# Sage

## Server

You can use this, or install it on your own from here: http://www.sagemath.org/

### 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?

### Symbolic calculations

1. Is it true that the square root of the square of a number is the number itself?
1. Use bool to convert to logical value
2. Is it true for real numbers? Positive numbers? (assume)
2. Prove (x-y)(x+y) == x^2-y^2
3. Prove (-1)^(2n) == 1 where n is integer!

### Sage functions, methods

1. Is 2018 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.