The Lagrange error bound practice problems
Bound the error of a Taylor polynomial with M|x − a|ⁿ⁺¹/(n + 1)!, or find the degree that makes it small enough. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle \sin{\left(x \right)} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{1000} \) at \( \displaystyle x = \frac{1}{2} \)?
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle e^{x} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{10000} \) at \( \displaystyle x = \frac{1}{10} \)?
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle e^{x} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{10000} \) at \( \displaystyle x = \frac{1}{5} \)?
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle \cos{\left(x \right)} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{1000000} \) at \( \displaystyle x = 1 \)?
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle e^{x} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{1000000} \) at \( \displaystyle x = \frac{1}{5} \)?
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle e^{x} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{10000} \) at \( \displaystyle x = \frac{1}{2} \)?
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle \sin{\left(x \right)} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{1000} \) at \( \displaystyle x = \frac{1}{10} \)?
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle \cos{\left(x \right)} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{10000} \) at \( \displaystyle x = \frac{1}{5} \)?
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle \sin{\left(x \right)} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{1000000} \) at \( \displaystyle x = \frac{1}{5} \)?
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle \cos{\left(x \right)} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{100000} \) at \( \displaystyle x = 1 \)?