In this lab sheet we will see how to carry out basic aspects of Calculus (limits, differentiation and integration) with Python.

Building blocks

1. TICKABLE Calculating limits.

A video describing the concept.

A video demo.

It is possible to calculate limits easily with SymPy. Let us consider the following function as a running example:

First let us define this function in Python as a function:

>>> import sympy as sym
>>> def f(x):
...     return 1 / x
>>> f(2)
0.5



We can now pass it symbolic variable:

>>> x  = sym.symbols('x')
>>> f(x)
1/x



We can compute the limits at $$\pm\infty$$:

>>> sym.limit(f(x), x, sym.oo)
0
>>> sym.limit(f(x), x, -sym.oo)
0
>>> sym.limit(f(x), x, +sym.oo)
0



We can also compute the limit at $$0$$ however we must be careful here (you will recall from basic calculus that the limit depends on the direction):

>>> sym.limit(f(x), x, 0)  # The default direction of approach is positive.
oo
>>> sym.limit(f(x), x, 0, dir="+")
oo
>>> sym.limit(f(x), x, 0, dir="-")
-oo



Experiment with caculating the limit of different functions.

2. TICKABLE Differentiation.

A video describing the concept.

A video demo.

We can use SymPy to carry out differentiation:

>>> sym.diff(f(x), x)
-1/x**2
>>> sym.diff(sym.cos(x), x)
-sin(x)



We can carry out various orders of differentiation. These all give the second order derivative of $$\cos(x)$$:

>>> sym.diff(sym.diff(sym.cos(x), x), x)
-cos(x)
>>> sym.diff(sym.cos(x), x, x)
-cos(x)
>>> sym.diff(sym.cos(x), x, 2)
-cos(x)



>>> y = sym.symbols('y')
>>> sym.diff(sym.cos(x) * sym.sin(y), x)  # Differentiating with respect to x
-sin(x)*sin(y)
>>> sym.diff(sym.cos(x) * sym.sin(y), y)  # Differentiating with respect to y
cos(x)*cos(y)



Finally it is also possible for SymPy to differentiate hypothetical functions (ones that we do not know anything about):

>>> g = sym.Function('g')  # We create g as a generic function
>>> sym.diff(g(x), x, 2)
Derivative(g(x), x, x)



Try and experiment with differentiating more complicated functions.

3. TICKABLE Integration.

A video describing the concept.

A video demo.

As well as differentiation it is possible to carry out integration.

We can do both definite and indefinite integrals:

>>> sym.integrate(f(x), x)  # An indefinite integral
log(x)
>>> sym.integrate(f(x), (x, sym.exp(1), sym.exp(5)))  # A definite integral
4



We can use this to verify the fundamental theorem of calculus for the specific example of our function $$f$$:

>>> sym.integrate(sym.diff(f(x), x), x)
1/x



We can also verify this for generic functions:

>>> g = sym.Function('g')
>>> sym.integrate(sym.diff(g(x), x), x) == g(x)
True
>>> sym.diff(sym.integrate(g(x), x), x) == g(x)
True



Experiment with integrating other functions.

4. TICKABLE Plotting. It is possible to use SymPy to create plots.

A video describing the concept.

A video demo.

To view these in jupyter we need to run this line:

%matplotlib inline


We can then create plots using the plot command:

>>> sym.plot(f(x), (x, -1, 1))
<...



Thatâ€™s not very clear, let us modify the limits on the y axis.

>>> sym.plot(f(x), (x, -1, 1), ylim=(-10, 10))
<...



We can also choose to modify the labels on the axes:

>>> sym.plot(f(x), (x, -1, 1), ylim=(-10, 10), xlabel=None, ylabel="$1/x$")
<...



Finally we can also combine various plots of different functions together. To do this we build each plot separately (telling Python not to show them) and then combine them:

>>> p1 = sym.plot(f(x), (x, -1, 1), show=False)
>>> p2 = sym.plot(x, (x, -1, 1), show=False, line_color="red")
>>> p1.extend(p2)
>>> p1.ylim = (-10, 10)
>>> p1.xlabel = None  # Try using strings here
>>> p1.ylabel = None
>>> p1.show()



It is also possible to save a file of the graph. Depending on how you name the file the type of tile will be different:

>>> p1.save("my_plot.pdf")  # Try .png etc...



Attempt to create plots of other functions.

5. TICKABLE: Worked example

A video describing the concept.

A video demo.

We are going to use the above to attempt to find and classify all points of inflection for the following quartic function:

First let us take a look at the function:

>>> def f(x):
...     return  -x ** 4 + 9 * x ** 2 + 4 * x - 12
>>> sym.plot(f(x), (x, -20, 20) , ylim=(-20, 20))
<...



Let us find the roots of the functions:

>>> sym.solveset(f(x), x)
{-2, 1, 3}



We have a quatric so one of those roots (of which there are only 3) must be repeated, let us see if we can factor our function:

>>> f(x).factor()
-(x - 3)*(x - 1)*(x + 2)**2



We see that (-2) is a repeated root.

Let us confirm the limiting behaviour of our function:

>>> sym.limit(f(x), x, sym.oo)
-oo
>>> sym.limit(f(x), x, -sym.oo)
-oo



Finally, we also see that there are 3 points of inflection so let us find them:

>>> poi = sym.solveset(sym.diff(f(x), x), x)
>>> poi
{-2, 1 + sqrt(6)/2, -sqrt(6)/2 + 1}



Let us now evaluate which of these gives a positive or negative value of the second derivative:

>>> for point in list(poi):  # We convert the poi to a list
...     print(point, (sym.diff(f(x), x, 2).subs({x: point})) > 0)
-2 False
1 + sqrt(6)/2 False
-sqrt(6)/2 + 1 True



We see that 2 points of inflection give negative second derivative (so they are local maxima), whereas $$-\sqrt{6}/2+1$$ is a local minimum. This confirms the plot.

Further work

These questions aim to push a bit further.

6. Consider the function below:

Identify the roots and limits (at $$\pm\infty$$) of the function and confirm this with a plot.

7. For the function of question 6, identify and classify all points of inflection.

8. There are various algebraic relationships on limits:

Confirm these for the following particular examples:

9. The point of this question is to investigate the definition of a derivative:

1. Consider $$f(x) = x^3 + 3x - 20$$;
2. Compute $$\frac{f(x+h)-f(x)}{h}$$;
3. Compute the above limit as $$h\to 0$$ and verify that this is the derivative of $$f$$.

Solutions

Solutions available.