2319-Basic examples of Sage.sagews

Basic examples of Sage

Multiplying Two Numbers

$\displaystyle 24536*5768$

24536*5768

141523648

Evaluating Sine

$\displaystyle \sin \frac {\pi}2$

sin(pi/2)

1

Evaluating Natural Logarithm

$\displaystyle \ln 2$

log(2)

log(2)

Finding a Numerical Approximation to Natural Logarithm

$\displaystyle \ln 2$

n(log(2))

0.693147180559945

Finding a Limit

$\displaystyle \lim_{x \to 0} \frac {\sin x}x$

limit(sin(x)/x, x=0)

1

Evaluating a Sum

$\displaystyle \sum_1^5 \sqrt{n^2 +1}$

sum(sqrt(n^2 +1) for n in [1..5])

sqrt(2) + sqrt(5) + sqrt(10) + sqrt(17) + sqrt(26)

Finding a Numerical Approximation to a Sum

$\displaystyle \sum_1^5 \sqrt{n^2 +1}$

n(sum(sqrt(n^2 +1) for n in [1..5]))

16.0346843392517

Finding the Derivative of a Function of One Variable

$\displaystyle \frac d{dx} [\sin (x^2 + 6x - 2)]$

diff(sin(x^2 + 6*x - 2), x)

2*(x + 3)*cos(x^2 + 6*x - 2)

Finding the Indefinite Integral of a Function of One Variable

$\displaystyle \int \ln x\, dx$

integral(ln(x), x)

x*log(x) - x

Finding the Definite Integral of a Function of One Variable

$\displaystyle \int_3^5 \ln x\, dx$

integral(ln(x), x, 3, 5)

-3*log(3) + 5*log(5) - 2

Finding a Numerical Approximation to a Definite Integral of a Function of One Variable

$\displaystyle \int_3^5 \ln x\, dx$

n(integral(ln(x), x, 3, 5))

2.75135269616617

Plotting the Graph of a Function of One Variable

$\displaystyle f(x) = 4x - 6x^{\frac 23} + 2$

x = var('x')
plot(4*x - 6*((x^2)^(1/3)) + 2, (x,-5,5))

Plotting the Graphs of Two Functions of One Variable

The graph of $\displaystyle f(x) = x^2$ is in blue.

The graph of $\displaystyle g(x) = - \cos x$ is in red.

x = var('x')
P = plot(x^2, (x,-5,5))
Q = plot(-cos(x), (x,-5,5), color='red')
P + Q

Plotting the Graph of a Function of Two Variables

$\displaystyle f(x, y) = x^2 + \frac {y^2}9$

x, y = var('x,y')
plot3d(x^2 + (1/9)*y^2, (x,-4,4), (y,-6,6), aspect_ratio=1)

Plotting the Graphs of Two Functions of Two Variables

The graph of $\displaystyle f(x, y) = x^2 + y^2$ is in blue.

The graph of $\displaystyle g(x, y) = 2x + 3y$ is in green.

x, y = var('x,y')
P = plot3d(2*x + 3*y, (x,-2,2), (y,-2,2), color='green')
Q = plot3d(x^2 + y^2, (x,-2,2), (y,-2,2))
P + Q

Plotting a Surface in $\mathbb{R}^3$

$\displaystyle \frac {x^2}9 - \frac {y^2}4 + 2z = 1$

x, y, z = var('x,y,z')
implicit_plot3d((1/9)*x^2 - (1/4)*y^2 + 2*z==1, (x, -30, 30), (y, -30,30), (z, -30,30), aspect_ratio=1)

Plotting a Function in Polar Coordinate

$r = 1 + \cos\theta$

polar_plot(1 + cos(x), (x, 0, 2*pi), aspect_ratio=1, color='red')

Plotting a Vector-Valued Function in $\mathbb{R}^3$

$\displaystyle r(t) = (\cos t, \sin t, t)$

t = var('t')
parametric_plot3d((cos(t), sin(t), t), (t,0,4*pi), color='purple', aspect_ratio=1)