# Informatics1-2017/Practice10

## Using Sage

### Sage server

If your browser finds the certificate untrustworthy, accept it manually!

### Command line

On leibniz write sage into a command prompt, this starts the sage interactive shell.

Most sage commands work here, for example:

23^19

1. Calculate the squareroot of 2017!
2. Caltulate the 4th root of 2017!
3. Calculate the 2017^6!
4. What's the remaider of 123*321 divided by 11?

### Text completion and help

Sage can complete your commands, try the following:

V = Vec[press TAB]

Sage completes the command to Vector, and offers additional options. Complete it to match the following:

V = VectorSpace(QQ,3)

This way V will be the 3 dimensional vector space over the rational field.

Now write V. and press TAB. Now we can see all the possible operations with V. Writing a questionmark at the end of a command gives a detailed description of the command. For example:

V.basis?

Try it:

V.basis()

## Sage notebook

• Go to the sage notebook page: notebook
• After logging in again with the new password, you can open a new notebook with New Worksheet.
• Name it something like Practical10

### Introduction

• You can write any sage command in the cells, even multiple ones, for example:
A = Matrix([[1, 1], [1, 0]])
B = Matrix([[-2, 0], [-1, 1]])
• Run the commands with SHIFT + ENTER. The commands run one after the other.
• Try writing just A or B in a cell and run it. Then try A*B.

### Using variables

1. Let Y be your year of birth, M the month, and D the day, create these variables.
2. How much is Y divided by D? Associate this value with the b variable.
3. Let r be the remainder of Y / M.
4. What's the difference r - b?

### Sage functions, methods

1. Is 2017 a prime? (use the is_prime() function)
2. Were you born on a prime day? (use the D variable!)
3. Solve the equation D*x^2 + M*x - b*r = 0 using the solve(fv, variable) function! (x needs to be a symbolic variable!)
4. Solve the equation numerically! Use the find_root(fv == 0, min, max) function, where min and max defines an interval where Sage looks for the solution.
5. Solve the above equation symbolically (make D, M, b, r symbolic variables, then use solve)!
6. Differentiate the function sin(x)cos(x)x^2.
7. Integrate the previous function.
8. Calculate the limit of (1 + 3/n)^4n, if n->oo
9. Let f be the following function: f = (x+2*y)^3
10. Substitute 3 into x; then 4 into x and 2 into y. What's the result? ( use the subs() method of f)
11. Expand f! (expand())
12. Using the above, calculate the Taylor series of sin(x)cos(x)x^2 up until the 4th member. (you can differentiate and integrate a function f by f.diff(x))

### Plotting with Sage (plot)

1. Plot a cosine curve from 0 to 4*pi!
2. Plot the (x-2)^2 + 3 polynomial from -2 to 4, color it green!
3. Plot next to the previous one (using the show function) the function x^3-3*x + 6 in red!
4. Plot a circle: cirlce((coordinates of the center), radius, optional). The "optional" can be: color, aspect_ratio=True so that the ratio of the x and y axis are kept, otherwise we might get an ellipse.