Math 100 Final - Lagrange Error Bound and Newton's Method

I dont remember this at all uh oh

"What is the formula for the Lagrange Error Bound of an $n$-th degree Taylor Polynomial?"

"$|R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1}$"

"In the Lagrange Error Bound formula

what does $M$ represent?"

"If you want to reduce the error of a Taylor approximation

what are two things you can do?"

"What is the iterative formula for Newton's Method?"

"$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$"

"What is the geometric interpretation of Newton's Method?"

"It uses the x-intercept of the tangent line at the current guess to find the next

"When does Newton's Method fail immediately?"

"When the derivative at the current guess is zero ($f'(x_n) = 0$)

"What is the primary use of Newton's Method?"

"To approximate the roots (zeros) of a function where $f(x)=0$ cannot be solved algebraically."

"What is the Lagrange Error Bound formula?"

"|R_n(x)| \le \frac{M}{(n+1)!} |x - a|^{n+1}"

"In the Error Bound formula

what derivative do you use to find $M$?"

"What interval do you check when finding the maximum value $M$?"

"The interval between the center $a$ and the point of approximation $x$. ($[a

"If you are approximating $\sin(x)$ or $\cos(x)$

what is a safe (and standard) value to choose for $M$?"

"If the $(n+1)$-th derivative is strictly **increasing** on the interval $[a

x]$

"If the $(n+1)$-th derivative is strictly **decreasing** on the interval $[a

x]$

"Why does increasing the degree $n$ of the polynomial usually reduce the error?"

"Because the factorial in the denominator $(n+1)!$ grows extremely fast