NAMSSN UI Chapter

A notebook is built from cells. Each input cell starts with In[ ]:= and produces an Out[ ]= when you press Shift+Enter. You will practise running simple expressions:

• 2 + 3
• 2^10
• Sin[Pi/4]

Goal: feel comfortable with “type → Shift+Enter → read output”.

Mathematica uses [ ] for function arguments and { } for lists. Function names start with capital letters:

• Sqrt[2], Exp[1], Log[10]
• {1, 2, 3, 4}
• Mean[{1,2,3,4}]

You do not need to memorise every function. The documentation is your friend:

• ?Integrate
• ??Plot
• Right-click a function name or use the built-in help browser.

You begin with D for derivatives and test standard patterns:

• D[Sin[x], x]
• D[Exp[-x^2], x]
• D[x^2 y^3, x], D[x^2 y^3, y]

You use Integrate and NIntegrate, and compare results:

• Integrate[1/(1 + x^2), {x, 0, 1}]
• NIntegrate[Exp[-x^2], {x, 0, 10}]
• Compare Integrate[f[x], {x, a, b}] with an approximation.

Goal: always ask “Do the exact and numeric answers agree to a good precision?”.

Use Limit to study behaviour near singularities or interesting points:

• Limit[Sin[x]/x, x -> 0]
• Limit[(1 + 1/n)^n, n -> Infinity]

You practise series expansions around a point:

• Series[Exp[x], {x, 0, 5}]
• Series[Sin[x]/x, {x, 0, 6}]
• Normal[%] to convert a series object into a polynomial.

You use Sum for closed forms and compare with partial sums:

• Sum[1/n^2, {n, 1, Infinity}]
• Table[Sum[1/n^2, {n, 1, N}], {N, 5, 50, 5}]

You start to treat Mathematica as a tool for transforms: substitutions, parameter differentiation, or integrating families of functions.

Example patterns (explored step by step in the subpage):

• Integrals depending on a parameter a and differentiating under the integral sign.
• Relating two integrals by a substitution and asking Mathematica to verify.

You build tables and try to recognise patterns before attempting a proof:

Table[Integrate[x^n, {x, 0, 1}], {n, 0, 6}]

You then compare the output with conjectured formulas.

Here you create a notebook that studies a family of integrals depending on parameters (e.g. an exponent or a shift), collects special cases, and records conjectured patterns.

The project subpage shows how to:

• State the family clearly at the top of the notebook.
• Use Table, Integrate, and numeric checks in a systematic way.
• Reserve a section for “Conjectures and questions”.

Finally, you practise turning your exploration into something readable by others: clear headings, commented code, well-chosen plots, and short summaries.

• Explain what the notebook is about in 3–5 sentences at the top.
• Write comments before each important block of code.
• Use plots only when they clarify a point.

Ambition for this lab: a NAMSSN UI student who completes Modules 0–3 and keeps using Mathematica on real course problems should be able to design useful notebooks for coursework, mini-projects, and contests.