update taylor exercise

fb216d4b · Robert Lanzafame · 1452892e · fb216d4b
Commit fb216d4b authored 6 months ago by Robert Lanzafame
--- a/book/numerical_methods/taylor-series-exercise.ipynb
+++ b/book/numerical_methods/taylor-series-exercise.ipynb
@@ -6,9 +6,18 @@
   "source": [
    "# Exercises on Taylor expansion\n",
    "\n",
-    "This page shows some exercises on calculating Taylor expansions. If you reload this page, you'll get new values.\n",
+    "Use this page to practice calculating Taylor expansions. This requires\n",
    "\n",
-    " Click `rocket` -->`Live Code` to start practising.\n"
+    "```{tip}\n",
+    "\n",
+    "This page presents several functions of $x$ that you should approximate with Taylor Series. You can click `rocket` -->`Live Code` to check your answers, however, note the following:\n",
+    "\n",
+    "1. You should work out the exercises \"by hand\" first (using paper or a digital notebook)\n",
+    "2. You can type your answer in the code cell and then click \"Check Answer\" to see if it is correct. Use basic Python syntax for representing the equation, for example, `x**2 + 1` for $x^2+1$; use `pi` or `sin()` and `cos()` for the trigonometric functions (not numpy!)\n",
+    "3. You can click the \"Show correct answer\" button to see the solution; however, note that the form of the equation will be different than that which you are expected to derive, so you will need to do some arithmetic to confirm you are correct.\n",
+    "\n",
+    "_The reason the Python syntax is different is because it uses Sympy, a library for symbolic mathematics._\n",
+    " ```\n"
   ]
  },
  {
@@ -213,6 +222,15 @@
    "check_answer(\"eq1_answer\",eq1_correct, check_equation)\n"
   ]
  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "```{tip}\n",
+    "Note that for Exercise 1, the equation is an even-order polynomial, which means that the Taylor Series should not introduce any error!\n",
+    "```"
+   ]
+  },
  {
   "cell_type": "markdown",
   "metadata": {},

 %% Cell type:markdown id: tags:

 # Exercises on Taylor expansion

-This page shows some exercises on calculating Taylor expansions. If you reload this page, you'll get new values.
+Use this page to practice calculating Taylor expansions. This requires

- Click `rocket` -->`Live Code` to start practising.
+```{tip}
+
+This page presents several functions of $x$ that you should approximate with Taylor Series. You can click `rocket` -->`Live Code` to check your answers, however, note the following:
+
+1. You should work out the exercises "by hand" first (using paper or a digital notebook)
+2. You can type your answer in the code cell and then click "Check Answer" to see if it is correct. Use basic Python syntax for representing the equation, for example, `x**2 + 1` for $x^2+1$; use `pi` or `sin()` and `cos()` for the trigonometric functions (not numpy!)
+3. You can click the "Show correct answer" button to see the solution; however, note that the form of the equation will be different than that which you are expected to derive, so you will need to do some arithmetic to confirm you are correct.
+
+_The reason the Python syntax is different is because it uses Sympy, a library for symbolic mathematics._
+ ```

 %% Cell type:code id: tags:thebe-remove-input-init

 ``` python
 import sympy as sp
 import numpy as np
 from sympy import pi, latex
 #from sympy.printing.mathml import mathml

 import operator
 import ipywidgets as widgets
 from IPython.display import Latex, Math
 sp.init_printing()
 #sp.init_printing(use_latex=True)


 check_equation = lambda eq1, eq2: sp.simplify(eq1 - eq2) == 0

 def check_answer(variable_name, expected, comparison=operator.eq):
    output = widgets.Output()
    correct_output = widgets.Output()
    button = widgets.Button(description="Check answer")
    show_correct_button = widgets.Button(description="Show correct answer")

    def _inner_check(button):
        with output:
            if comparison(globals()[variable_name], expected):
                output.outputs = [{'name': 'stdout', 'text': 'Correct!',
                                   'output_type': 'stream'}]
                correct_output.clear_output()  # Clear the correct answer display if they got it right
            else:
                output.outputs = [{'name': 'stdout', 'text': 'Incorrect!',
                                   'output_type': 'stream'}]

    def _show_correct_answer(button):
        with correct_output:
            correct_output.clear_output()  # Clear previous outputs
            print(f"The correct answer is in python notation: {expected}")
            print(f"Or in mathematical notation:")
            display(expected)


    button.on_click(_inner_check)
    show_correct_button.on_click(_show_correct_answer)

    display(button, output, show_correct_button, correct_output)



 ```

 %% Cell type:markdown id: tags:


 ## Exercise 1

 Calculate the taylor series expansion of:

 %% Cell type:code id: tags:thebe-remove-input-init

 ``` python
 x, y = sp.symbols('x, y')
 a_1 = sp.Integer(np.random.randint(2,6))
 b_1 = sp.Integer(np.random.randint(-10,10))
 c_1 = sp.Integer(np.random.randint(-5,5))
 eq1_original = a_1 * x**2 + b_1*x
 eq1_correct = sp.series(eq1_original,x,c_1)
 eq1_answer = 0
 display(eq1_original)
 #display(eq1_correct)
 ```

 %% Cell type:markdown id: tags:

 around:

 %% Cell type:code id: tags:thebe-remove-input-init

 ``` python
 display(sp.Eq(x,c_1))
 ```

 %% Cell type:markdown id: tags:

 Discard any $O(x^3)$ terms.

 Fill in your answer and run the cell before clicking 'Check answer'.

 %% Cell type:code id: tags:auto-execute-page,disable-download-page

 ``` python
 eq1_answer =
 ```

 %% Cell type:code id: tags:thebe-remove-input-init

 ``` python
 check_answer("eq1_answer",eq1_correct, check_equation)
 ```

 %% Output





 %% Cell type:markdown id: tags:

+```{tip}
+Note that for Exercise 1, the equation is an even-order polynomial, which means that the Taylor Series should not introduce any error!
+```
+
+%% Cell type:markdown id: tags:
+
 ## Exercise 2

 Calculate the taylor series expension of:

 %% Cell type:code id: tags:thebe-remove-input-init

 ``` python
 a_2 = sp.Integer(np.random.randint(1,7))
 c_2 = sp.Integer(np.random.randint(-5,5))
 eq2_original = a_2*sp.tan(x)
 display(eq2_original)
 eq2_correct = sp.series(eq2_original,x,c_2*sp.pi,3).removeO()
 #display(eq2_correct)
 eq2_answer = 0
 ```

 %% Cell type:markdown id: tags:

 around:

 %% Cell type:code id: tags:thebe-remove-input-init

 ``` python
 display(sp.Eq(x,c_2*sp.pi))
 ```

 %% Cell type:markdown id: tags:

 discard any $O(x^3)$ terms.

 Fill in your answer and run the cell before clicking 'Check answer'. Furthermore, use `pi` for $\pi$:

 %% Cell type:code id: tags:

 ``` python
 eq2_answer =
 ```

 %% Cell type:code id: tags:thebe-remove-input-init

 ``` python
 check_answer("eq2_answer",eq2_correct, check_equation)
 ```

 %% Cell type:markdown id: tags:

 ## Exercise 3

 Calculate the taylor series expension of:

 %% Cell type:code id: tags:thebe-remove-input-init

 ``` python
 a_3 = sp.Integer(np.random.randint(1,10))
 c_3 = sp.Integer(np.random.randint(-1,1))
 eq3_original = a_3 / (1 - x)
 display(eq3_original)
 eq3_correct = sp.series(eq3_original,x,c_3,3).removeO()
 #display(eq3_correct)
 eq3_answer = 0
 ```

 %% Cell type:markdown id: tags:

 around:

 %% Cell type:code id: tags:thebe-remove-input-init

 ``` python
 display(sp.Eq(x,c_3))
 ```

 %% Cell type:markdown id: tags:

 discard any $O(x^3)$ terms.

 Fill in your answer and run the cell before clicking 'Check answer':

 %% Cell type:code id: tags:

 ``` python
 eq3_answer =
 ```

 %% Cell type:code id: tags:thebe-remove-input-init

 ``` python
 check_answer("eq3_answer",eq3_correct, check_equation)
 ```