User:Egm6341.s11.team1.arm/HW4

Problem 9
If $$\displaystyle G(t)$$ is given by Eq9.2

$$\displaystyle e(t)$$ is defined as

Therefore,

and

By Rolle's Theorem, this means that there exists a point $$\displaystyle \xi_1$$ such that

Looking at the $$\displaystyle G^{(1)}(t)$$,

As shown in Lecture 20 notes ,

Taking the derivative of $$ \displaystyle e(t) $$,

Since $$\displaystyle G^{(1)}(0) = 0$$ and $$\displaystyle G^{(1)}(\xi_1) = 0$$, by Rolle's Theorm there exists a point $$\displaystyle \xi_2$$ such that

Using the solution from Problem 4.3,

Since $$\displaystyle G^{(2)}(0) = 0$$ and $$\displaystyle G^{(2)}(\xi_2) = 0$$, by Rolle's Theorm there exists a point $$\displaystyle \xi_3$$ such that

Evaluating $$\displaystyle G^{(3)}(t)$$at $$\displaystyle \xi_3$$,

Using the solution from Problem 4.4,

And employing the Derivative Mean Value Theorem

Evaluating for $$\displaystyle e(1)$$, the proof breaks down because $$\displaystyle \xi_3$$ does not cancel out and remains unknown:

If $$\displaystyle G(t)$$ is given by Eq9.3

Analysis is similar to that of Eq9.4 - 9.17.

Eq9.17 then becomes:

The proof again breaks down when solving for $$\displaystyle e(1)$$ because $$\displaystyle \xi_3$$ does not cancel out:

Part B
Inserting Eq9.18 into the third derivative of Eq9.1,

Evaluating at $$\displaystyle t=0$$,

Since $$\displaystyle G^{(3)}(0)=0$$ and $$\displaystyle G^{(3)}(\xi_3)=0$$, by Rolle's Theorem, there exists a point $$\displaystyle \xi_4$$ such that

Evaluating the derivative,

Evaluating the derivative of Eq9.18,

Inserting into Eq9.26 and evaluating at $$\displaystyle t=0$$,

The proof can not be continued at this point since $$\displaystyle G^{(4)}(t)$$ has only one known zero because

Part 1
From Wolfram Alpha ,

Also, the asymptotic error estimate is defined for the trapezoidal rule in Atkinsion eq. 5.1.9 as:

Evaluating 10.2 with the appropriate derivatives for $$\displaystyle f$$,

Using the MATLAB code below, the following table was generated:

 Matlab code: 

Part 2
From Wolfram Alpha ,

Also, the asymptotic error estimate is defined for the trapezoidal rule in Atkinsion eq. 5.1.9 as:

Evaluating 10.2 with the appropriate derivatives for $$\displaystyle f$$,

Using the MATLAB code below, the following table was generated:

 Matlab code: 

Part 3
From Wolfram Alpha ,

Using the MATLAB code below, the following table was generated:

 Matlab code: