Calculus week

First meeting

After this meeting students should:

  • Understand how to use the Sympy library to carry out basic Calculus tasks

  • Know what they need to do to prepare for their third tutorial.

Problem

Explain to students that we will be solving the following problem:

Consider the function \(f(x)= x ^ 3 - ax ^ 2 + bx - 5\)

  1. Given that \(\frac{df}{dx}|_{x=0}=0\), \(\frac{d^2f}{dx^2}|_{x=0}=5\) and that \(b>0\) find the values of \(a\) and \(b\).

  2. For the specific values of \(a\) and \(b\) find:

    1. \(\lim_{x\to 0}f(x)\);

    2. \(\lim_{x\to \infty}f(x)\);

    3. \(\int f(x) dx\);

    4. \(\int_{5}^{\pi} f(x) dx\).

Solution

Group exercise (breakout rooms of 3): ask students to spend 5 minutes writing a plan to tackle that problem (not necessarily carrying out each step).

Clearly write down these steps:

  1. Differentiate and write system of equations and solve them for \(a\) and \(b\).

  2. Take the limits and calculate the integrals.

Now show how to get code to do this:

>>> import sympy as sym  # Note that we are using an alias
>>> x = sym.Symbol("x")
>>> a = sym.Symbol("a")
>>> b = sym.Symbol("b")
>>> expression = x ** 3 - a * x ** 2 + b * x - 5
>>> expression
-a*x**2 + b*x + x**3 - 5

So far we have not done anything different to what we saw in the Algebra chapter. Now to differentiate:

>>> derivative = sym.diff(expression, x)
>>> derivative
-2*a*x + b + 3*x**2

This can now be used to give us our first equation:

>>> first_equation = sym.Eq(derivative.subs({x: 0}), 0)
>>> first_equation
Eq(b, 0)

Now let us get the second derivative:

>>> second_derivative = sym.diff(expression, x, 2)
>>> second_derivative
2*(-a + 3*x)

and get the second equation:

>>> second_equation = sym.Eq(second_derivative.subs({x: 0}), 5)
>>> second_equation
Eq(-2*a, 5)

Now to solve the first equation to obtain a value for \(b\):

>>> sym.solveset(first_equation, b)
FiniteSet(0)

Now to substitute that value for $a$ (note that this is not necessary) and solve the second equation for $a$:

>>> second_equation = second_equation.subs({b: 0})
>>> second_equation
Eq(-2*a, 5)

We can solve this:

>>> sym.solveset(second_equation, a)
FiniteSet(-5/2)

Substituting these back:

>>> expression = expression.subs({b: 0, a: - sym.S(5) / 2})
>>> expression
x**3 + 5*x**2/2 - 5

We can confirm our findings:

>>> sym.diff(expression, x).subs({x: 0})
0
>>> sym.diff(expression, x, 2).subs({x: 0})
5

Now we can compute the limits and integrals:

>>> sym.limit(expression, x, 0)
-5
>>> sym.limit(expression, x, sym.oo)
oo
>>> sym.integrate(expression, x)
x**4/4 + 5*x**3/6 - 5*x
>>> sym.factor(sym.integrate(expression, (x, 5, sym.pi)))
(-5 + pi)*(3*pi**3 + 25*pi**2 + 125*pi + 565)/12

Come back: with time take any questions.

Point at resources.

After class email

Send the following email after class:

Hi all,

A recording of today's class is available at <>.

In this class I went over a demonstration of using Python to solve a
calculus problem. I carried out the following mathematical techniques:

- Differentiation
- Limits
- Integrations

In preparation for your tutorial tomorrow please work through the third
chapter of the Python for mathematics book:
https://vknight.org/pfm/tools-for-mathematics/03-calculus/introduction/main.html

Please get in touch if I can assist with anything,
Vince

Second meeting

I will work through the following problem:

Consider the functions \(f(x) = x ^ 3 + 3x - 3\) and \(g(x) = \cos(x) \sin(x)\).`

  1. Create a variable turning_points_of_f which has value the turning points of \(f(X)\).

  2. Create a variable turning_points_of_g which has value the turning points of \(g(X)\).

  3. Create a variable max_of_f_on_unit_circle which has the maximum value of \(f\) for \(x\in[0, 2\pi]\).

  4. Create a variable max_of_g_on_unit_circle which has the maximum value of \(g\) for \(x\in[0, 2\pi]\).

  5. Which function has the maximum value?

The solution approach:

>>> import sympy as sym
>>> x = sym.Symbol("x")
>>> f = x ** 3 + 3 * x - 3
>>> g = sym.cos(x) * sym.sin(x)
>>> turning_points_of_f = sym.solveset(sym.diff(f, x), x)
>>> turning_points_of_f
FiniteSet(I, -I)
>>> turning_points_of_g = sym.solveset(sym.diff(g, x), x)
>>> turning_points_of_g
Union(ImageSet(Lambda(_n, _n*pi + pi/4), Integers), ImageSet(Lambda(_n, _n*pi + 3*pi/4), Integers))
>>> max_of_f_on_unit_circle = max(f.subs({x: 0}), f.subs({x: 2 * sym.pi}))
>>> max_of_f_on_unit_circle
-3 + 6*pi + 8*pi**3
>>> max_of_g_on_unit_circle = max(g.subs({x: 5 * sym.pi / 4}), g.subs({x: 3 * sym.pi / 4}), g.subs({x: 7 * sym.pi / 4}), g.subs({x: sym.pi / 4}))
>>> max_of_g_on_unit_circle
1/2
>>> float(max_of_f_on_unit_circle)
263.8997...

Highlight that the answer to the final question is thus \(f\).

Note that max is not a function that has been specifically been seen before but that’s not unexpected.

Now explain what the kernel is. Draw a picture showing the notebook separated from the terminal. Analogy of a brain.

Restart the kernel, show that the last command does not work.

Rerun the cells but include a mistake:

sym.solveset = (sym.diff(f, x), x)

Note the type of error we then get. And show that here we want the kernel to forget everything.

If there is time demonstrate plotting.

Discuss that sympy has some basic plotting but that I do not recommend it.

Point at matplotlib and numpy chapters and also specify that we will be using other things that are seen in future chapters (list comprehensions).:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> domain = np.linspace(0, 2 * np.pi, 100)
>>> f_image = [f.subs({x: x_value}) for x_value in domain]
>>> g_image = [g.subs({x: x_value}) for x_value in domain]
>>> plt.figure()  
>>> plt.plot(domain, f_image, label="$f(x)$")  
>>> plt.plot(domain, g_image, label="$g(x)$")  
>>> plt.legend()