Exercise with assistance given 3. May:

PS: If the english here gives you trouble, please ask the overstudass (anderlin@stud.ntnu.no).

Problem 1

Exam TMA4130 Calculus 4N December 2003 Problem 6. In english, this is asking you to do one iteration with Newton's method, using the given initial values.

Problem 2

Given \(f(x)=e^{-x^2}\).

• a) Find a aproximation to the integral \(I=\int_{0.0}^{0.8} f(x) \mathrm{d}x\) by using the
  • i) Trapezoidal rule
  • ii )Simpson's rule
  Use \( h=0.2\) in both cases.
• b) How many equally large intervalls \(n\) does the trapezoidal rule need if the total error should not exceed \(10^{-5}\)?

Problem 3

To simulate the thermic properties of car brakes ("Bremseskive"), a numeric aproximation to the middle temperatur is given by

\( T = \frac{\int_{r_e}^{r_0} T(r)r \mathrm{d} r}{\int_{r_e}^{r_0}r \mathrm{d} r}\)

where \( T(r) \) is the temepratur at different locations (on "Bremsekloss"). \(r_e = 9.38\) cm and \( r_0 = 14.58\) cm. \( T(r) \) for some values of \(r\) is given in the following table. These values are found by numeric solutions of the heat equation.

\( r \)(cm)	\( T(r) \)(°C)
9.38	338
9.90	423
10.42	474
10.94	506
11.46	557
11.98	573
12.50	601
13.02	622
13.54	651
14.06	661
14.58	671

Use these values to find a approximation to the middle temperature \(T\) (on "Bremsekloss").

Problem 4

Kreyszig 8. edition, 18.1: Problem 5 og 11

Problem 5

We shall look at iteratove methods for solving the system

\( \left[ {4\atop {1\atop 0}} {1\atop {4\atop 1}} {0\atop {1\atop 4}}\right] \left[ {u_1\atop {u_2\atop u_3}} \right] = \left[ {5\atop {6\atop 5}} \right] \)

• a) Do two iteration using the Jacobi method.
• b) Do two iterations using the Gauss Seidel method

Use in both cases the start vector \([0; 0; 0]^T\).

Deadline 9. May. Good luck!