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Interpolation and Taylor approximation

Fitting polynomials through points, and approximating functions near a centre.

Through n + 1 points passes exactly one polynomial of degree n (Lagrange). Near a point, the Taylor polynomial matches the derivatives. Picture it: the dashed polynomial hugging the curve near the centre and peeling away. Think it: approximation theory asks for the best polynomial over a whole interval, Chebyshev, not Taylor.

ឧទាហរណ៍​ដែល​បាន​ធ្វើ · taylor series of e^x

Calculus · Taylor-Series

Taylor series of e^(x)

e^{x}

ជំហាន​ដោយ​ជំហាន

  1. f(x) = e^{x}

    Taylor series about x = 0: f(x) = Σ fᵏ(x₀)/k! · (x − x₀)ᵏ.

  2. f^{(0)}(0) = e^{x}\big|_{x=0} = 1

    Derivative 0 at the centre.

  3. f^{(1)}(0) = e^{x}\big|_{x=0} = 1

    Derivative 1 at the centre.

  4. f^{(2)}(0) = e^{x}\big|_{x=0} = 1

    Derivative 2 at the centre.

  5. f^{(3)}(0) = e^{x}\big|_{x=0} = 1

    Derivative 3 at the centre.

  6. f^{(4)}(0) = e^{x}\big|_{x=0} = 1

    Derivative 4 at the centre.

  7. f^{(5)}(0) = e^{x}\big|_{x=0} = 1

    Derivative 5 at the centre.

  8. \frac{x^{5}}{120} + \frac{x^{4}}{24} + \frac{x^{3}}{6} + \frac{x^{2}}{2} + x + 1

    Assemble the terms up to degree 5.

បង្ហាញ​ចម្លើយ
\frac{x^{5}}{120} + \frac{x^{4}}{24} + \frac{x^{3}}{6} + \frac{x^{2}}{2} + x + 1 + O((x-0)^{6})

ឥឡូវ​អ្នក ជ្រើស​បញ្ហា ឬ​វាយ​បញ្ចូល ឬ​គូរ​របស់​អ្នក​ផ្ទាល់ ។ ជំហាន​នីមួយៗ រូបភាព ចម្លើយ​ត្រូវ​បាន​លាក់​រហូត​ដល់​អ្នក​សួរ ។

រក្សា​ការ​ធ្វើការ​របស់​អ្នក​ផ្ទាល់

គណនី​ឥត​គិត​ថ្លៃ​បន្ថែម​ចំណាំ​លើ​មេរៀន​នីមួយៗ កំណត់​ហេតុ​នៃ​អ្វី​ដែល​អ្នក​បាន​បញ្ចប់ បញ្ហា​ដែល​អ្នក​បាន​ដោះស្រាយ​នៅ​កន្លែង​មួយ និង​គ្រូ​ដែល​អ្នក​អាច​សួរ​អំពី​ទំព័រ​នេះ ។ គណិតវិទ្យា​ដោយ​ខ្លួន​វា​គឺ​បើក​ចំហ​សម្រាប់​អ្នក​ទាំងអស់​គ្នា ចុះឈ្មោះ​ចូល ឬ​មិន​ចុះឈ្មោះ​ចូល ។

ចុះឈ្មោះ ចូល

និមិត្ត​សញ្ញា​ដែល​បាន​ប្រើ​នៅ​ទីនេះ

ចុច​និមិត្ត​សញ្ញា​ណាមួយ​សម្រាប់​ការ​កំណត់​ពេញលេញ រូបភាព និង​តួអក្សរ​នីមួយៗ​មាន​ន័យ​អ្វី ។

របៀប​: Interpolation and Taylor approximation

  1. Taylor series about x = 0: f(x) = Σ fᵏ(x₀)/k! · (x − x₀)ᵏ.
  2. Derivative 0 at the centre.
  3. Derivative 1 at the centre.
  4. Derivative 2 at the centre.
  5. Derivative 3 at the centre.
  6. Derivative 4 at the centre.
  7. Derivative 5 at the centre.
  8. Assemble the terms up to degree 5.

សំណួរ​ដែល​មនុស្ស​សួរ

Why not just solve exactly?

Most equations have no closed-form solution at all, and many that do are unusable in practice. A numerical method delivers as many correct digits as you need, and a good one tells you how many that is.

Why can Newton's method fail?

If it starts where the tangent is nearly flat it shoots far away; near a repeated root it slows to a crawl; and with several roots it may land on the wrong one. A bracketing method like bisection is slower but cannot fail.

បន្ថែម​ទៀត​ក្នុង Numerical Methods

Root finding: bisection and Newton's methodNumerical integration: trapezoid and SimpsonFloating point and error