Quiz Chapter 01.07: Taylor Series Revisited

1. The coefficient of the x^{5} term in the Maclaurin polynomial for \sin \left(2x\right) is

2. Given f \left( 3 \right) = 6, {f}' \left( 3 \right) = 8, and {f}'' \left( 3 \right) = 11, and that all other higher order derivatives of f \left( x \right) are zero at x=3, assuming the function and all its derivatives exist and are continuous between x=3 and x=7, the value of f \left( 7 \right) is

3. Given that y \left( x \right) is the solution to \dfrac{dy}{dx}=y^{3}+2, y\left( 0 \right) = 3, the value of y\left( 0.2 \right) from a second order Taylor polynomial is

4. The series

• • • \, \displaystyle\sum_{n=0}^{\infty}\left( -1 \right)^{n} \dfrac{x^{2n}}{\left( 2n \right)!}4^{n}

is a Maclaurin series for the following function

5. The function

• • • erf\left( x \right) = \dfrac{2}{\sqrt{\pi}} \displaystyle\int_{0}^{x} e^{-t^{2}}dt

Is called the error function. It is used in the field of probability and cannot be calculated exactly for finite values of x. However, one can expand the integrand as a Taylor polynomial and conduct integration. The approximate value of erf using the first three terms of the Taylor series around t=0 is

6. Using the remainder of Maclaurin polynomial of n^{th} order for f \left( x \right) defined as

• • • R_n\left ( x \right ) = \dfrac{x^{n+1}}{\left ( n+1 \right )!}f^{\left ( n+1 \right )}\left ( c \right ), n \geq 0, 0 \leq c \leq x

the least order of the Maclaurin polynomial required to get an absolute true error of at most 10^{-6} in the calculation of \sin \left( 0.1 \right) is (do not use the exact value of \sin \left( 0.1 \right) or \cos \left( 0.1 \right) to find the answer, but the knowledge that \left | \sin \left( x \right) \right | \leq 1 and \left | \cos \left( x \right) \right | \leq 1).