Scientific Computing - Quiz 2 - Question 1,2

Problem:

Question 1:

Consider the function $f(t)=\cos(t)$. What is the error bound associated with expanding the function $f(t+\Delta t)$ about $t$using $\Delta t=0.1$. In particular, compute the error bound when truncated at the second-derivative (keep only the first two significant-digits).

Question 2:

Consider the function $f(t)=\cos(t)$. What is the error bound associated with expanding the function $f(t+\Delta t)$ about $t$ using $\Delta t=0.1$. In particular, compute the error bound when truncated at the third-derivative (keep only the first two significant-digits).

Solution:

Consider the Taylor expansions:

For question 1:

$\cos(t + \Delta t) = \cos(t) - \Delta t \sin(t) - \frac{\Delta t^2}{2}\cos(c)$
So the error bound is the maximum value of the last term, which is $\frac{0.1^2}{2} = 0.005$

For question 2:

$\cos(t + \Delta t) = \cos(t) - \Delta t \sin(t) - \frac{\Delta t^2}{2}\cos(t) + \frac{\Delta t^3}{6}\sin(c)$
So the error bound is the maximum value of the last term, which is $\frac{0.1^3}{6} = 0.00017$