# Fall 2008 Final

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## 1Question 1¶

### 1.1Solution¶

#### 1.1.1Part (a)¶

State fixed point theorem

see htsefp

Show fixed point

see htsefp

#### 1.1.3Part (b) (ii)¶

Show that this scheme satisfies conditions for FPT

see htsefp

#### 1.1.4Part (b) (iii)¶

Order of convergence?

see htsefp

#### 1.1.5Part (b) (iv)¶

Compute $x_n$

see htsefp

Relative error?

see htsefp

## 2Question 2¶

### 2.1Solution¶

#### 2.1.1Part (a)¶

Zeroth divided diff

see htsefp

#### 2.1.2Part (b)¶

Define interpolating polys in Newton form

see htsefp

#### 2.1.3Part (c)¶

Show existence of $\xi$ etc

see htsefp

#### 2.1.4Part (d)¶

Construct table of divided diffs, find interpolating polys

see htsefp

## 3Question 3¶

### 3.1Solution¶

#### 3.1.1Part (a)¶

Use Taylor series to show something

see htsefp

#### 3.1.2Part (b)¶

Use Richardson extrap to find some formula

see htsefp

#### 3.1.3Part (c)¶

Find some approximation using Richardson extrap formula above

see htsefp

## 4Question 4¶

### 4.1Solution¶

#### 4.1.1Part (a)¶

Define deg of acc

see htsefp

Find deg of acc

see htsefp

#### 4.1.3Part (c)¶

Find constant $k$

see htsefp

#### 4.1.4Part (d)¶

Approximate $\ln(2)$

see htsefp

## 5Question 5¶

Simpson's rule

### 5.1Solution¶

#### 5.1.1Part (a)¶

State composite rule

see htsefp

#### 5.1.2Part (b)¶

Show composite rule satisfies something

see htsefp

#### 5.1.3Part (c) (i)¶

Evaluate $I_h(f)$ for some $h$

see htsefp

#### 5.1.4Part (c) (ii)¶

Value of $h$ required to get a certain deg of precision?

see htsefp

## 6Question 6¶

Runge-Kutta

### 6.1Solution¶

#### 6.1.1Part (a)¶

Define/find local truncation error, find order of method

see htsefp

#### 6.1.2Part (b) (i)¶

Show that $w_{i+1}$ satisfies some identity

see htsefp

#### 6.1.3Part (b) (ii)¶

Conditions on $h$ to get $\displaystyle \lim_{i \to \infty} w_i = 0$?

see htsefp

## 7Question 7¶

Linear shooting stuff, skipping