)
m = -1 + 6 – 8
m = -3
Jadi nilai m = -3

Pembagian Dengan (x –a)(x – b)
Bentuk pembagiannya dapat ditulis sebagai

P(x) = (x – a)(x – b)H(x) + S(x)

berarti: untuk x = a , P(a) = S(a) dan untuk x = b,P(b) = S(b)
Catatan: S(x) berderajat 1, misal px + q

Contoh5:
Suku banyak (x4 – 3x3 – 5x2 + x – 6) dibagi (x2 – x – 2), sisanya sama dengan….
Jawab:
Bentuk pembagian ditulis: P(x) = (x2 – x – 2)H(x) + S(x)
Karena pembagi berderajat 2 maka sisa = S(x) berderajat 1
misal: sisanya px + q
sehingga bentuk pembagian ditulis:
Fx4 – 3x3 – 5x2 + x – 6 = (x2 – x – 2)H(x) + px + q
Fx4 – 3x3 – 5x2 + x – 6 = (x + 1)(x – 2)H(x) + px + q
P(x) dibagi (x + 1) bersisa P(-1)
P(x) dibagi (x – 2) bersisa P(2)
P(-1) = (-1)4 – 3(-1)3 – 5(-1)2 + (-1) – 6
= 1 + 3 – 5 – 1 – 6 = -8
P(2) = 24 – 3.23 – 5.22 + 2 – 6
= 16 – 24 – 20 + 2 – 6 = -32
P(x) = px + q
P(-1) = -p + q = -8
P(2) = 2p + q = -32 _
-3p = 24 ® p = -8
p = -8 disubstitusi ke
–p + q = -8
8 + q = -8 ® q = -16
Sisa: px + q = -8x + (-16) Jadi sisa pembagiannya: -8x -16

Contoh 6:
Suatu suku banyak bila dibagi oleh x + 2 bersisa -13, dibagi oleh x – 3 sisanya 7.
Suku banyak tersebut bila dibagi oleh x2 – x - 6 bersisa….
Jawab:
Misal sisanya: S(x) = ax + b,
P(x): (x + 2) Þ S(-2) = -13 ® -2a + b = -13
P(x): (x – 3) Þ S(3) = 7 ® 3a + b = 7 _
-5a = -20® a = 4
a = 4 disubstitusi ke -2a + b = -13
 -8 + b = -13
 b = -5
Jadi sisanya adalah: ax + b = 4x - 5

Contoh 7:
Jika suku banyak
P(x) = 2x4 + ax3 - 3x2 + 5x + b dibagi oleh (x2 – 1) memberi sisa 6x + 5, maka a.b=….
Jawab :
P(x) = 2x4 + ax3 - 3x2 + 5x + b
P(x) : (x2 – 1) Þ sisa = 6x + 5
Pembagi : (x2 -1) = (x + 1)(x – 1)
Maka:
P(x):(x + 1) Þ sisa =P(-1)
P(-1) = 2(-1)4 + a(-1)3 – 3(-1)2 + 5(-1) + b = 6(-1) + 5
2 - a - 3 – 5 + b = – 6 + 5
-a + b - 6 = -1
-a + b = 5…………….(1)
P(x):(x – 1) Þ sisa =P(1)
P(1) = 2 (1)4 + a(1)3 – 3(1)2 + 5(1) + b = 6(1) + 5
2 + a - 3 + 5 + b = 6 + 5
a + b + 4 = 11
a + b = 7…………………...(2)
-a + b = 5.…(1)
a + b = 7….(2) +
2b = 12
® b = 6
b = 6 disubstitusi ke a + b = 7
a + 6 = 7
a = 1
Jadi a.b = 1.6 = 6

Contoh 8
Jika suku banyak x3 – x2 + px + 7 dan sukubanyak 2x3 + 3x2 - 4x – 1 dibagi (x + 1)
akan diperoleh sisa yang sama, maka nilai p sama dengan….
Jawab:
x3 – x2 + px + 7 dibagi (x + 1)
Sisanya P(-1) = -1 -1 – p + 7
= 5 - p
2x3 + 3x2 - 4x – 1 dibagi (x + 1)
Sisanya P(-1) = -2 + 3 + 4 – 1
= 4
Karena sisanya sama,
Berarti 5 – p = 4
- p = 4 – 5
Jadi p = 1

Contoh 9
Jika suku banyak x3 – 7x + 6 dan sukubanyak x3 – x2 – 4x + 24 dibagi (x + a) akan diperoleh sisa yang sama, maka nilai a sama dengan….
Jawab:
x3 – 7x + 6 dibagi (x + a)
Sisanya P(-a) = a3 – 7a + 6
x3 – x2 – 4x + 24 dibagi (x + a)
Sisanya P(-a) = a3 – a2 – 4a + 24
Sisanya sama berarti:
a3 – 7a + 6 = a3 – a2 – 4a + 24
a2 – 7a + 4a + 6 – 24 = 0
a2 – 3a – 18 = 0
(a + 3)(a – 6) = 0
a = -3 atau a = 6
Jadi nilai a = - 3 atau a = 6

Contoh 10:
Jika suku banyak
P(x) = 2x3 + ax2 - bx + 3 dibagi oleh (x2 – 4) memberi sisa x + 23, maka a + b=….
Jawab :
P(x) = 2x3 + ax2 - bx + 3
P(x) : (x2 – 4) Þ sisa = x + 23
Pembagi : (x2 – 4) = (x + 2)(x – 2)
Maka:
P(x):(x + 2) Þ sisa = P(-2)
-16 + 4a + 2b + 3 = (-2) + 23
4a + 2b = 21 + 13
4a + 2b = 34….(1
P(x) = 2x3 + ax2 - bx + 3
P(x) : x2 - 4 Þ sisa = x + 23
Pembagi : x2 -1 = (x + 2)(x – 2)
Maka:
P(x):(x – 2) Þ sisa =P(2)
16 + 4a – 2b + 3 = 2 + 23
4a – 2b + 19 = 25
4a – 2b = 25 – 19
4a – 2b = 6….(2)

4a + 2b = 34.…(1)
4a – 2b = 6….(2) +
8a = 40
® a = 5
a = 5 disubstitusi ke 4a – 2b = 6
20 – 2b = 6
- 2b = -14 ® b = 7
Jadi a + b = 5 + 7 = 12

TEOREMA FAKTOR

Jika f(x) adalah sukubanyak; (x – k) merupakan faktor dari f(x) jika dan hanya jika
f(k) = 0
Artinya: Jika (x – k) merupakan faktor, maka nilai f(k) = 0 sebaliknya, jika f(k) = 0 maka (x – k) merupakan faktor

Contoh 1:
Tunjukan (x + 1) faktor dari x3 + 4x2 + 2x – 1

Jawab:
(x + 1) faktornya, berarti P(-1) = 0
P(-1) = (-1)3 + 4(-1)2 + 2(-1) – 1
= -1 + 4 – 2 – 1 = 0
Jadi, (x + 1) adalah faktornya.
Cara lain untuk menunjukan (x + 1) adalah faktor dari x3 + 4x2 + 2x – 1 adalah dengan
pembagian horner:
1 4 2 -1
-1 -1 -3 1 +

1 3 -1 0

Karena sisa pembagiannya 0 maka (x + 1) meripakan factor dari x3 + 4x2 + 2x – 1

Contoh 2:
Tentukan faktor-faktor dari P(x) = 2x3 – x2 – 7x + 6
Jawab:
Misalkan faktornya (x – k), maka nilai k yang mungkin adalah pembagi bulat dari 6, yaitu
pembagi bulat dari 6 ada 8 yaitu: ±1, ±2, ±3, dan ±6. Nilai-nilai k itu kita substitusikan
ke P(x), misalnya k = 1 diperoleh:
P(1) = 2.13 – 1.12 – 7.1 + 6
= 2 – 1 – 7 + 6
= 0
Oleh karena P(1) = 0, maka (x – 1) adalah salah satu factor dari P(x) = 2x3 – x2 -7x + 6
Untuk mencari faktor yang lain, kita tentukan hasil bagi P(x) oleh (x – 1) dengan
pembagian horner:
Koefisien sukubanyak P(x) = 2x3 – x2 – 7x + 6 adalah 2 -1 -7 6

2 -1 -7 6
1 2 1 -6
+
2 1 - 6 0

Hasil baginya: H(x) = 2x2 + x - 6
Karena hasil baginya adalah H(x) = 2x2 + x – 6 = (2x – 3)(x + 2) dengan demikian
2x3 – x – 7x + 6 = (x – 1)(2x2 + x – 6)
2x3 – x – 7x + 6 = (x – 1)(2x – 3)(x + 2)
Jadi faktor-faktornya adalah (x – 1), (2x – 3 ) dan (x + 2)

Contoh 3:
Diketahui (x – 2) adalah factor P(x) = 2x3 + x2 - 7x - 6. Salah satu faktor yang lainnya
adalah…. a. x + 3
b. x – 3
c. x – 1
d. 2x – 3
e. 2x + 3
P(x) = 2x3 + x2 - 7x – 6 berarti koefisien P(x) adalah 2 1 -7 -6 k = 2

2 1 -7 -6
2 4 10 6 +

2 5 3 0

Hasil baginya: H(x) = 2x2 + 5x + 3
= (2x + 3)(x + 1)
Jadi faktor yang lain adalah 2x + 3

Contoh 4:
Sukubanyak f(x) = x3 - ax2 + bx – 2 mempunyai faktor (x – 1). Jika dibagi oleh (x + 2) bersisa -36, maka nilai a + b adalah….
a. 5 b. 6 c. 7 d.8 e.9
Jawab:
Sukubanyak f(x) = x3 - ax2 + bx – 2
(x – 1) faktor f(x) → f(1) = 0
1 – a + b – 2 = 0
-a + b = 1….(1)
dibagi (x + 2) bersisa -36, f(-2) = -36
(-2)3 – a(-2)2 + b(-2) – 2 = -36
- 8 – 4a – 2b – 2 = -36
- 4a – 2b = -36 + 10
-4a – 2b = -26
2a + b = 13….(2)
Persamaan (1): -a + b = 1
Persamaan (2): 2a + b = 13 -
-3a = -12
a = 4
b = 1 + 4 = 5
Jadi nilai a + b = 4 + 5 = 9

Akar-akar Rasional Persamaan Sukubanyak

Salah satu penggunaan teorema faktor adalah mencari akar-akar sebuah persamaan sukubanyak, karena ada hubungan antara faktor dengan akar-akar persamaan sukubanyak
Jika P(x) adalah sukubanyak; (x – k) merupakan faktor dari P(x) jika dan hanya jika k akar dari persamaan P(k) = 0
k disebut akar atau nilai nol dari persamaan sukubanyak: P(x) = 0

Teorema Akar-akar Rasional
Jika P(x) = anxn + an-1xn-1 + …+ a1x + ao dan (x – k) merupakan faktor dari P(x) maka
K merupakan akar dari P(x).

Contoh 1:
Tunjukan -3 adalah salah satu akar dari x3 – 7x + 6. Kemudian tentukan akar-akar yang lain.
Jawab:
Untuk menunjukan -3 akar dari P(x), cukup kita tunjukan bahwa P(-3) = 0
P(x) = x3 – 7x + 6.
P(-3) = (-3)3 – 7(-3) + 6
= -27 + 21 + 6
= 0
Oleh karena P(-3) = 0, maka -3 adalah akar dari Persamaan P(x) = x3 – 7x + 6 = 0
Untuk menentukan akar-akar yang lain, kita tentukan terlebih dahulu hasil bagi
P(x) = x3 – 7x + 6 dengan x + 3 dengan pembagian Horner sebagai berikut
P(x) = x3 – 7x + 6
berarti koefisien P(x) adalah 1 0 -7 6 dengan k = -3

1 0 -7 6
-3 -3 9 -6
+
1 -3 2 0

Hasil baginya: H(x) = x2 – 3x + 2
= (x – 1)(x – 2)
sehingga persamaan sukubanyak tsb dapat ditulis menjadi (x + 3)(x – 1)(x – 2) = 0.
Jadi akar-akar yang lain adalah x = 1 dan x = 2

Contoh 2:
Banyaknya akar-akar rasional dari persamaan x4 – 3x2 + 2 = 0 adalah….
a. 4 b. 3 c. 2 d.1 e.o
Jawab:
Karena persamaan sukubanyak berderajat 4, maka akar-akar rasionalnya paling banyak ada 4 yaitu faktor-faktor bulat dari 2. Faktor-faktor bulat dari 2 adalah 1, -1, 2 dan -2
Dari 4 kemungkinan yang akan menjadi akar-akar rasional persamaan sukubanyak tsb,
kita coba nilai 1
Koefisien x4 – 3x2 + 2 = 0 adalah 1, 0, -3, 0, dan 2

1 0 -3 0 2
1 1 1 -2 -2
+
1 1 2 -2 0

Ternyata P(1) = 0, berarti 1 adalah akar rasionalnya,
Selanjutnya kita coba -1.
Koefisien hasil bagi: 1,1,-2, dan -2

1 1 -2 -2
-1 -1 0 2
+
1 0 -2 0

Ternyata P(-1) = 0, berarti -1 adalah akar rasionalnya, Sehingga:
(x – 1)(x + 1)(x2 – 2) = 0
(x2 – 2) difaktorkan lagi menjadi (x - √2)(x + √2) = 0
Berarti akar yang lain: √2 dan -√2, tapi bukan bilangan rasional.
Jadi akar-akar rasionalnya hanya ada 2 yaitu 1 dan -1.

Jumlah dan Hasil Kali Akar-akar Persamaan Sukubanyak
Jika akar-akar Persamaan Sukubanyak: ax3 + bx2 + cx + d = 0 adalah x1, x2, dan x3 maka
x1 + x2 + x3 = -b
a
x1.x2 + x1.x3 + x2.x3 = c
a
x1.x2.x3 = -d
a
Contoh 1:
Jumlah akar-akar persamaan x3 – 3x2 + 2 = 0 adalah….
Jawab:
a = 1, b = -3, c = 0, d = 2
x1 + x2 + x3 = -b/a = -3/1 = 3
Contoh 2:
Hasilkali akar-akar persamaan 2x3 – x2 + 5x – 8 = 0 adalah….
Jawab:
a = 2, b = -1, c = 5, d = -8
x1.x2.x3 = c/a = 5/2

Contoh 3:
Salah satu akar persamaan x3 + px2 – 3x – 10 = 0 adalah -2 Jumlah akar-akar persamaan
tersebut adalah….

Jawab:
-2 adalah akar persamaan x3 + px2 – 3x - 10 = 0 → -2 memenuhi persamaan tsb.
sehingga: (-2)3 + p(-2)2 – 3(-2) - 10 = 0
-8 + 4p + 6 – 10 = 0
-8 + 4p + 6 – 10 = 0
4p – 12 = 0 ® 4p = 12® p = 3
Persamaan tersebut: x3 + 3x2 – 3x – 10 = 0
Jumlah akar-akarnya: x1 + x2 + x3 = -b/a = -3

Contoh 4:
Akar-akar persamaan x3 – 4x2 + x – 4 = 0 adalah x1, x2, dan x3. Nilai x12 + x22 + x32 =….
x1 + x2 + x3 = 4
x1x2 + x1x3 + x2x3 = 1
Jadi:
x12 + x22 + x32 = (x1 + x2 + x3)2 - 2(x1x2 + x1x3 + x2x3)
= 42 – 2.1
= 16 – 2
= 14

III. Latihan
Jawablah pertanyaan di bawah dengan benar
1. Nilai sisa dari f(x)=x4+x3-2x2+x+2 jika dibagi x+2 adalah…
2. Hasil bagi dan sisa dari 2x2-5x2+2x-4 dibagi x+2 adalah….
3. Nilai sisa dari f(x)=3x3+x2+x+2 jika dibagi 3x-2 adalah…
4. Hasil bagi dari x5 - 32 adalah….
x-2
5. Diketahui suku banyak f(x)=5x3-4x2+3x-2 Nilai dari 5f(4)-4f(3) adalah….
6. Jika f(x) = 4x2-12x3+13x2-8x+a habis dibagi (2x-1), maka nilai a adalah….
7. Jika x3-4x2+px+6 dan x2+3x-2 dibagi (x+1) memberikan sisa yang sama, nilai p
adalah…
8. Suku banyak F(X) jika dibagi oleh (x-3) sisanya 8 dan jika dibagi oleh (x-2)
sisanya -7.Maka jika suku banyak itu dibagi oleh x2-x-6, sisanya adalah….

IV. Tes Formatif
( Terlampir)
V. Daftar pustaka
Tim penulis MGMP Matematika SMA kota Semarang, Matematika SMA / MA XI A IPA, ( Semarang : CV. Jabbaar Setia, 2008)
Tim penyusun KREATIF Matematika, Matematika SMA/MA kelas XI IPA semester gasal, ( Klaten, Viva Pakarindo, 2007)
Simangunsong Wilson, Matematika dasar, ( Jakarta: Erlangga, 2005)






Jawablah pertanyaan di bawah dengan benar
1. Hasil bagi dan sisa dari
2x2-5x2+2x-4 dibagi x+2
Adalah….
a. 2x2-9x+20 sisa -44
b. 2x2-9x+20 sisa -24
c. 2x2-9x+20 sisa -14
d. 2x2-9x+20 sisa -14
e. 2x2-9x+20 sisa -14
Pembahasan:
Maka:
-2 2 -5 2 -4
-4 18 -40 +
2 -9 20 -44
Jadi hasil baginya 2x2-9x+20
Sisa -44
Kunci a
2. Nilai sisa dari
f(x)=x4+x3-2x2+x+2
jika dibagi x+2 adalah…

a. -6 d. 0
b. -4 e. 2
c. -2
Pembahasan:
Ambil koefisiennya
Maka:
-2 1 1 -2 1 2
-2 2 0 -2 +
1 -1 0 1 0
Jadi hasil baginya x3 - x2 + 1
Sisa “0”
Kunci d


6. Nilai sisa dari
f(x)=x4+x3-2x2+x+2
jika dibagi x+2 adalah…

a. -6 d. 0
b. -4 e. 2
c. -2

7. Nilai sisa dari
f(x)=3x3+x2+x+2
jika dibagi 3x-2 adalah…

a. -1 d. 3
b. 1 e. 4
c. 2

Pembahasan:

f(x)=3x3+x2+x+2
Maka:
3 1 1 2
2 2 2 +
3 3 3 4
Sisa 4
Kunci e


7. Nilai sisa dari
f(x)=3x3+x2+x+2
jika dibagi 3x-2 adalah…

a. -1 d. 3
b. 1 e. 4
c. 2

8. Hasil bagi dari adalah….


Pembahasan:


Maka:
2 1 0 0 0 0 -32
2 4 8 16 32 +
1 2 4 8 16 0
Jadi hasil baginya
x4+2x3+4x2+8x+16
Kunci e


8. Hasil bagi dari adalah….


9. Diketahui suku banyak
f(x)=5x3-4x2+3x-2 Nilai dari
5f(4)-4f(3) adalah….
a. 900
b. 902
c. 904
d. 906
e. 908

Pembahasan:
f(x)=5x3-4x2+3x-2, untuk x=4 f(4)
maka: 4 5 -4 3 -2
20 64 268 +
5 16 67 266
Jadi f(4) = 226
Untuk x=3 f(3)
3 5 -4 3 -2
15 33 108 +
5 11 36 106
Jadi f(3) = 106
Maka nilai 5f(4) – 4f(3) adalah…
= 5(266) – 4(106)
= 1330 – 424
= 906
Kunci d

9. Diketahui suku banyak
f(x)=5x3-4x2+3x-2 Nilai dari
5f(4)-4f(3) adalah….
a. 900
b. 902
c. 904
d. 906
e. 908

10. Jika f(x) = 4x2-12x3+13x2-8x+a
habis dibagi (2x-1), maka nilai a
adalah….

a. 10
b. 8
c. 6
d. 4
e. 2

Pembahasan:
f(x) = 4x2-12x3+13x2-8x+a
f(x) habis dibagi (2x-1) untuk x =

4 -12 13 -8 a
2 -5 4 -2 +
4 -10 8 -4 a-2

f( ) = a-2 = 0
a = 2
Kunci e

10. Jika f(x) = 4x2-12x3+13x2-8x+a
habis dibagi (2x-1), maka nilai a
adalah….

a. 10
b. 8
c. 6
d. 4
e. 2

11. Jika x3-4x2+px+6 dan
x2+3x-2 dibagi (x+1) memberikan
sisa yang sama, nilai p adalah…
a. -5 d. 3
b. -3 e. 5
c. 1


Pembahasan:
x3-4x2+px+6 dibagi (x+1)
Maka
f(-1)=(-1)3-4(-1)2+p(-1)+6
f(-1)=-1-4-p+6
f(-1)=1-p


G(x)=x2+3x-2 dibagi (x+1)
Maka
G(-1)=(-1)2+3(-1)-2
G(-1)=1-3-2
G(-1)=-4

F(-1)=G(-1)
1-p = -4-1
-p = -5
p = 5

Kunci e

11. Jika x3-4x2+px+6 dan
x2+3x-2 dibagi (x+1) memberikan
sisa yang sama, nilai p adalah…
a. -5 d. 3
b. -3 e. 5
c. 1


12. Suku banyak F(X) jika dibagi oleh
(x-3) sisanya 8 dan jika dibagi oleh
(x-2) sisanya -7. Maka jika suku
banyak itu dibagi oleh x2-x-6,
sisanya adalah….
a. 3x+1
b. 3x-1
c. x-3
d. x+3
e. 1-3x


Pembahasan:
F(x) = (x2-x-6)H(x)+3
F(x) = (x-3)(x+2)H(x)ax+b
F(3) = 0.H(x)+3a+b=8
F(-2) = 0.H(x)+(-2a)+b=-7
Jadi
3a+b=8
-2a+b=-7 -
5a = 15
a = 3

3a +b=8
3(3)+b=8
b=8-9
b=-1
Jadi f(x) dibagi x2-x-6 tersisa….
ax+b = 3x-1
Kunci b

12. Suku banyak F(X) jika dibagi oleh
(x-3) sisanya 8 dan jika dibagi oleh
(x-2) sisanya -7. Maka jika suku
banyak itu dibagi oleh x2-x-6,
sisanya adalah….
a. 3x+1
b. 3x-1
c. x-3
d. x+3
e. 1-3x

STATISTIK

STATISTIK

Defenisi :
Salah satu definisi menyebutkan bahwa statistik adalah metode ilmiah untuk menyusun, meringkas, menyajikan dan menganalisa data, sehingga dapat ditarik suatu kesimpulan yang benar dan dapat dibuat keputusan yang masuk akal berdasarkan data tersebut.

Jika suatu kesimpulan data sudah dihimpun, pada statistika deskriptif kita hendak menyimpulkan data itu dalam beberapa hal. Pertama kita hendak membuat tabel, misalnya tabel frekuensi, tabel frekuensi kumulatif dan lain-lain yang mengatur data kasar itu. Juga kita akan melihat diagram atau grafik yang dapat memberi gambaran mengenai keseluruhan data itu, misalnya diagram lambang (piktogram), diagram batang, diagram lingkaran, histogram, ogive dan lain-lain. Kemudian kita hendak menghitung karakteristik data yang dapat mencakup semua data itu, misalnya rata-rata, median, modus dan lain-lain.


Histogram dan Poligon :

HISTOGRAM dan POLIGON FREKUENSI adalah dua grafik yang menggambarkan distribusi frekuensi.

HISTOGRAM terdiri dari persegi panjang yang alasnya merupakan panjang kelas interval, sedangkan tingginya sama dengan frekuensi masing-masing kelas interval.

POLIGON FREKUENSI adalah suatu garis putus putus yang menghubungkan titik tengah ujung batang histogram. Biasanya ditambah dua segmen garis lain yang menghubungkan titik tengah ujung batang pertama dan terakhir dengan titik tengah kelas yang paling ujung dimana frekuensinya bernilai nol.


Ukuran Pemusatan Data :
Untuk sekelompok data yang diperoleh, yaitu x1, x2, x3, . . . . . . , x maka dapat ditentukan:

1. RATA-RATA (MEAN) (notasi: x dibaca : x bar)
_
x = (x1+x2+.....+xn)/n = å xi / n = å (fi.xi) / n dimana åfi = n

~
2. MEDIAN (notasi: x )
Adalah nilai tengah dari data yang telah diurutkan menurut besarnya.

Dengan ketentuan:
Jika banyak data ganjil, maka median adalah nilai tengah dari data yang telah diurutkan.

(Data ke (n+1)/2 )

^
3. MODUS (notasi : x)
Adalah nilai data yang sering muncul (mempunyai frekuensi terbesar). Modus dapat ada ataupun tidak ada. Kalaupun ada dapat lebih dari satu.

Contoh:

Diketahui data
7, 9, 8, 13, 12, 9, 6, 5 n = 8

1. Rata-rata
_
x = (5+6+7+8+9+9+12+13)/8 = 8,625


2. Median
Data diurutkan terlebih dahulu menjadi
5 6 7 8 9 9 12 13
~
x = (8+9)/2 = 8,5


3. Modus
^
x = 9

• 
  

Pengertian Limit Fungsi

PENGERTIAN LIMIT FUNGSI

LIMIT FUNGSI: Mendekati hampir, sedikit lagi, atau harga batas
Limit fungsi:Suatu limit f(x) dikatakan mendekati A {f(x) → A} sebagai suatu limit.
Bila x mendekati a {x→a}Dinotasikan Lim F(x) = A
x→a
Langkat-langkah mengerjakan limit fungsi (supaya bentuk tak tentu dapat dihindari) adalah ….
 Subtitusi langsung.
 Faktorisasi.
 Mengalikan dengan bilangan sekawan.
 Membagi dengan variabel pangkat tertinggi.

SIFAT-SIFAT LIMIT FUNGSI
Berapa teorema limit:
Bila Lim f(x) = A dan Lim g(x) = B
x → a x →a
Maka
1. Lim [k.f(x)] = k Lim f(x)
x→a x→a

= k. A

2. Lim [f(x)+g(x)] = Lim f(x) + Lim g(x)
x→a x→a x→a

= A + B

3. Lim [f(x) x g(x)]
x→a

= Lim f(x) x Lim g(x)
x→a x→a

= A x B

4. Lim f(x) Lim f(x)
x→a g(x) = x→a . = A
Lim g(x) B
x→a
n n n
5. Lim f(x). = Lim f(x) = A
x→a x→a
n n n
6. Lim √ f(x) = √ Lim f(x) = √ A
x→a x→a

Soal latihan:
1. Nilai dari Lim 3x adalah….
x→2
a. 1
b. 2
c. 3
d. 4
e. 6
Pembahasan 1: Lim 3x = 3(2) = 6
x→2
Pembahasan 2:Lim 3x = 3 Lim x = 3(2) = 6
x→2 x→2

2. Nilai dari Lim (2x+4) adalah….
x→2
a. -2
b. 2
c. 4
d. 6
e. 8
Pembahasan:
Lim (2x+4) = 2(2) + 4 = 4 + 4 = 8
x→2
3. Nilai dari Lim [6x-2x] adalah….
x → 3
a. -6
b. 8
c. 12
d. 14
e. 16
Pembahasan 1: Lim [6x-2x] = Lim 4x = 4(3) = 12
x→3 x→3

Pembahasan 2: Lim [6x-2x] = Lim 6x – Lim 2x
x→3 x→3 x→3
= 6(3) – 2(3)
= 18 – 6 = 12

LIMIT FUNGSI BENTUK TAK TENTU
Limit fungsi bentuk 0
0
Jika f(x) = (x-a).h(x)
g(x) = (x-a).k(x)

Maka: Lim f(x) = Lim (x-a).h(x) = Lim h(x) = h(a)
x→a g(x) x→a (x-a).k(x) x→a k(x) k(a)

Limit Fungsi Bentuk ~
~
Jika diketahui limit tak hingga (~)

Sebagai berikut: Lim axn + bxn-1 + cxn-2 + …+ d = R
x→~ pxm + qxm-1 + rxm-2 + … + s
Maka:
1. R= 0 jika nm

Limit Fungsi Bentuk (~ - ~)

a. Lim √ ax +b - √ px +q = R
x→~
Maka: 1. R= ~ jika a>p
2. R= 0 jika a=p
3. R= -~ jika a

p
2. R = b-q jika a=p
2√a
3. R= -~ jika a

< m sehingga nilai R = 0 8. Nilai dari Lim 2x2 + 5x – 12 adalah…. x→-4 3x2 – 13x - 4 Pembahasan: Lim 2x2 + 5x – 12 x→-4 3x2 – 13x - 4 = Lim (2x – 3) (x – 4) x→-4 (3x + 1) (x – 4) = Lim (2x – 3) x→-4 (3x + 1) = 2(-4) – 3 = 11 3(-4 ) + 1 13 9. Nilai dari Lim 2x2 + 4x – 10 adalah…. x→~ 4x2 + 7 Pembahasan: Pangkat diatas = Pangkat dibawah Maka 2 = 1 4 2 LIMIT FUNGSI TRIGONOMETRI Rumus limit fungsi trigonometri 1. Lim x = 1 diperoleh lim sin x = 1 x→0 sin x x→0 x 2. Lim tan x = 1 diperoleh lim x = 1 x→0 x x→0 tan x Akibatnya : 1. lim sin ax = 1 x→0 ax 2. lim ax = 1 x→0 sin ax 3. lim tan ax = 1 x→0 ax 4. lim ax = 1 x→0 tan ax Contoh : 1. lim sin 3x = . lim 3 sin 3x = 3 lim sin 3x . = 3 . 1 = 3 x→0 2x x→0 2 3x 2 x→0 3x 2 2 2. lim 4x = . lim 4 5x = 4 lim 5x = 4 x→0 tan 5x x→0 5 tan 5x 5 x→0 tan x 5 3. lim sin 3x = lim 3 sin 3x . 7x = 3 lim sin 3x lim 7x x→0 tan 7x x→0 7 3x tan 7x 7 x→0 3x x→0 tan 7x = 3 . 1 . 1 7 = 3 7 4. lim 1 – cos 2x = lim 1 – ( 1 – 2 sin 2 x) x→0 3x2 x→0 3x2 = lim 2 sin 2x x→0 3x2 = 2 lim sin x 2 3 x→0 x2 III. Latihan Jawablah pertanyaan di bawah dengan benar 1. Nilai dari Lim x4 – 3x2 + 4x adalah…. x→0 2x3 – x2 - 2x 2. Nilai dari Lim x2 – 4 adalah…. x→2 x2 + x - 6 3. Nilai dari Lim 4x2 + 3x - 6 adalah …. x→~ 2x2 – 8x -1 4. Nilai dari Lim √ 4x2 – 2x + 6 - √ 4x2 + 2x -1 adalah…. x→~ 5. Nilai dari Lim (8x – 2)2 adalah…. x→~ (4x + 1)2 6. Nilai dari Lim x2 – x adalah…. x→0 x2 + 2x 7. Nilai dari Lim 6x3 - 4x2 + 2x – 1 adalah…. x→~ 3x4 – 2x3 + 5x + 2 8. Nilai dari Lim 2x2 + 5x – 12 adalah…. x→-4 3x2 – 13x - 4 9. Nilai dari Lim 2x2 + 4x – 10 adalah…. x→~ 4x2 + 7 10. lim 1 – cos x = … x→0 x tan x 11. lim 4 x cot x adalah … x→0 3 12. lim sin (a + x) – sin (a – x ) adalah … x→0 x IV. . Tes Formatif ( Terlampir) V. Daftar pustaka Tim penulis MGMP Matematika SMA kota Semarang, Matematika SMA / MA XI A IPA, ( Semarang : CV. Jabbaar Setia, 2008) Tim penyusun KREATIF Matematika, Matematika SMA/MA kelas XI IPA semester gasal, ( Klaten, Viva Pakarindo, 2007) Simangunsong Wilson, Matematika dasar, ( Jakarta: Erlangga, 2005) x
SIFAT-SIFAT LIMIT FUNGSI
Berapa teorema limit:
Bila Lim f(x) = A dan Lim g(x) = B
x → a x →a
Maka
1. Lim [k.f(x)] = k Lim f(x)
x→a x→a

= k. A

2. Lim [f(x)+g(x)] = Lim f(x) + Lim g(x)
x→a x→a x→a

= A + B

3. Lim [f(x) x g(x)]
x→a

= Lim f(x) x Lim g(x)
x→a x→a

= A x B

4. Lim f(x) Lim f(x)
x→a g(x) = x→a . = A
Lim g(x) B
x→a
n n n
5. Lim f(x). = Lim f(x) = A
x→a x→a
n n n
6. Lim √ f(x) = √ Lim f(x) = √ A
x→a x→a

Soal latihan:
1. Nilai dari Lim 3x adalah….
x→2
a. 1
b. 2
c. 3
d. 4
e. 6
Pembahasan 1: Lim 3x = 3(2) = 6
x→2
Pembahasan 2:Lim 3x = 3 Lim x = 3(2) = 6
x→2 x→2

2. Nilai dari Lim (2x+4) adalah….
x→2
a. -2
b. 2
c. 4
d. 6
e. 8
Pembahasan:
Lim (2x+4) = 2(2) + 4 = 4 + 4 = 8
x→2
3. Nilai dari Lim [6x-2x] adalah….
x → 3
a. -6
b. 8
c. 12
d. 14
e. 16
Pembahasan 1: Lim [6x-2x] = Lim 4x = 4(3) = 12
x→3 x→3

Pembahasan 2: Lim [6x-2x] = Lim 6x – Lim 2x
x→3 x→3 x→3
= 6(3) – 2(3)
= 18 – 6 = 12

LIMIT FUNGSI BENTUK TAK TENTU
Limit fungsi bentuk 0
0
Jika f(x) = (x-a).h(x)
g(x) = (x-a).k(x)

Maka: Lim f(x) = Lim (x-a).h(x) = Lim h(x) = h(a)
x→a g(x) x→a (x-a).k(x) x→a k(x) k(a)

Limit Fungsi Bentuk ~
~
Jika diketahui limit tak hingga (~)

Sebagai berikut: Lim axn + bxn-1 + cxn-2 + …+ d = R
x→~ pxm + qxm-1 + rxm-2 + … + s
Maka:
1. R= 0 jika nm

Limit Fungsi Bentuk (~ - ~)

a. Lim √ ax +b - √ px +q = R
x→~
Maka: 1. R= ~ jika a>p
2. R= 0 jika a=p
3. R= -~ jika a

p
2. R = b-q jika a=p
2√a
3. R= -~ jika a

< m sehingga nilai R = 0 8. Nilai dari Lim 2x2 + 5x – 12 adalah…. x→-4 3x2 – 13x - 4 Pembahasan: Lim 2x2 + 5x – 12 x→-4 3x2 – 13x - 4 = Lim (2x – 3) (x – 4) x→-4 (3x + 1) (x – 4) = Lim (2x – 3) x→-4 (3x + 1) = 2(-4) – 3 = 11 3(-4 ) + 1 13 9. Nilai dari Lim 2x2 + 4x – 10 adalah…. x→~ 4x2 + 7 Pembahasan: Pangkat diatas = Pangkat dibawah Maka 2 = 1 4 2 LIMIT FUNGSI TRIGONOMETRI Rumus limit fungsi trigonometri 1. Lim x = 1 diperoleh lim sin x = 1 x→0 sin x x→0 x 2. Lim tan x = 1 diperoleh lim x = 1 x→0 x x→0 tan x Akibatnya : 1. lim sin ax = 1 x→0 ax 2. lim ax = 1 x→0 sin ax 3. lim tan ax = 1 x→0 ax 4. lim ax = 1 x→0 tan ax Contoh : 1. lim sin 3x = . lim 3 sin 3x = 3 lim sin 3x . = 3 . 1 = 3 x→0 2x x→0 2 3x 2 x→0 3x 2 2 2. lim 4x = . lim 4 5x = 4 lim 5x = 4 x→0 tan 5x x→0 5 tan 5x 5 x→0 tan x 5 3. lim sin 3x = lim 3 sin 3x . 7x = 3 lim sin 3x lim 7x x→0 tan 7x x→0 7 3x tan 7x 7 x→0 3x x→0 tan 7x = 3 . 1 . 1 7 = 3 7 4. lim 1 – cos 2x = lim 1 – ( 1 – 2 sin 2 x) x→0 3x2 x→0 3x2 = lim 2 sin 2x x→0 3x2 = 2 lim sin x 2 3 x→0 x2 III. Latihan Jawablah pertanyaan di bawah dengan benar 1. Nilai dari Lim x4 – 3x2 + 4x adalah…. x→0 2x3 – x2 - 2x 2. Nilai

Active And Passive Voice

ACTIVE AND PASSIVE VOICE

Kalimat Aktif dan Kalimat Pasif

Kata kerja transitif mempunyai dua voice (ragam gramatikal), aktif dan pasif.

1) Bentuk aktif adalah orang, binatang, atau benda yang ditunjukkan oleh subjek dikatakan

melakukan sesuatu pada yang lain.

Contoh: Karim killed a tiger. Karim membunuh seekor harimau

2) Bentuk pasif adalah orang, binatang atau benda dikatakan menderita sesuatu dari sesuatu yang lain.

Contoh: A tiger was killed by Karim. Seekor harimau dibunuh oleh Karim

Bentuk pasif :

To Be + Past Participle

Aturan-aturan :

a) Kata kerja transitif tidak digunakan dalam bentuk pasif, kecuali kalau kata kerja itu menggunakan cognate object dalam bentuk aktif.

Aktif : She sang a fine song. Ia menyanyikan sebuah nyanyian yang merdu

Pasif : A fine song was sung by her. Sebuah nyanyian yang merdu dinyanyikan olehnya

b) Bilamana kalimat diubah dari bentuk aktif ke pasif, objek untuk kata kerja aktif menjadi subjek untuk kalimat kerja pasif.

objek untuk kata kerja aktif :

Aktif: Linda can make tarts. Linda dapat membuat kue tart

Subjek untuk kata kerja pasif :

Pasif: Tarts can be made by Linda

c) Retained object (objek yang tetap dipakai/dipertahankan dalam pasif)

Dua buah objek dalam kalimat aktif, ketika diubah menjadi kalimat pasif, masih tetap ada sebuah objek dipertahankan, objek ini dinamakan retained object. Objek ini mungkin objek tak langsung dari kata kerja aktif atau objek langsung dari kata kerja aktif.

Objek tak langsung dari kata kerja aktif

Kata Kerja aktif Kata kerja pasif

We gave him a prize A prize was given him by us

Objek langsung dari kata kerja aktif

Kata Kerja aktif Kata kerja pasif

We gave him a prize He was given a prize by us

Berikut contoh-contoh kalimat aktif yang dirubah menjadi kalimat pasif dalam bentuk tenses :

1) Simple present

Aktif

John bites Mary

John doesn’t bite Mary

Does John bite Mary?

What does John do?

Who bites Mary?

Who does John bite?



Pasif

Mary is bitten by John

Mary isn’t bitten by John

Is Mary bitten by John?

What is done by John?

Who is Mary bitten by?

Who is bitten by John?

2) Simple continuous

Aktif

John is biting Mary

John isn’t biting Mary

Is John biting Mary?

What is John doing?

Who is biting Mary?

Who is John biting?



Pasif

Mary is being bitten by John

Mary isn’t being bitten by John

Is Mary being bitten by John?

What is being done by John?

Who is Mary being bitten by?

Who is being bitten by John?

3) Present perfect

Aktif

John has bitten Mary

John hasn’t bitten Mary

Has John bitten Mary?

What has John done?

Who has bitten Mary?

Who has John bitten?



Pasif

Mary has been bitten by John

Mary hasn’t been bitten by John

Has Mary been bitten by John?

What has been done by John?

Who has Mary been bitten by?

Who has been bitten by John?

4) Present perfect continuous

Aktif

John has been biting Mary

John hasn’t been biting Mary

Has John been biting Mary?

What has John been doing?

Who has been biting Mary?

Who has John been biting?



Pasif

Mary has been being bitten by John

Mary hasn’t been being bitten by John

Has Mary been being bitten by John?

What has been being done by John?

Who has Mary been being bitten by?

Who has been being bitten by John?

5) Simple past

Aktif

John bit Mary

John didn’t bite Mary

Did John bite Mary?

What did John do?

Who bit Mary?

Who did John bite?


Pasif

Mary was bitten by John

Mary wasn’t bitten by John

Was Mary bitten by John?

What was done by John?

Who was Mary bitten by?

Who was bitten by John?

6) Past continuous

Aktif

John was biting Mary

John wasn’t biting Mary

Was John biting Mary?

What was John doing?

Who was biting Mary?

Who was John biting?



Pasif

Mary was being bitten by John

Mary wasn’t being bitten by John

Was Mary being bitten by John?

What was being done by John?

Who was Mary being bitten by?

Who was being bitten by John?

7) Past perfect

Aktif

John had bitten Mary

John hadn’t bitten Mary

Had John bitten Mary?

What had John done?

Who had bitten Mary?

Who had John bitten?



Pasif

Mary had been bitten by John

Mary hadn’t been bitten by John

Had Mary been bitten by John?

What had been done by John?

Who had Mary been bitten by?

Who had been bitten by John?

8) Past perfect continuous

Aktif

John had been biting Mary

John hadn’t been biting Mary

Had John been biting Mary?

What had John been doing?

Who had been biting Mary?

Who had John been biting?



Pasif

Mary had been being bitten by John

Mary hadn’t been being bitten by John

Had Mary been being bitten by John?

What had been being done by John?

Who had Mary been being bitten by?

Who had been being bitten by John?

9) Future

Aktif

John will bite Mary

John won’t bite Mary

Will John bite Mary?

What will John do?

Who will bite Mary?

Who will John bite?



Pasif

Mary will be bitten by John

Mary won’t be bitten by John

Will Mary be bitten by John?

What will be done by John?

Who will Mary be bitten by?

Who will be bitten by John?

10) Future continuous

Aktif

John will be biting Mary

John won’t be biting Mary

Will John be biting Mary?

What will John be doing?

Who will be biting Mary?

Who will John be biting?



Pasif

Mary will be being bitten by John

Mary won’t be being bitten by John

Will Mary be being bitten by John?

What will be being done by John?

Who will Mary be being bitten by?

Who will be being bitten by John?

11) Future perfect

Aktif

John will have bitten Mary

John won’t have bitten Mary

Will John have bitten Mary?

What will John have done?

Who will have bitten Mary?

Who will John have bitten?



Pasif

Mary will have been bitten by John

Mary won’t have been bitten by John

Will Mary have been bitten by John?

What will have been done by John?

Who will Mary have been bitten by?

Who will have been bitten by John?

12) Future perfect continuous

Aktif

John will have been biting Mary

John won’t have been biting Mary

Will John have been biting Mary?

What will John have been doing?

Who will have been biting Mary?

Who will John have been biting?



Pasif

Mary will have been being bitten by John

Mary won’t have been being bitten by John

Will Mary have been being bitten by John?

What will have been being done by John?

Who will Mary have been being bitten by?

Who will have been being bitten by John?

Kata-kata kerja transitif kadang-kadang mempunyai arti pasif walaupun bentuk kalimatnya adalah aktif :

a) Dengan komplemen

Sugar tastes sweet (pasif: sugar is sweet when it is tasted). Gula manis rasanya (gula manis bila

dirasakan)

b) Tanpa komplemen

The books is printing (pasif: the book is being printed). Buku itu sedang dicetak

The cows are milking (pasif: the cows are being milked). Sapi-sapi itu sedang diperah

Kesimpulan :

TENSES


ACTIVE


PASSIVE

Simple Present

Present Continuous

Present Perfect

Past Tense

Past Continuous

Simple Future

Be going to

Past perfect

Future perfect


Mary

Mary

Mary

Mary

Mary

Mary

Mary Mary

Mary


Helps

is helping

has helped

helped

was helping

will help

is going to help

had helped

will have helped


John

John

John

John

John

John

John

John

John


John

John

John

John

John

John

John

John

John


is helped

is being helped

has been helped

was helped

was being helped

will be helped

is going to be helped

had been helped

will have been helped


by Mary

by Mary

by Mary

by Mary

by Mary

by Mary

by Mary

by Mary

by Mary

Procedure, Narrative And Expressions

Procedure, Narrative And Expressions

Procedure

How to make Lemonade

Ingredients:

For each glass use:

- 2 tablespoons of lemon juice.

- 2 tablespoons of sugar.

- 1 glass of water.

Methods:
1. Slice a lemon in half and squeeze the juice into a cup.
2. Take out the seeds.
3. Pour two tablespoons of juice into glass.
4. Add sugar.
5. Add water and stir well.
6. Taste the lemonade. You may want to add more sugar or more lemon to make it taste just right.
7. Put it in ice cubes. A drop of red food coloring will make pink lemonade.
A CONE-SHAPED BASKET
Materials:
- Heavy paper
- Ribbon or string
- A plate
- A pencil
- Paste
Method:
1. Use a half circle of paper to make the cone basket.
2. Draw a whole circle on paper using a plate as the pattern to make a half circle.
3. Cut out the circle and fold it in half.
4. Cut the two halves apart along the fold.
5. Twist the half circle into a cone shape and it in place.
6. Use a ribbon or a string for the handle. Paste the ends of the ribbon in place.
7. Decorate your cone basket.
How to make Popcorn crunch
Materials:
- 1,5 cups of sugar wheat cereal
- 1 cup of golden syrup flaked almond
- 0,5 cup of butter
- 8 cup of popcorn already pop
- 2 cups of puffed
- 1 cup of toasted
- 1 teaspoon of vanilla
- 0,75 teaspoon of cinnamon
Time: 10 minutes
How to make it:
Place sugar and golden syrup in a heatproof dish, stir and cook until sugar is dissolved (approximately four minutes on high).
Add butter and cook for six minutes.
While this is cooking, place popcorn, puff wheat cereal, and almonds into a separate bowl.
Add cinnamon and vanilla to golden syrup mixture, combine syrup with popcorn, cereal and almonds and spread over a lighty greased 25 cetimeters x 30 centimeters baking tray.
Allow to cool and then cut into pieces.
Store in an airtight container.
HOW TO MAKE PEANUT CRUNCH
What you’ll need :
v 1 cup of peanuts
v 3 cups of brown sugar
v 2 tablespoons of vinegar
v 1 cup of water
What to do :
Place the sugar, water and vinegar into a large saucepan.
Stir slowly over a low heat until the sugar is disolved
Add peanuts , increase the heat and allow to boil
Remove from the heat when the nuts have craked and the mixture appears golden brown
Allow bubbles to settle
Spoon into small paper patty cases or pour the mixture into a flat greased pan and mark into bite-size pieces.
Recount
CLASS PICNIC
Last Friday our school went to Centennial Park for a picnic
First our teachers marked the rolls and the we got on the buses. On the buses, everyone was chatting and eating. When we arrived at the park, some students played cricket, some played cards but others went for a walk with the teachers. At lunchtime, we sat together and had our picnic. Finally, at two o’clock we left for school.
We had a great day.
3. Descriptive ‘ISSIS’ Cafe.
‘ISSIS’ is Javanese word meaning ‘cool’. So, besides the food, ISSIS Café offers a spacious, fully air-conditioned, cozy place.
Located at Jl. Cilacap No. 8, Jakarta Pusat, ISSIS Café is famous for its European food, especially steak, barbecue ribs, salad, and soup. You might find this kind of food anywhere else, but there is no other place that offers great meals at better prices than ISSIS Café.
You can enjoy a delicious imported sirloin steak for only Rp. 25,000 and ice cappuccino for only Rp. 5,500. There is also a salad bar with eight different vegetables. You can make your own salad which you can eat as much as you like, for only Rp. 12,000. The customers are mostly college students, office workers, and families.
4. News Item Undersea earthquake strikes off Maluku
JAKARTA (AP): A strong earthquake struck in eastern Indonesian waters on Tuesday, a meteorological agency said. A local official said there was no threat of a destructive tsunami, and no damage or causalities were immediately reported.
The quake, which had a preliminary magnitude of 6.1, was centered beneath the Banda Sea around 188 kilometers (117 miles) southwest of Ambon, the capital of Maluku province, the U.S. Geological Survey said on its Web site.
The tremor was not felt by residents in the region and there were no reports of damage or casualties, said Aprilianto, an official at a Jakarta-based local Meteorological and Geophysics Agency.
5. Report Australia
Australia is a large continent. It has six states and two territories.
The capital city of Australia is Canberra. It is in the Australian Capital Territory.
The population of Australia is about 20 million. The first inhabitants to live in Australia were Aboriginal people. After that people came from all over the world. The main language is English, however many other languages are spoken.
There are many plants and animals that are only found in Australia, e.g. kangaroos, platypuses, gum trees and Waratahs.
The main products and industries are wool, minerals, oil, coal, cereals and meat.
Some famous landmarks are the Harbour Bridge, the Opera House and Uluru (Ayers Rock).
2. Drugs
Drugs are chemical substances. There are three different types of drugs: stimulants, depressants and hallucinogens.
Stimulants speed up the central nervous system. They increase heart rate, blood pressure and breathing. Examples are caffeine, nicotine, amphetamines, ecstasy and cocaine.
Depressants slow down the central nervous system. They decrease heart and breathing rates. Alcohol, heroin and analgesics are common examples of these types of drugs.
Hallucinogens change mood, thought and senses. LSD is the most well-known example of this type of drug.
3. A traditional market
A traditional market is the type of market where people can bargain the prices. The items sold in traditional market are basically the same. They are fruits, vegetables, meat and fish, spices, dry good and household items. At the glances, the market may seem to be disorganized mess.
Surrounding the market there are many small scale traders, usually selling fruits. This traders can not afford the cost of renting a stall inside the market.
On the first floor of the market, there are permanent kiosks and stall selling textile, stationery, clothing, electronic goods, household appliances, gold shops, etc.
On the second floor, people can buy meat and fish, fruits, vegetables, and dry goods. The sellers sell fruits and vegetables through the middle area. Meanwhile they sell dry goods in the edge area of the second floor
6. Hortatory Exposition Good morning, ladies and gentlemen
Thank you very much for the House of Representatives which had invited me to give the speech concerning about the mystery sinetrons shown in many television presently. My name is Budi Santoso, a lecturer at University of Indonesia majoring in mass communication. Here, I represent the academics point of view about the subject we discuss this morning.
As we know, there are many mystery sinetrons shown on Television stations presently. The sinetrons depict horrible scenes about the spirit world. It is described that spirits often disturb people by frightening them in the darkness, when they walk alone at night or at the cemetery. The spirits are pictured as frightening appearance such as white clothes flying corpse, shattered face copse etc. the show must be abandoned for several reasons.
Firstly, it make a wrong perception of people especially children and women to do activity at nights, for example going to the wells, even cooking at kitchen alone. How do you fell if you always live in anxiety.
Thirdly, such kind of sinetrons waste out time to think unreal phenomena while we are facing many kinds of living problems.
In brief, for the reason, I think television station must stop showing mystery sinetrons. They are bad influences for people, frightening our children and destroying their belief to god.
Thank you very much for your attention.
7. Analytical Exposition SHOULD CHILDREN WEAR HATS AT SCHOOL?
(Statement of position):
I believe that you should always wear a hat at school when you are playing outside , to stop you from getting sunburn.
(Argument 1):
Firstly, if you don’t wear a hat, you will get sunburn ant the sunburn is painful.
(Argument 2):
Secondly, sunburn could lead to skin cancer. Sunburn can lead to health problems later in life. Many older people suffer from skin cancer which can kill them.
(Reinforcement of position statement):
In my opinion all school students should wear hats.
8. Spoof One day, two villagers went to Jakarta. They went to the biggest mall and saw shiny silver walls that could open and move apart and back together. They were amazed when an old lady rolled in to the small room and the doors closed. A minute later, the doors opened and a young beautiful lady stepped out. The father said to his son “Go, get your mother now.”
The ending of the story is funny because they thought that the doors can change an old lady into a young beautiful lady. Whereas the doors were actually elevator doors.
9. Explanation How to Fly a Hot Air Balloon
A hot air balloon consists of a basket, four big gas tanks, a burner and the balloon or ‘envelope’
First, four nylon poles are put into sockets on top of the basket. The burner is then put on top of the poles. Next, the cables are connected to the burner frame. The cables also go under the basket in order to hold everything together.
After this, the hoses from the full gas tanks must be connected to the burner so that pilot can test it.
Next, the mouth of the balloon is held open by two people while it is filled with cold air from the fan until it is quite fat and tight.
Now for the difficult bit. The pilot lies on the ground, half in the basket, turns on the gas burner, and points the flame into the ‘mouth’ of the balloon. This is so that the balloon slowly stands up.
When the it is ready to go, a bit more air is heated up the in the balloon. This results in the air in the balloon to be hot enough to get the balloon to rise off the ground
10. Discussion Euthanasia
Euthanasia is the act of intentionally causing the painless death of a sick person. In terms of a physician’s actions, it can be passive in that a physician plays no direct role in the death of the person or it can be active in that the physician does something directly to cause the death. Now the question: Do you think it is right for a physician to refuse to participate in active euthanasia?
It is NEVER right for a physician or any one else to deliberately hasten a person’s death. This includes all forms of euthanasia-active and passive. To deliberately withhold food/fluids is to subject the person to a painful death-it is NOT a humane death. We are not in a position to determine the worth of a life. Every person has a soul-it is up to God to determine when he will take that soul from the shell that is the human body. We all have a duty to support life with ordinary means-food/fluids but we are not required to use extrordinary means-aggressive life support,dialysis,etc indefinitely.
O.K. now, euthanasia (I think) is a type of suicide, whether or not you are sick. Physicians absolutley have the choice of NOT participating, in fact it should not be legal!!! Now if you put someone to death who was sick, or heck they dont even have to be sick, but you would look at your self everyday and be reminded of it. i bet you would feel like a murderer. If people want to die they can commit suicide in their own homes, makin someone else do it is not going to make it any better morally. Euthanasia is a sad, sad deal, please try to stop it.
I believe if the person wants to die to end their life they should have the right to choose.I do not believe that the person should beable to have themselves killed if they’re not really suffering. Yes the person does have the right to kill him or herself. BUt the authourity stands in the way of that. If this is such a free country why can’t anyone participate in EUTHANASIA?
If we have the right (at least in the U.S) to do whatever we want to with our lives, whether it be rich and famous or an alcoholic crack head, why is it that we can not decide when our life should end? It is ours, if we can kill the life that grows inside us, we should be allowed to also destroy the life that harbors others. Especially if that person is in pain and requests that some end their suffering. I don’t think you can allow one law with out the either. A life is a life, right?
http://www-hsc.usc.edu/~mbernste/ethics.euthanasia.htm
11. Review Get Married
Illustrating the current situation happening in Indonesia, Get Married presents the figures of unemployment. A few big-name celebrities show up in cameo roles. The movie tells about a true friendship of four youngsters Mae (Nirina Zubir), Guntoro (Desta ‘Club Eighties’), Eman (Aming), dan Beni (Ringgo Agus Rahman) who judge themselves as the most frustrated people in Indonesia. Soon, they turn out to be street kids and spend most of their times at street, bullying people who pass by.
.
Suddenly, it comes to a moment when Mae is persuaded to grant her parents’ wish to have a grandchild. Mae’s parents, (Meriam Bellina dan Jaja Mihardja) firmly state that Mae must get married in a little while. Soon, they are busy finding candidates who would marry their only daughter. However, along the process of finding the right one for Mae, the three male friends of Mae turn out to be brutal evaluators for the candidates. In the mean time, Mae falls badly in love with Rendy (Richard Kevin), a rich, handsome and kind-hearted man. Unfortunately Rendy, Mae, Guntoro, Eman, Beni are brought into a misunderstanding, and soon fights break out between the two groups of Mae and Rendy.
Written based on some of youngsters’ real-life brotherhood experiences—this story will stir you to your emotional core while bringing out your sense of brotherhood. There are a lot of little things and big things that make this movie worth watching. The story is good, the banter is great, the relationships between the characters are great, and it’s a fun time at the movies. While some of the jokes are amusing, some of the fights go on a few bit too long.
Adapted from: http://maliablog.wordpress.com
12. Narratives Snow Maiden
Once upon a time there lived a couple in a village. They had got married for a long time, but so far they did not have a baby yet. Every single minute they prayed to God, begged for a baby, but it never came true.
One day, they went to snow mountain. They made a girl from snow and they dressed her beautifully. When it got dark, they decided to go home and left the snow girl alone. The following morning, someone knocked the door. \”Any body home?\”, she said. The old woman inside opened the door and asked, \”Who are you?\” The girl said \”I\’m Snow Maiden, your daughter\”. The old woman was surprised and happy. \”Oh really? Thanks God! Come in, please!\”
Since that meeting, they lived happily. Snow Maiden was beautiful, kind, diligent and helpful. Her parents and all of her friends loved her very much One day, Snow Maiden played with her friends. They played fire. At first, Snow Maiden just looked at their play. Suddenly, her friends asked her to jump on the fire. Of course she refused it because one thing that made her afraid was the fire. It\’s because Snow Maiden was made of snow, so she should avoid the fire. But her friends kept on forcing her to jump on. Finally, she could not do anything then she did it. She jumped on the fire and she melted. Her friends was so sorry about this, they cried and cried hoping Snow Maiden could live again, but it was useless. Snow Maiden would not be back anymore.
Her mother tried to entertain Snow Maiden\’s friends and asked them to make a new Snow Maiden. They went to a snow mountain and started making it. They expected to have the new Snow Maiden. Days passed but their dreams never came true.
Malin Kundang
Once upon time, there was a boy named Malin Kundang. He lived with his mother in a very poor condition. They looked fire wood in the forest nearly to make a living. Malin Kundang was so unsatisfied with their bad luck. That’s why he decided to go to another city to look for a better life.
Not long afterwards, Malin asked his mother’s permission to go to a big city. He promised to come back soon with much money. His mother permitted him and always prayed for him. In his journey, Malin Kundang joined a merchant in a big ship.
Actually, Malin was a diligent boy. He worked hard to get much money and everything changed. He became a rich merchant. His business partner asked him to marry his daughter. Malin agreed. Then Malin and his business partner’s daughter got married. They had honeymoon and traveled all over the world.
Many years later, Malin’s ship anchored in his village. Many villagers welcome his arrival and admired his glorious ship. Furthermore, they wanted to see his success. Malin’s mother heard that her son had come back. She was very glad and eager to see him. With a happy smile in her lips, she went to the seashore to meet her son. Do you know what happened when she met Malin? He pretended that he didn’t know her. Of course she was very very sad and disappointed.
In her desperation she cried to God to punish Malin. She cursed Malin Kundang and his ship to be a stone. Since then, people can see the big stone in the beach.
Keterangan : Diolah dari berbagai sumber
5. Functional Skills
FUNCTIONAL SKILL
1. Offering Help or Things (menawarkan bantuan / menawarkan sesuatu)
Untuk menawarkan bantuan, dapat digunakan ungkapan-ungkapan berikut:
- May I help you?
- Can I help you?
- Could I help you?
- How can I be of assistance to you?
- How can I be of help to you?
- What can I help you - What can I do for you?
- How can I assist you?
- How can I help you?
- Let me help you?
- Do you want me to help you?
- Shall I …?
Cara memberi tawaran seperti menawarkan makanan atau minuman dalam bahasa Inggris lazimnya dengan menggunakan ungkapan Would you like…?, Would you care for …?, why don’t you have…?, How about having …? May I offer you …?
Contoh:
Tawaran Respon
- Would you like some bread? Yes, please.
- Would you care for some coffee? No, thanks. I don’t drink coffee.
- Why don’t you have some biscuit, please? Thanks, I’d love to.
Jawaban untuk menerima tawaran antara lain: Yes please, Sure, Why not, Ofcourse, Certainly, I’d love to, It’s a good idea, That’s great. Untuk menolak tawaran digunakan ungkapan seperti: No, thanks, Please don’t bother, I’d love to but…, That’s great but…
1. 2. Introducing (memperkenalkan)
Memperkenalkan Dri Memperkenalkan Orang Lain
- I’d like to introduce myself.
- My I introduce myself?
- Let me introduce myself!
- I want to introduce myself - I’d like you to meet … (nama)
- This is my friend/boss/etc…(nama)
- Have you met…(nama)?
- May I introduce you to …(nama/jabatan)
- Let me introduce you to ….
- I want to introduce you to ….
1. Inviting (mengundang/mengajak)

Undangan/Ajakan Menolak Menerima
- let’s + V1
- Why don’t we …?
- How about…?
- I’d like to invite you to…
- Would you like to…?
- I wonder if you’d like to - I’m sorry I can’t
- I’d like to but…
- I’m afraid I can’t
- No, let’s not. - I’d love to
- I’d like very much
- I’d be happy/glad to
accept
- Yes, I’d be delighted to.
- That’s good ide
1. Expressing Thanks (terimakasih)
Ungkapan Respon
Thank you
Thank you very much
Thanks.
Thank you very much for… (kata benda)
I’m grateful for…(kata benda/noun) You are welcome.
That’s all right
Not at all
Don’t mention it
Thet’s all right
Any time
1. Congratulations (ucapan selamat)
Ungkapan Respon
Congratulations
Congratulations on …
I’d like to congratulate you.
I’d like to congratulate you on…
It was great to hear…
It was to hear about….
Happy birthday to you.
Happy new year.
Good luck!
Have a nice holiday Thank you
Thank you and the same to you
Thank you. I need it.
Thank you very much.

1. Sympathy (menyatakan rasa simpati)
Ungkapan-ungkapan perasaan simpati atas mala petaka/musibah yang dialami orang lain diantaranya:
• I’m sorry to hear that
• Oh, that’s too bad.
• How awful!
• How terrible!
• Poor!
1. Pleasure, Displeasure (senang & tidak senang)
Pleasure/senang Displeasure/tidak senang
It’s really delightful/Iam delighted
I’m satisfied
That’s great
That’s wonderful
It’s really a great pleasure I’m dissatisfied
We are fed up with…
I feel dosappointed
She is extremely displeased

1. Satisfaction, Dissatisfaction ( kepuasan, ketidakpuasan)
Ketika kita akan mengungkapkan kepuasan atas kerja seseorang, kita dapat gunakan ungkapan:
• Well done!
• Great! Good work
• I am satisfied with your work
• You did well
• Your job is satisfactory
• I am so happy about this
• I’m glad to what you’ve done
• It’s really satisfying
Katika kita akan mengungkapkan ketidakpuasan atas kerja seseorang, kita dapat gunakan:
• I’m not satisfied with work
• You haven’t done well enough
• I am really dissappointed
• Sorry, but your work is not satisfactory
• Oh, no!
• It’s not very nice
• It’s really not good enough
1. Asking & Giving Opinion (meminta & memberi pendapat)
Asking Opinion Giving opinion
How was the trip?
How do you like your new house?
How do you think of Rina’s idea?
How do you feel about this dicition?
What is your opinions of the movie?
What are your feelings about it? I think (that)….
In my opinion….
As I see, …
If you ask me, I feel…

10. Agreement/approval, Disagreement/disapproval (setuju, tidak setuju)
Ketika kita merasa sependapat dengan opini orang lain, kita bisa mengatakan:
• So do I
• Yes, I agree with you
• It is certainly
• Exactly
• That’s what I want to say
• I am with you
• I am on your side
Ketika kita merasa tidak sependapat dengan opini orang lain, kita bisa mengatakan:
• Well, I don’t think so
• I don’t think that is true
• I disagree with …
• I wouldn’t say that
• Exactly not
• I can’t say so
• On contrary
• I don’t buy that idea
11. Fear, Anciety (ungkapan ketakutan, kegelisahan)
Fear Respon
I am afraid
I am feared
I am scared
I am terrified
The sound is horrifying Don’t be afraid
There is nothing to be afraid of
It is nothing

Anciety Respon
I am worried about…
I am anxious to know about…
I wondered if…
That made me worried
I have been thinking about ….
I am afraid if… Take is easy
Calm down
I know you are worried but…
It is not a big deal
Don’t worry
Stay cool
12. Pain, Relief (ungkapan kesakitan, kelegaan)
Pain Relief
Ouch!
That was hurt
It is painful
It hurts me
I’ve got a backache/toothache/stomachache
I feel sore all over
My eyes hurt I’m very relieved to hear…
Finally, it was over
I feel relieved
I feel much better
I’m glad it’s over
That’s a great relief
I’m extremely glad to hear…
Thank goodness for that
Marvellous
What a relief!
13. Like/Love & Dislike/Hate (suka/cinta & tidak suka/benci)
Like Dislike
I love it
I like it
I am keen on it
I am crazy about it
We all enjoy
(benda/noun/gerund)…is my cup of tea I don’t really like it
I dislike it
I am not really interested in…
I can’t enjoy…
(benda/noun/gerund)…is not my cup of tea
I can’t stand
I hate it
14. Embarrassment & Annoyance (Ungkapan rasa malu, kejengkelan)
Embarrassment Annoyance
I am embarrassed
I feel ashamed
Oh my God
Shame on me
I don’t feel comfortable
I feel awkward I am annoyed
I had enough with it
I can’t bear it any longer
You made me annoyed
You are such a pain in the neck
You made me sick
15. Request (permintaan)
Request Acceptance Refusal
Would it be possible for you to
Would you be so kind as to
Would you…,please?
Would you mind …?
Any chance of…
Can you…? I should be delighted to come
By all means
I have no objection
I’d be happy to
Sure
Yeah
OK
No problem
Mmm I regret to say that we find ourselves unable to go
I’m afraid it’s not possible
I’m afraid not
Sorry
No, I won’t
Not likely
You must be joking
16. Complaint, Blame (keluhan,menyalahkan)
Complaint Blame
I’m not at all satisfied with the service
I really do/must objec to the service
I take great exception to…
I want to complain about…
This is crazy! You’re the one to blame
It’s your fault!
It’s your mistake!
You’re wrong

17. Regret, Apology (penyesalan, meminta maaf)
Regret Apology
Much to my regret
Sadly, I ….
Unfortunately
I’m terribly sorry
I honestly regret that I …
Sorry, I … Please accept my apologies for what I did
Please forgive me for what I did
I am extremely sorry
I really must apologies
May I offer you my sincerest apologies?
18. Possibility & Impossibility (kemungkinan & ketidakmungkinan)
Menyatakan Kemungkinan Menanyakan Kemungkinan
I think there is possibility to …
I sassume/believe…
In all probability,…
it is going to be possible for me to…
that will probably …
it’s quite possible … Do you think he/it could…?
Would you say we’re capable of…?
Are you capable of…?
Are you able to…?
Do you have any experience of…?
Can you…?
Do you know how to…?
Do you think you can…?