Exercise 1

1. Arithmetic operations:

a) 3^6/(3^6 - 1) compare with (1 - 1/3^6 )^-1
b) 2 √7 - 2 (√7 + 2)2 - 1
c) Area = πr2 with radius r = π^1/3 - 1

2. Exponential and logarithms:

a) e^3 , ln e^3 , log10 e^3 , log10(10^5)
b) e^π√163
c) solve 3^x = 17 for x and check the result (soluton x = ln 17 / ln 3 , check by substitution)

3. Trigonometry:

a) sin π/6 , cos π , tan π/2
b) sin^2 π/6 + cos^2 π/6
c) y = cosh^2 x - sinh^2 x for x = 32π

4. Complex numbers:

a) (3 + 2i)/( 3 - 2i) , check the result by hand calculation
b) e^i/π5 , check the Euler-formula eix = cos x + isin x by computing both sides of the equation
c) execute the commands exp(pi/2*i) and exp(pi/2i) explain the difference

Exercise 2

1. Equation of straight line:

Compute the y-coordinate of the straight line
y = mx + c
with slope m = 0.5 and the intercept c = -2 at the following x-coordinates
x = 0, 1.5, 3, 4, 5, 7

2. Multiply, divide, and exponentiate vectors:

Create a vector t with 10 elemts: 1, 2, 3, . . . , 10 and compute the following quantities:

x = tsin(t), y = (t - 1)/( t + 4) , z = sin(t^2)/t^2

3. Points on a circle:

All points with coordinates:

x = r cos θ , y = r sin θ , r = const

lie on a circle with radius r, i.e., they satisfy the equation
x^2 + y^2 = r^2

a) Create a column vector for θ with the elements

0, π/4, π/2, 3π/4, π, 5π/4

b) Take r = 2 and compute the column vectors x and y.

c) Check that x and y satisfy the equation of circle, by computing the radius

r = (x^2 + y^2)^1/2

4. Matrices and vectors::

a) Create a vector and a matrix with the following commands:

v=0:0.2:12;

M=[sin(v); cos(v)];

b) Find the size of v and M.

c) Extract the first 10 elenents of each row of the matrix, and display them as column vectors.

Exercise 3.

1. On-line help:
Type help function to get on-line help on function. Read through the help file.

2. Convert temperature:
Write a function that outputs a conversion-table for Celsius and Fahrenheit temperatures. The input of the function should be two numbers: Ti and Tf, specifying the lower and upper range of the table in Celsius. The output should be a two column matrix: the first column showing the temperature in Celsius from Ti to Tf in the increments of 1oC and the second column showing the corresponding temperatures in Fahrenheit.

a) create a column vector C from Ti to Tf

b) calculate the corresponding numbers in Fahrenheit: F= 9/5C + 32

c) make a final matrix temp=[C F]

3. Calculate factorials:

Write a function factorial to compute the factorial n! for an y integer n.The input should be the number n and the output n!. You might have to use a for loop or a while loop to do the calculation. (You can use the built-in function prod to calculate factorials. For example n!=prod(1:n). In this exercise use this function only for control.)

4. Compute the cross product:

Write a function file crossprod to compute the cross product of two vectors u, and v, given u = [u1, u2, u3], v = [v1, v2, v3], and u × v = [u2v3-u3v2, u3v1-u1v3, u1v2 -u2v1]. Check your function by taking cross products of pairs of unit vectors: (i,j), (i, k), etc. i = [1, 0, 0], j = [0, 1, 0], k = [0, 0, 1]. (Do not use the built-in function cross here.)

5. Sum

a geometric series: Write a function to compute the sum of the geometric series
1 + r + r2 + r3 + . . . + rn

for a given r and n. Thus the input to the function must be r and n and the output must be the sum of the series.

Exercise 4.

1. A simple sine plot:

Plot y = sin x, 0 ≤ x ≤ 2π, taking 100 linearly spaced points in the given intervall. Label the axis and put "Plot created by yourname" in the title.

2. Line-styles:

Make the same plot as before, but rather than displaying the graph as a curve, show the unconnected data points. Display the data points by small circles. Now combine the two plots in one command, to show the line through the data points as well as the distinct data points.

3. An exponentially decaying sine plot:
Plot y= e-0.4x sinx, 0 ≤x≤ 4π, taking 10, 50, and 100 points in the interval.

4. Space curves:

Use the command plot3(x,y,z) to plot the circular helix
x(t) = sin t,
y(t) = cost, z(t) = t, 0 ≤ x ≤ 20.

5. On-line-help:

Type help plot on the Matlab promt and hit return. If too much text flashes by the screen, type more on, hit return, and then type help plot again. This should give you paged output. Read through the on-line help. To move the next page of the screen output, simply press the space bar.

6. Log scale plots:

Create a vector x = 0 : 10 : 1000.
Plot x vs.x3 using the three log scale plot commands. Hint: First, compute y = x3 and then, use semilogx(x,y) etc.

7. Overlay plots:

Plot y = cos x and z = 1 - 1/2x^2 + 1/24x^4 for 0 ≤x≤π on the same plot.
Hint: You can use i) plot(x,y,x,z,':'), ii) hold on command after the first plot command . . . , iii) line(x,y,'...') after the first plot command . . .
Try the command legend('cos(x)','z').