Maclaurin Polynomial
Definition
A Maclaurin polynomial is a type of Taylor polynomial that is centered at the point x = 0. It is used to approximate a function by adding up terms of different powers of x.
Analogy
Think of a Maclaurin polynomial as building blocks for approximating a function. Each term in the polynomial represents a different size block, and when you stack them together, they form an approximation of the original function.
Related terms
Taylor Series: A Taylor series is an infinite sum of terms that represent the values of all derivatives of a function at a given point. It can be used to approximate functions around any point, not just x = 0.
Power Series: A power series is an infinite sum of terms where each term contains powers of x multiplied by coefficients. It can be used to represent functions as well as approximate them.
Remainder Term: The remainder term in a Taylor or Maclaurin polynomial represents the difference between the actual value of the function and its approximation using only a finite number of terms from the series.
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Practice Questions (9)
- The third Maclaurin polynomial for sin š„ is given by p(x) = x - x^3/3!. If this polynomial is used to approximate sin(0.1), what is the Lagrange error bound?
- What is the maximum error of a 4th degree Maclaurin polynomial to approximate cos(Ļ/4)?
- Find the third-order Maclaurin polynomial for $e^{5x}$.
- What are the first three terms of the Maclaurin polynomial for $cos(2\pi x)$?
- Find the first 3 non-zero terms of the Maclaurin polynomial for $ln(1+4x)$.
- What is the third-order Maclaurin polynomial for $\frac{1}{1+x}ln(1+x)$?
- Find the first three non-zero terms of the Maclaurin polynomial for $arctan(2x)$.
- What is the fourth-degree term of the Maclaurin polynomial for $cos^2(x)$?
- What is the seventh-degree term of the Maclaurin polynomial for $sin^3(x)$?
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