Quiz Chapter 01.07: Taylor Series Revisited

 MULTIPLE CHOICE TEST TAYLOR SERIES INTRO TO SCIENTIFIC COMPUTING

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 $\left | \cos \left( x \right) \right | \leq 1$).