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!