from sympy import *
from IPython.display import display

x,y,z, t,n = symbols('x y z t n')

Limites

expr = x**2 + y**2

Limit(expr,x,-oo)

$\displaystyle \lim_{x \to -\infty}\left(x^{2} + y^{2}\right)$

display(Limit(expr,x,0,'-'))
print()
display(Limit(expr,x,0,'-').doit())

$\displaystyle \lim_{x \to 0^-}\left(x^{2} + y^{2}\right)$

$\displaystyle y^{2}$

limit(expr,y,2)

$\displaystyle x^{2} + 4$

display(Limit(sin(x)/x,x,0))
print()
display(Limit(sin(x)/x,x,0).doit())

$\displaystyle \lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$

$\displaystyle 1$

---

Expansão de Taylor

expr = sin(x)

expr

$\displaystyle \sin{\left(x \right)}$

expr.series(x)

$\displaystyle x - \frac{x^{3}}{6} + \frac{x^{5}}{120} + O\left(x^{6}\right)$

expr.series(x,x0=2)

$\displaystyle \sin{\left(2 \right)} + \left(x - 2\right) \cos{\left(2 \right)} - \frac{\left(x - 2\right)^{2} \sin{\left(2 \right)}}{2} - \frac{\left(x - 2\right)^{3} \cos{\left(2 \right)}}{6} + \frac{\left(x - 2\right)^{4} \sin{\left(2 \right)}}{24} + \frac{\left(x - 2\right)^{5} \cos{\left(2 \right)}}{120} + O\left(\left(x - 2\right)^{6}; x\rightarrow 2\right)$

expr.series(x,n=10)

$\displaystyle x - \frac{x^{3}}{6} + \frac{x^{5}}{120} - \frac{x^{7}}{5040} + \frac{x^{9}}{362880} + O\left(x^{10}\right)$

---

Derivadas

expr = x**5 + y**3 + z

diff(expr,y)

$\displaystyle 3 y^{2}$

Derivative(expr,x)

$\displaystyle \frac{\partial}{\partial x} \left(x^{5} + y^{3} + z\right)$

---

Derivative(expr,x,y,z)

$\displaystyle \frac{\partial^{3}}{\partial z\partial y\partial x} \left(x^{5} + y^{3} + z\right)$

---

Sentido da função

expr = x**2 + y**2 - 5

plotting.plot3d(expr)

<sympy.plotting.plot.Plot at 0x2abc83efd90>

display(maximum(expr,x))
display(maximum(expr,x,Interval(-2,2)))

$\displaystyle \infty$

$\displaystyle y^{2} - 1$

display(minimum(expr,x))
display(minimum(minimum(expr,y),x))

$\displaystyle y^{2} - 5$

$\displaystyle -5$

---

expr = x**2

plotting.plot(expr)

<sympy.plotting.plot.Plot at 0x2abcb700f70>

print(is_increasing(expr))
print(is_increasing(expr,Interval.open(0,oo)))
print(is_strictly_increasing(expr,Interval(0,oo)))
print(is_increasing(expr,Interval.Lopen(-2,2)))

False
True
False
False

print(is_decreasing(expr))
print(is_decreasing(expr,Interval.open(0,oo)))
print(is_strictly_decreasing(expr,Interval.open(-oo,0)))
print(is_decreasing(expr,Interval.Ropen(-2,2)))

False
False
True
False

print(is_monotonic(expr,Reals))
print(is_monotonic(expr,Interval(2,5)))
print(is_monotonic(expr,Interval(-2,5)))

False
True
False

---

expr = ln(x+1)

plotting.plot(expr)

singularities(expr,x)

$\displaystyle \left\{-1\right\}$

---

Cálculo de derivada

expr = x**5 + y**3 + z
deriv = Derivative(expr,x,2)

display(deriv)

print()

display(deriv.doit())

print()

display(deriv.doit().subs(x,-1))

$\displaystyle \frac{\partial^{2}}{\partial x^{2}} \left(x^{5} + y^{3} + z\right)$

$\displaystyle 20 x^{3}$

$\displaystyle -20$

---

diff(expr,x) → $\displaystyle 5 x^{4}$

diff(expr,x,x) → $\displaystyle 20 x^{3}$

diff(expr,x,x,x) → $\displaystyle 60 x^{2}$

diff(expr,x,x,x,x) → $\displaystyle 120 x$

---

diff(expr,x) → $\displaystyle 5 x^{4}$

diff(expr,x,2) → $\displaystyle 20 x^{3}$

diff(expr,x,3) → $\displaystyle 60 x^{2}$

diff(expr,x,4) → $\displaystyle 120 x$

---

Integrais

expr = sin(x) * cos(x)

Integral(expr)

$\displaystyle \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx$

Integral(expr,x,y)

$\displaystyle \iint \sin{\left(x \right)} \cos{\left(x \right)}\, dx\, dy$

Integral(expr,x,y,z)

$\displaystyle \iiint \sin{\left(x \right)} \cos{\left(x \right)}\, dx\, dy\, dz$

Integral(expr,(x,0,5))

$\displaystyle \int\limits_{0}^{5} \sin{\left(x \right)} \cos{\left(x \right)}\, dx$

Integral(expr,(y,-oo,oo))

$\displaystyle \int\limits_{-\infty}^{\infty} \sin{\left(x \right)} \cos{\left(x \right)}\, dy$

Integral(expr,(y,-oo,oo),(x,0,E))

$\displaystyle \int\limits_{0}^{e}\int\limits_{-\infty}^{\infty} \sin{\left(x \right)} \cos{\left(x \right)}\, dy\, dx$

---

Cálculo de integral

Integral(expr,(x,0,5)).doit()

$\displaystyle \frac{\sin^{2}{\left(5 \right)}}{2}$

integrate(x**5)

$\displaystyle \frac{x^{6}}{6}$

expr = Integral(x**2,(x,0,1))
expr.doit()

$\displaystyle \frac{1}{3}$

display(expr.as_sum())

print()

display(expr.as_sum(5,evaluate=False))

print()

display(expr.as_sum(5))

print()

display(expr.as_sum().expand())

$\displaystyle \frac{\frac{1}{4 n} - \frac{\frac{n^{2}}{2} + \frac{n}{2}}{n^{2}} + \frac{\frac{n^{3}}{3} + \frac{n^{2}}{2} + \frac{n}{6}}{n^{2}}}{n}$

$\displaystyle \frac{\sum_{k=1}^{5} \left(\frac{k}{5} - \frac{1}{10}\right)^{2}}{5}$

$\displaystyle \frac{33}{100}$

$\displaystyle \frac{1}{3} - \frac{1}{12 n^{2}}$

---

Integral de linha

C = Curve([E**t + 1, E**t - 1], (t, 0, ln(2)))
print('lenght:',C.length)
C

lenght: sqrt(2)

$\displaystyle Curve\left(\left( e^{t} + 1, \ e^{t} - 1\right), \left( t, \ 0, \ \log{\left(2 \right)}\right)\right)$

pl = plotting.plot_parametric(E**t + 1,E**t - 1, (t, 0, ln(2)),
                              xlim=(1.5,3),
                              ylim=(-1,2)
                             )

line_integrate(x + y, C, [x, y])

$\displaystyle 3 \sqrt{2}$