Materi 1: Pengantar Metode Numerik
Summary
TLDRThis transcript is an introduction to numerical methods in mathematics, particularly useful when analytical methods fail. The speaker emphasizes the importance of self-study during the pandemic, encouraging students to utilize online resources. The difference between analytical methods, which yield precise answers, and numerical methods, which provide approximations, is explained. Using a quadratic equation as an example, the speaker contrasts how simple problems can be solved analytically while more complex problems require numerical methods. Students are encouraged to grasp this foundation before advancing to specific numerical techniques.
Takeaways
- 📚 Numerical methods are mathematical techniques used to solve problems that can't be solved analytically.
- 💡 Students need to be more independent in learning, especially during the pandemic, by utilizing online resources such as YouTube and websites.
- 🧮 Numerical methods involve approximations using decimal numbers, unlike analytical methods that provide exact solutions.
- 📖 Analytical methods, such as solving quadratic equations using factorization or the quadratic formula, are exact and often taught in high school.
- 🔄 When equations become more complex (involving algebra, trigonometry, or logarithms), analytical methods are not sufficient.
- ✖️ Numerical methods are useful when dealing with more complicated equations, like combinations of algebraic and transcendental equations.
- ❓ In cases where analytical methods fail, numerical methods are used to find approximate solutions to equations.
- 📐 The example of a quadratic equation shows that analytical methods can solve for roots exactly, but this becomes difficult with more complex problems.
- 🔢 Numerical methods provide approximate answers by calculating values iteratively, which is useful when exact solutions are hard to obtain.
- 📝 The video encourages students to review the material carefully and to make summaries or notes for better understanding.
Q & A
What is the main topic of the video?
-The video is an introduction to numerical methods, particularly their application in solving mathematical problems that cannot be solved analytically.
Why is independent learning emphasized in the video?
-Independent learning is emphasized due to the limitations of online conferences during the pandemic, such as high data usage and the complexity of learning mathematics through lectures. Students are encouraged to read books and use online resources to supplement their learning.
What is the difference between numerical methods and analytical methods?
-Analytical methods provide exact solutions using formulas and well-established rules, such as solving quadratic equations. Numerical methods, on the other hand, provide approximate solutions, often using iterative techniques and involving decimal numbers.
When are numerical methods used in mathematics?
-Numerical methods are used when analytical methods cannot be applied, especially when dealing with complex equations involving combinations of algebra, trigonometry, logarithms, and exponential functions.
Can you give an example of when an analytical method can be used?
-An example of an analytical method is solving a quadratic equation like x² - 5x + 6 = 0. This can be solved using factoring, completing the square, or the quadratic formula.
Why can't analytical methods always be applied to complex equations?
-Analytical methods cannot always be applied when equations involve a mixture of different mathematical elements, such as algebra combined with trigonometry or logarithms, making them too complex to solve using simple formulas.
What is meant by 'X pembuat nol' (X that makes zero)?
-'X pembuat nol' refers to the value(s) of X that, when substituted into the equation, make the equation equal to zero. These values are called the roots or solutions of the equation.
How does the example of solving x² - 5x + 6 = 0 illustrate analytical methods?
-In the example x² - 5x + 6 = 0, analytical methods such as factoring are used to find the solutions, which are x = 3 and x = 2. Substituting these values into the equation results in the equation equaling zero, confirming that they are the correct solutions.
What is a key characteristic of numerical methods?
-A key characteristic of numerical methods is that they provide approximate solutions involving many decimal places, unlike analytical methods which provide exact results.
What will be covered in the next part of the lesson according to the video?
-The next part of the lesson will cover specific numerical methods used to solve more complex equations, where analytical methods cannot be applied.
Outlines
📚 Introduction to Numerical Methods
The speaker begins by emphasizing the importance of self-directed learning, particularly during the pandemic, where students must rely on resources like books and online materials such as YouTube. The speaker discusses the limitations of online conferencing for learning mathematics, due to high data usage and the complexity of the subject. Numerical methods are introduced as mathematical techniques used to solve problems that cannot be addressed analytically. The speaker promises to explain these methods through video and direct writing, hoping the students will grasp the concepts.
🔢 Numerical Methods Explained
Numerical methods are described as approximations used to solve mathematical problems involving decimals, such as 1.33 or 0.25678, when analytic methods fail. The speaker explains that analytic methods, which students learned in high school, provide exact solutions, as in the case of solving quadratic equations. They recall the three methods used in high school—factoring, completing the square, and the ABC formula—to find the exact roots of a quadratic equation. However, for more complex problems involving multiple equations or a combination of algebra, trigonometry, and other elements, analytic methods become insufficient, and numerical methods must be employed.
Mindmap
Keywords
💡Numerical Methods
💡Analytical Methods
💡Approximation
💡Quadratic Equation
💡Factorization
💡Root-finding
💡Decimal Numbers
💡Exponential Functions
💡Combination of Functions
💡Self-directed Learning
Highlights
Introduction to Numerical Methods in the context of pandemic-induced self-learning for students.
Importance of self-directed learning and the need for students to engage with various learning materials.
Discussion on the limitations of learning through video conferences due to data consumption.
Introduction to the concept of Numerical Methods as a branch of mathematics.
Explanation of Numerical Methods as a means to solve mathematical problems that cannot be solved analytically.
Definition of 'approximation' in the context of Numerical Methods involving decimal numbers.
Conditions under which Numerical Methods are used when Analytical Methods fail.
Definition and examples of Analytical Methods learned in high school.
Illustration of how to solve a quadratic equation using Analytical Methods.
Comparison between the certainty of results from Analytical Methods versus the approximations from Numerical Methods.
Challenges in solving complex mathematical problems that combine algebra, trigonometry, and logarithms.
Explanation of 'root finding' or 'zero finding' in the context of solving equations.
Example of solving a specific equation to find the roots using factorization.
Mistake correction in the calculation during the example solution.
Final solution to the example equation, finding the roots x = 3 and x = 2.
Discussion on the limitations of Analytical Methods for complex problems and the necessity of Numerical Methods.
Preview of upcoming content on the forms and applications of Numerical Methods to solve complex equations.
Instruction to students to review and summarize the content for better understanding before proceeding.
Transcripts
Halo Bismillahirohmanirohim
Assalamualaikum warahmatullahi
wabarakatuh pertemuan kita pertama kali
ini tentang daftar metode numerik atau
pengantar metode numerik dalam berbagai
kesempatan
KMU untuk masa pandemi saat ini
sebenarnya kemandirian belajar mahasiswa
itu sangat dibutuhkan artinya anak-anak
ibu semua mahasiswa Matematika harus
mengupayakan diri untuk banyak membaca
buku-buku atau bahan-bahan materi
pembelajaran melalui internet baik itu
YouTube atau web dan seterusnya karena
kalau mengharapkan pembelajaran secara
conference dio konfrens itu satu akan
memakan banyak kuota yang kedua kalau
kita cuman ngoceh aja mungkin bisa pakai
video conference seperti ceramah Tapi
karena kita harus belajar matematika
Nia
Hai ada sulit sebenarnya karena itu
tujuannya untuk memahamkan kalian semua
tapi kita coba aja kali ini Ibu coba
dengan video
nulisan Ibu langsung semoga pahamnya Oke
jadi metode numerik itu sebenarnya
adalah metode matematika
Hai yang digunakan untuk menyelesaikan
persoalan matematika persoalan
matematika yang seperti apa soal
matematika yang tidak bisa diselesaikan
secara analitik Oke kita lanjut tadi
kita berhenti dulu karena adzan Nah jadi
sekarang coba lihat nah metode numerik
adalah metode penyelesaian matematika
yang bersifat hampiran nah Apa itu
hampiran hampiran itu dia melibatkan
banyak bilangan desimal seperti 1,3 33
atau 0,25 678 bla bla dan seterusnya
jadi dia melibatkan banyak sekali
bilangan desimal nah kapan metode menu
mereka itu dipakai Nah jadi metode
numerik itu dipakai saat metode analitik
tidak bisa dipakai nah Apa itu metode
analitik nah metode analitik itu metode
yang biasa kita pakai Sari
gitu jadi di SMA kita belajar banyak
materi salah-salah materi-materi itu
semua punya cara penyelesaiannya sendiri
misal trigonometri kita punya identitas
trigonometri kita punya rumus sin cos
tangen dan seterusnya semua yang sudah
kita pelajari di SMA itu semua disebut
dengan metode analitik karena dia punya
jawaban yang pasti nah misalnya nih
sekali gampang
ke dalam persamaan kuadrat nah dalam
persamaan kuadrat boleh dilihat untuk
soal seperti ini x kuadrat kurang 5 x +
6 = 0 itu seperti yang kalian pahami itu
waktu di SMA kita bisa menyelesaikannya
dengan tiga cara untuk mencari x1 dan x2
untuk mencari akar-akar penyelesaian
namanya jadi bisa pakai faktoran bisa
pakai kuadrat sempurna ataupun rumus abc
Nah jadi dengan menggunakan tiga cara
ini kita peroleh hasil yaitu x1 dan x2
kan gitu nah itu yang disebut dengan
cara my ataupun metode analitik nah
bagaimana kalau soalnya tidak seperti
ini ini kan satu jenis nih persamaan
kuadrat nah bagaimana kalau soalnya
banyak persamaan di dalam misalnya di
seperti ini ada aljabar x kuadrat ada 2x
Sin X gabungan aljabar dengan
trigonometri kemudian ada X
dan atau eh bukan logaritma jadi ada
tiga persamaan dalam satu soal terus ada
seperti soal yang minggu yang kemarin di
Bekasi e pangkat x kurang 4x itu
gabungan eh pangkat eksponen dengan
aljabar 4x ada juga e pangkat min x
dengan x Nah jadi ini adalah
Hai gabungan dari beberapa persamaan Nah
untuk soal-soal Seperti ini cara apa
yang bisa kita pakai untuk mencari x-nya
Hai atau untuk mencari yang disebut
dengan x pembuat nol
Hai nah gitu
yo Come kira-kira Oke kita lanjut tadi
ibu sebut X pembuat nol Apa itu X
membuat 03 pembuat nol itu mirip seperti
FX = 0 nah dalam hal ini seperti soal
kita ini FX kuadrat berarti soal FX
kuadrat ini ini kan soalnya fx = x
kuadrat kurang 5 x + 6 ya kan nah kita
pindahin kemarin jadi FX =
Hai x kuadrat kurang 5 x + 6 artinya
kalau ini adalah FX = 0 maka kita harus
mencari X berapa aja yang membuat
persamaan ini menjadi
Hai nah dalam hal ini kalau kita pakai
rumus pemfaktoran kita dapat X1 berapa
12.53 X
Hai kurang dua ya kan Tinggal kita
peroleh klik satu samadengan 3/5 nya
sama dengan dua saat kita substitusi ini
ke dalam persamaan berarti F3 =
Hai x kuadrat kurang 5 x + 6 maka disini
tiga kuadrat dikurang 5 dikali tiga
ditambah enam berapakah ini
a dikurang 10 ditambah enam berapakah
knapa nih 9 kurang 10
un1x salah ya
di tunggu tunggu tunggu
hai oke sama tadi nih 9 dikurang 15
ditambah enam berarti sama dengan nol
nah ini yang disebut dengan x pembuat
nol berarti F3 itu saat di TV3 itu saat
disubstitusikan ke persamaan aslinya x
kuadrat kurang 5 x + 6 membuat persamaan
ini menjadi nol itu yang disebut dengan
akar-akar
Hai penyelesaian
Hai jadi akar-akar Penyelesaian dari
soal ini adalah x = 3 dan x = 2 Nah itu
kalau soalnya persamaan kuadrat
Bagaimana kalau soalnya seperti ini
Hai Oke kita bisa dipikirkan kira-kira
Nah jadi ini yang disebut dengan metode
analitik yang hasilnya itu pasti jelas
rumusnya apa Nah kalau soalnya sudah
begini maka tidak bisa lagi menggunakan
metode analitik jadi metode yang dipakai
metode numerik nah sampai di sini paham
kira-kira seperti
ke-8 dipahami sekali lagi jadi setelah
paham ini kita akan lanjut ke
Hai metode numerik itu apa aja bentuknya
untuk menyelesaikan persamaan seperti
ini Nah nanti kita akan lanjut video ini
berhenti sampai di sini dulu silahkan di
cerna baik-baik silakan dibuat resumen
nya atau catatannya dengan baik kalau
sudah paham kita lanjut oke ini
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