mathgenerator.calculus

  1import random
  2from scipy.integrate import quad
  3import sympy
  4
  5
  6def definite_integral(max_coef=100):
  7    r"""Definite Integral of Quadratic Equation
  8
  9    | Ex. Problem | Ex. Solution |
 10    | --- | --- |
 11    | The definite integral within limits $0$ to $1$ of the equation $28x^2 + 32x + 66 = $ | $91.33$ |
 12    """
 13    def integrand(x, a, b, c):
 14        return a * x**2 + b * x + c
 15
 16    a = random.randint(0, max_coef)
 17    b = random.randint(0, max_coef)
 18    c = random.randint(0, max_coef)
 19
 20    result = quad(integrand, 0, 1, args=(a, b, c))[0]
 21    solution = round(result, 2)
 22
 23    problem = f"The definite integral within limits $0$ to $1$ of the equation ${a}x^2 + {b}x + {c} = $"
 24    return problem, f'${solution}$'
 25
 26
 27def power_rule_differentiation(max_coef=10, max_exp=10, max_terms=5):
 28    r"""Power Rule Differentiation
 29
 30    | Ex. Problem | Ex. Solution |
 31    | --- | --- |
 32    | Differentiate $1x^{5} + 4x^{7} + 4x^{4}$ | $5x^{4} + 28x^{6} + 16x^{3}$ |
 33    """
 34    numTerms = random.randint(1, max_terms)
 35    problem = "Differentiate $"
 36    solution = "$"
 37
 38    for i in range(numTerms):
 39        if i > 0:
 40            problem += " + "
 41            solution += " + "
 42        coefficient = random.randint(1, max_coef)
 43        exponent = random.randint(1, max_exp)
 44
 45        problem += f'{coefficient}x^{{{exponent}}}'
 46        solution += f'{coefficient * exponent}x^{{{exponent - 1}}}'
 47
 48    return problem + '$', solution + '$'
 49
 50
 51def power_rule_integration(max_coef=10, max_exp=10, max_terms=5):
 52    r"""Power Rule Integration
 53
 54    | Ex. Problem | Ex. Solution |
 55    | --- | --- |
 56    | Integrate $9x^{6} + 2x^{6} + 4x^{3}$ | $\frac{9}{6}x^{7} + \frac{2}{6}x^{7} + \frac{4}{3}x^{4} + C$ |
 57    """
 58    numTerms = random.randint(1, max_terms)
 59    problem = "Integrate $"
 60    solution = "$"
 61
 62    for i in range(numTerms):
 63        if i > 0:
 64            problem += " + "
 65            solution += " + "
 66        coefficient = random.randint(1, max_coef)
 67        exponent = random.randint(1, max_exp)
 68
 69        problem += f'{coefficient}x^{{{exponent}}}'
 70        solution += rf'\frac{{{coefficient}}}{{{exponent}}}x^{{{exponent + 1}}}'
 71
 72    solution += " + C"
 73
 74    return problem + '$', solution + '$'
 75
 76
 77def stationary_points(max_exp=3, max_coef=10):
 78    r"""Stationary Points
 79
 80    | Ex. Problem | Ex. Solution |
 81    | --- | --- |
 82    | $f(x)=6x^3 + 6x^2 + x + 8$ | ${- \frac{1}{3} - \frac{\sqrt{2}}{6}, - \frac{1}{3} + \frac{\sqrt{2}}{6}}$ |
 83    """
 84    solution = ''
 85
 86    # A constant function has no stationary points, and the answer will be Reals. e.g.: 
 87    #   x = sympy.symbols('x')
 88    #   s = sympy.stationary_points(1 + 0 * x , x)
 89    #   assert s == sympy.Reals
 90    while solution == sympy.Reals or len(solution) == 0:
 91        x = sympy.symbols('x')
 92        problem = 0
 93        for exp in range(max_exp + 1):
 94            coefficient = random.randint(0, max_coef)
 95            problem += coefficient * pow(x, exp)
 96        solution = sympy.stationary_points(problem, x)
 97
 98    problem = 'f(x)=' + str(problem).replace('**', '^').replace('*', '')
 99    return f'${problem}$', f'${sympy.latex(solution)[6:-8]}}}$'
100
101
102def trig_differentiation():
103    r"""Trigonometric Differentiation
104
105    | Ex. Problem | Ex. Solution |
106    | --- | --- |
107    | $\frac{d}{dx}(\csc)=$ | $-\csc \cdot \cot$ |
108    """
109    pairs = {
110        r'\sin': r'\cos',
111        r'\cos': r'-\sin',
112        r'\tan': r'\sec^{{2}}',
113        r'\cot': r'-\csc^{{2}}',
114        r'\sec': r'\sec \cdot \tan',
115        r'\csc': r'-\csc \cdot \cot'
116    }
117    problem = random.choice(list(pairs.keys()))
118    solution = f'${pairs[problem]}$'
119    problem = rf'$\frac{{d}}{{dx}}({problem})=$'
120
121    return problem, solution