Numerical Applications

Maple

1. Write a function to convert degrees into radians. Test it with the following values:

   30, 45, 90, 180, 720, 1 degree

2. Find the first three derivatives of the following functions:
   
   $$x^3 - 3x^2 +3$$
   
   
   
   
   $$\sin(x) e^{-2x^2}$$
   
   
   
   
   $$\frac{3x^5-4}{\sqrt{1-x^2}} \tan(x)$$
   
   
   and integrate the results again.

3. Compute the taylor series of the function
   
   $$\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x exp(-t^2) dt$$
   
   
   up to the order $N=8$.
   Plot $\mathrm{erf}(x)$ and the approximating taylor ploynoms.

Matlab

1. Write a function to convert degrees into radians. Test it with the following values:

   30, 45, 90, 180, 720, 1 degree

2. Create a vector that contains the values of the function
   
   $$f(x) = \sin(x) e^{-0.1x^2}$$
   
   
   in the interval -10 .. 10, and plot it. Create a plot of the subintervall -1 .. 1.

   Repeat it with the function
   

   $$g(x) = \sin(\frac{1}{x})$$
   
   
   and have a closer look at the region around 0.

3. The phobos image contains three neighbouring columns, which are brighter than their surroundings. Find them and correct them by scaling them with an approprate factor.
   Create and plot a large image made up of the following four images:

   -

   phobos with bad spots,

   -

   bad spots,

   -

   phobos without bad spots,

   -

   phobos with corrected line.

   Start with the original image.