PreTeXt Book Examples An example to get you started

\(f(x) = x^2 + 2x + 1\)

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\(f(x) = x^2 - 2x + 1\)
 #paired
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\(f'(x) = 2x + 2\)

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\(f'(x) = 2x - 2\)
 #paired
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\(f'(x) = 0 \Rightarrow x = -1\)

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\(f'(x) = 0 \Rightarrow x = 1\)
 #paired
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\(f(-1) = 0\) is the minimum.

Prove that \(\sqrt{2}\) is irrational.

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Assume \(\sqrt{2} = \frac{p}{q}\) in lowest terms.

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Assume \(\sqrt{2} = \frac{p}{q}\) where \(p, q\) share a factor.
 #paired
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Then \(2 = \frac{p^2}{q^2}\), so \(p^2 = 2q^2\).

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So \(p^2\) is even, hence \(p\) is even.

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Write \(p = 2k\); then \(4k^2 = 2q^2\), so \(q^2 = 2k^2\).

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Write \(p = 2k\); then \(2k^2 = q^2\).
 #paired
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So \(q\) is also even — contradiction.

\(\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}\)

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\(\int_{-\infty}^{\infty} e^{-x^2}\,dx = \pi\)
 #paired
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Let \(I = \int_{-\infty}^{\infty} e^{-x^2}\,dx\).

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Let \(I = \int_{0}^{\infty} e^{-x^2}\,dx\).
 #paired
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Then \(I^2 = \iint e^{-(x^2+y^2)}\,dx\,dy\).

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Convert to polar: \(I^2 = \int_0^{2\pi}\int_0^\infty e^{-r^2}r\,dr\,d\theta = \pi\).

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Convert to polar: \(I^2 = \int_0^{\pi}\int_0^\infty e^{-r^2}r\,dr\,d\theta = \pi/2\).
 #paired

def factorial(n):
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    if n == 0:
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    if n == 1: #paired
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        return 1
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    return n * factorial(n - 1)

def merge_sorted_lists(list_one, list_two):
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    result = []
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    result = list() #paired
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    i, j = 0, 0
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    while i < len(list_one) and j < len(list_two):
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    while i <= len(list_one) and j <= len(list_two): #paired
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        if list_one[i] <= list_two[j]:
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        if list_one[i] < list_two[j]: #paired
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            result.append(list_one[i]); i += 1
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        else:
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            result.append(list_two[j]); j += 1
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    return result + list_one[i:] + list_two[j:]

def swap(a, b):
    return b, a
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def swap(a, b):
    a, b = b, a #paired
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x = 3
y = 7
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x, y = swap(x, y)
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x, y = swap(y, x) #paired
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print(x, y)

class Stack:
    def __init__(self):
        self.items = []
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    def push(self, item):
        self.items.append(item)
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    def push(self, item):
        self.items.insert(0, item) #paired
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    def pop(self):
        return self.items.pop()
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    def pop(self):
        return self.items.pop(0) #paired

import math
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import sys
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print(math.pi) #tag:; depends:import;
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print(sys.version) #tag:; depends:import;

for i in range(3):
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    for j in range(3):
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        for k in range(3):
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            if i != j != k:
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            if i == j == k: #paired
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                print(i, j, k)

results = []
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for a in range(5):
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    for b in range(5):
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        for c in range(5):
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            for d in range(5):
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                total = a + b + c + d
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                total = a * b * c * d #paired
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                if total == 10:
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                if total > 10: #paired
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                    results.append((a, b, c, d))
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count = len(results)
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print(count)

a = 10
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a = 0 #paired
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b = 20
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b = 0 #paired
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c = a + b
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print(c)