David's Sage Notebook Demo

Some basic calculations

(5/2+7/2)/(5/9)

54/5

sqrt(5)

sqrt(5)

sqrt(5.)

2.23606797749979

sin(pi/3)

sqrt(3)/2

show(sin(pi/3))

\frac{\sqrt{ 3 }}{2}

sin(10)

sin(10)

N(sin(10))

-0.544021110889370

N(sin(10),digits=50)

-0.54402111088936981340474766185137728168364301291622

Some solving

x = var('x')
solve(x^2 + 3*x + 2==0, x)

[x == -2, x == -1]

var('x, y')
solve([x+y==6, x-y==4], x, y)

[[x == 5, y == 1]]

x = var('x')
find_root(0.1*x + sin(x)==0, 0.01, 2*pi)

5.6792077963144036

u = var('u')
diff(sin(u),u)

cos(u)

x = var('x')
integrate(sin(x^2),x)

\frac{{\sqrt{ \pi } \left( {\left( {\sqrt{ 2 } i} + \sqrt{ 2 } \right) \text{erf} \left( \frac{{\left( {\sqrt{ 2 } i} + \sqrt{ 2 } \right) x}}{2} \right)} + {\left( {\sqrt{ 2 } i} - \sqrt{ 2 } \right) \text{erf} \left( \frac{{\left( {\sqrt{ 2 } i} - \sqrt{ 2 } \right) x}}{2} \right)} \right)}}{8}

t = var('t')
x = function('x',t)
desolve(diff(x,t) == -x + 1,[x,t])

{{e}^{-t} \left( {e}^{t} + c \right)}

Graphing

x = var('x')
c = plot(cos(x),(x,-5,5),rgbcolor=(0,1,0))
s = plot(sin(x),(x,-5,5),rgbcolor=(1,0,0))
show(c+s)

var('x')
parametric_plot((cos(5*x),sin(3*x)),0,2*pi)

Interaction

@interact
def _(n=(-5,5,.25)):
    x = var('x')
    c = plot(3*sin(x),(x,-5,5),rgbcolor=(0,0,1))
    L = plot(3*cos(n)*(x-n)+3*sin(n),(x,-5,5),rgbcolor=(1,0,0))
    p = circle((n,3*sin(n)),.1,rgbcolor=(0,1,0))
    show(c+L+p,ymin=-3,ymax=3)

@interact
def _(n=(1,(1..10))):
    show(expand((x+1)^n))

html('<h2>Tangent line grapher</h2>')
@interact
def tangent_line(f = input_box(default=sin(x)), xbegin = slider(0,10,1/10,0), xend = slider(0,10,1/10,10), x0 = slider(0, 1, 1/100, 1/2)):
    prange = [xbegin, xend]
    x0i = xbegin + x0*(xend-xbegin)
    var('x')
    df = diff(f)
    tanf = f(x0i) + df(x0i)*(x-x0i)
    fplot = plot(f, prange[0], prange[1])
    print 'Tangent line is y = ' + tanf._repr_()
    tanplot = plot(tanf, prange[0], prange[1], rgbcolor = (1,0,0))
    fmax = f.find_maximum_on_interval(prange[0], prange[1])[0]
    fmin = f.find_minimum_on_interval(prange[0], prange[1])[0]
    show(fplot + tanplot, xmin = prange[0], xmax = prange[1], ymax = fmax, ymin = fmin)