2016-04-25.sagews

Math 480: Open Source Mathematical Software

2016-04-25

William Stein

Lectures 13: Symbolic Calculus (part 1 of 3)

Today:

1. (John Jeng) Update on peer grading (due Friday at 6pm)

2. Turn on screen cast

3. New homework assignment (due Friday at 6pm)

4. Today: symbolic calculus (part 1 of 3)

Some Resources for Symbolic Calculus in Sage

• Basic Sage Calculus Tutorial (written by a UW undergrad): http://www.sagemath.org/calctut/

• Sage for Undergraduates has a TON: http://www.gregorybard.com/Sage.html

• Sage reference manual: http://doc.sagemath.org/html/en/reference/calculus/index.html

• Sage Calculus "thematic tutorial": http://doc.sagemath.org/html/en/prep/Calculus.html

Short crash course

• defining symbolic variables

• defining symbolic functions

• plot

• differente

• integrate

• finding zeros

reset()

y + theta

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'y' is not defined

# x is predefined; can define other variables...

%var y, theta, aleph

show(x+y+theta +aleph)

$\displaystyle \mathit{aleph} + \theta + x + y$

var('y, theta')

(y, theta)

f(z, william) = z*(z+1)*william
f(10)

110*william

z + william

william + z

type(f)

<type 'sage.symbolic.expression.Expression'>

f.parent()

Callable function ring with argument z

f * f

z |--> (z + 1)^2*z^2

show(f)

$\displaystyle z \ {\mapsto}\ {\left(z + 1\right)} z$

# GOTCHA!
z = .5
f(z) = z*(z+1)  # this line whacks the previous defn of z!!
print z

z

z = .5
def f(z):
    print z
print z

0.500000000000000

f(z) = z*(z+1)
g = plot(f, (0, 3), color='chartreuse', thickness=10, zorder=10)  # has a bazillion options
show(g + text('hi there class!', (2,6), zorder=11))

diff(x^5 + 2*x + pi, x)

5*x^4 + 2

f = sin(x)*cos(x)*tan(x)
g = integrate(f, x)
show(g)

$\displaystyle \frac{1}{2} \, x - \frac{1}{4} \, \sin\left(2 \, x\right)$

h = g.differentiate(x)
show(h)

$\displaystyle -\frac{1}{2} \, \cos\left(2 \, x\right) + \frac{1}{2}$

k = h - f
show(k)

$\displaystyle -\cos\left(x\right) \sin\left(x\right) \tan\left(x\right) - \frac{1}{2} \, \cos\left(2 \, x\right) + \frac{1}{2}$

k

-cos(x)*sin(x)*tan(x) - 1/2*cos(2*x) + 1/2

plot(k, 0, 1)

k.si
︠228efa99-f7b6-47c7-9774-eb894d2acffes︠
k.simplify_full()

0

sin(x).find_root(2,5)

3.1415926535898335

N(sin(1/2))

0.479425538604203

complex_plot(x^2, (-1,1), (-1,1))