Function Limits and Important Limits

Python facilitates the computation of mathematical concepts such as limits, derivatives, integrals, and differential equations. This section explores using smybolic mathematics to evaluate limits.

Example 1: Computing Left and Right Limits

Consider the function $ y = \arctan(1/x) $ at $ x = 0 $. We compute its left and right limits:

import matplotlib.pyplot as plt
import numpy as np
import sympy as sp

x = sp.Symbol('x')
expr = sp.atan(1 / x)
left_limit = sp.limit(expr, x, 0, dir='-')
right_limit = sp.limit(expr, x, 0, dir='+')
print(f"Left limit of {expr}: {left_limit}")
print(f"Right limit of {expr}: {right_limit}")

# Plotting the function
x_vals = np.arange(-6, 6, 0.01)
y_vals = np.arctan(1 / x_vals)
plt.plot(x_vals, y_vals)
plt.title('y = arctan(1/x)')
plt.show()

The output shows:

Left limit of atan(1/x): -pi/2
Right limit of atan(1/x): pi/2

Example 2: Validating Two Key Limits

We verify two fundamental limits using Python:

import matplotlib.pyplot as plt
import numpy as np
import sympy as sp

x = sp.Symbol('x')
f1 = sp.sin(x) / x
f2 = (1 + 1 / x) ** x
limit1 = sp.limit(f1, x, 0)
limit2 = sp.limit(f2, x, 'oo')
print(f"First limit {f1}: {limit1}")
print(f"Second limit {f2}: {limit2}")

# Plotting for visual confirmation
x1 = np.arange(-3, 3, 0.01)
x2 = np.arange(0.01, 100, 0.1)
y1 = np.sin(x1) / x1
y2 = (1 + 1 / x2) ** x2

plt.figure(figsize=(12, 5))
plt.subplot(121)
plt.plot(x1, y1)
plt.title('y = sin(x)/x')
plt.subplot(122)
plt.plot(x2, y2)
plt.title('y = (1+1/x)^x')
plt.show()

Results:

sin(x)/x First limit: 1
(1 + 1/x)**x Second limit: E

Derivatives and Differentiation

Example 1: Derivative and Tangent Line

Compute the derivative of $ f(x) = 2x^3 + 3x^2 - 12x + 7 $ and find the tangent line at $ x = -1 $:

import matplotlib.pyplot as plt
import numpy as np
import sympy as sp

x = sp.Symbol('x')
f = 2 * x ** 3 + 3 * x ** 2 - 12 * x + 7
f_prime = sp.diff(f)
print(f"Derivative of {f}: {f_prime}")

slope = f_prime.subs(x, -1)
y_value = f.subs(x, -1)
tangent_line = slope * (x + 1) + y_value
print(f"Tangent line equation: {tangent_line}")

# Plotting function and tangent
x_vals = np.arange(-4, 3, 0.01)
y_func = 2 * x_vals ** 3 + 3 * x_vals ** 2 - 12 * x_vals + 7
y_tangent = 8 - 12 * x_vals
plt.plot(x_vals, y_func, x_vals, y_tangent)
plt.title('Function and Tangent Line')
plt.show()

Output:

Derivative of 2*x**3 + 3*x**2 - 12*x + 7: 6*x**2 + 6*x - 12
Tangent line equation: 8 - 12*x

Example 2: Higher Order Derivatives

Calculate up to the 11th derivative of $ y = x^{10} + 2(x - 10)^9 $:

import sympy as sp

x = sp.Symbol('x')
y = x ** 10 + 2 * (x - 10) ** 9
for n in range(1, 12):
    y = sp.diff(y)
    print(f"{n}th derivative: {y}")

Example 3: Implicit Differentiation

Find the derivative of an implicit function defined by $ 2x^2 - 2xy + y^2 + x + 2y + 1 = 0 $:

import sympy as sp

x, y = sp.symbols('x y')
z = 2 * x ** 2 - 2 * x * y + y ** 2 + x + 2 * y + 1
implicit_derivative = -sp.diff(z, x) / sp.diff(z, y)
print(f"Implicit derivative: {implicit_derivative}")

Result:

Implicit derivative: (-4*x + 2*y - 1)/(-2*x + 2*y + 2)

Example 4: Parametric Derivatives

For parametric equations $ x = e^t \cos(t) $ and $ y = e^t \sin(t) $, compute the derivative:

import sympy as sp

t = sp.Symbol('t')
x = sp.exp(t) * sp.cos(t)
y = sp.exp(t) * sp.sin(t)
parametric_derivative = sp.diff(y, t) / sp.diff(x, t)
simplified = sp.simplify(parametric_derivative)
print(f"Parametric derivative: {parametric_derivative}")
print(f"Simplified form: {simplified}")

Output:

Parametric derivative: (exp(t)*sin(t) + exp(t)*cos(t))/(-exp(t)*sin(t) + exp(t)*cos(t))
Simplified form: tan(t + pi/4)

Example 5: Partial Derivatives

Compute partial derivatives of $ z = \sin(xy) + \cos^2(xy) $:

import sympy as sp

x, y = sp.symbols('x y')
z = sp.sin(x * y) + (sp.cos(x * y)) ** 2
d1 = sp.diff(z, x)
d2 = sp.diff(z, y)
d3 = sp.diff(z, x, 2)
d4 = sp.diff(sp.diff(z, x), y)
print(f"First partial: {d1}")
print(f"Second partial: {d2}")
print(f"Third partial: {d3}")
print(f"Fourth partial: {d4}")

Integration Calculations

Example 1: Definite and Indefinite Integrals

Evaluate $ \int \sqrt{4 - x^2} dx $ and $ \int_1^2 \sqrt{4 - x^2} dx $:

import sympy as sp

x = sp.Symbol('x')
y = sp.sqrt(4 - x ** 2)
indef_integral = sp.integrate(y, x)
def_integral = sp.integrate(y, (x, 1, 2))
print(f"Indefinite integral: {indef_integral}")
print(f"Definite integral: {def_integral}")

Output:

Indefinite integral: x*sqrt(4 - x**2)/2 + 2*asin(x/2)
Definite integral: -sqrt(3)/2 + 2*pi/3

Example 2: Triple Integral

Compute the triple integral over a region bounded by $ z = \sqrt{2 - x^2 - y^2} $ and $ z = \sqrt{x^2 + y^2} $:

import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.mplot3d import Axes3D

# Plotting surfaces
x = np.arange(-1, 1, 0.05)
y = np.arange(-1, 1, 0.05)
x, y = np.meshgrid(x, y)
z1 = np.sqrt(x ** 2 + y ** 2)
z2 = np.sqrt(2 - x ** 2 - y ** 2)
ax = Axes3D(plt.figure())
ax.plot_surface(x, y, z1)
ax.plot_surface(x, y, z2)
plt.show()

# Converting to cylindrical coordinates
import sympy as sp

r, s, z = sp.symbols('r s z')
f = (r ** 2 + z) * r
integral_result = sp.integrate(
    sp.integrate(sp.integrate(f, (z, r, sp.sqrt(2 - r ** 2))), (r, 0, 1)), 
    (s, 0, 2 * sp.pi)
)
print(f"Triple integral result: {integral_result}")

Differential Equations

Example 1: General Solution

Solve the differential equation $ y' + 2xy = xe^{-x^2} $:

import sympy as sp

x = sp.Symbol('x')
f = sp.Function('f')
y = f(x)
equation = sp.Eq(y.diff(x) + 2 * x * y, x * sp.exp(-x ** 2))
general_solution = sp.dsolve(equation, y)
print(f"General solution: {general_solution}")

Output:

General solution: Eq(f(x), (C1 + x**2/2)*exp(-x**2))

Example 2: Particular Solution

Find the particular solution of $ xy' + y - e^{-x} = 0 $ with $ y(1) = 2e $:

import sympy as sp

x = sp.Symbol('x')
f = sp.Function('f')
y = f(x)
equation = sp.Eq(x * y.diff(x) + y - sp.exp(-x), 0)
particular_solution = sp.dsolve(equation, y, ics={f(1): 2 * sp.exp(1)})
print(f"Particular solution: {particular_solution}")

Output:

Particular solution: Eq(f(x), ((1 + 2*exp(2))*exp(-1) - exp(-x))/x)