Math — Calculus

June 17, 2026·Reha Tuncer·Math

calculus derivatives integrals math python polynomials

Math — Calculus

A progressive journey through single-variable and multivariable calculus — from summation and product notation to derivatives, integrals, and Python implementations of polynomial differentiation and integration.

Learning Objectives

#	Concept
1	Evaluate summations using $\Sigma$ (sigma) notation
2	Evaluate products using $\Pi$ (pi) notation
3	Compute derivatives of polynomial, exponential, and logarithmic functions
4	Compute partial derivatives of multivariable functions
5	Evaluate indefinite and definite integrals
6	Evaluate double integrals over rectangular regions
7	Implement polynomial derivative calculation in Python
8	Implement polynomial integral (antiderivative) calculation in Python
9	Apply the closed-form formula for $\sum_{i=1}^{n} i^2$

Module Structure

This module contains two types of tasks:

Type	Files	Description
Computation Exercises	0–8, 11–16	Hand-calculated answers stored as single-number files
Python Implementations	9, 10, 17	Functions that compute sums, derivatives, and integrals programmatically

Computation Exercises (Tasks 0–8, 11–16)

These tasks build calculus intuition through manual computation. Each file stores the numeric answer.

Task	File	Topic	Concept Tested
0	`0-sigma_is_for_sum`	Sigma notation	$\sum_{i=2}^{5} i$
1	`1-seegma`	Sigma notation	$\sum_{k=1}^{4} k^2$
2	`2-pi_is_for_product`	Pi notation	$\prod_{i=1}^{4} i$
3	`3-pee`	Pi notation	Product with expressions
4	`4-hello_derivatives`	Derivatives	$f'(x)$ for polynomial $f$
5	`5-log_on_fire`	Derivatives	Derivative of $\ln(x)$ and $e^x$
6	`6-voltaire`	Derivatives	Derivative at a specific point
7	`7-partial_truths`	Partial derivatives	$\frac{\partial f}{\partial x}$
8	`8-all-together`	Mixed derivatives	Combined partial derivative evaluation
11	`11-integral`	Indefinite integrals	$\int x^n \, dx$
12	`12-integral`	Indefinite integrals	$\int e^x \, dx$
13	`13-definite`	Definite integrals	$\int_a^b f(x) \, dx$
14	`14-definite`	Definite integrals	Area under a curve
15	`15-definite`	Definite integrals	Definite integral with substitution
16	`16-double`	Double integrals	$\iint f(x,y) \, dx \, dy$

Computation Exercises Approach

These 14 tasks build calculus intuition through hand computation before code — each file stores a single numeric answer, and the real learning happens in the manual work that produced it.

Purpose: Before implementing differentiation and integration in Python (Tasks 9, 10, 17), you work through the core calculus operations by hand. This mirrors the machine learning workflow: understand the math on paper first, then automate it. The manual computation builds the intuition that makes the code meaningful rather than mechanical.

Learning progression:

Stage	Tasks	Topic	What you practice
1	0–1	Sigma ( $\Sigma$ ) notation	Summing sequences: $\sum_{i=a}^{b} f(i)$ — the foundation of series and integrals
2	2–3	Pi ( $\Pi$ ) notation	Multiplying sequences: $\prod_{i=a}^{b} f(i)$ — used in likelihood products and combinatorial formulas
3	4–6	Derivatives	Power rule $\frac{d}{dx}x^n = nx^{n-1}$ , derivative of $\ln x$ , derivative of $e^x$ , evaluating at a point
4	7–8	Partial derivatives	Treating other variables as constants: $\frac{\partial}{\partial x}f(x,y)$ , mixed partials
5	11–12	Indefinite integrals	Reverse power rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ , $\int e^x dx = e^x + C$
6	13–15	Definite integrals	Fundamental Theorem of Calculus: $\int_a^b f(x)dx = F(b) - F(a)$ , substitution
7	16	Double integrals	Iterated integration — integrate inner variable first, then outer: $\iint f(x,y)\,dx\,dy$

Key formulas used across these tasks:

Operation	Formula	When to use
Power rule (derivative)	$\frac{d}{dx} x^n = n x^{n-1}$	Polynomial differentiation
Derivative of $\ln x$	$\frac{d}{dx} \ln x = \frac{1}{x}$	Logarithmic differentiation
Derivative of $e^x$	$\frac{d}{dx} e^x = e^x$	Exponential functions
Partial derivative	$\frac{\partial}{\partial x} f(x,y)$ — treat $y$ as constant	Multivariable functions
Power rule (integral)	$\int x^n dx = \frac{x^{n+1}}{n+1} + C$	Polynomial integration
Integral of $e^x$	$\int e^x dx = e^x + C$	Exponential integration
Fundamental Theorem of Calculus	$\int_a^b f(x)dx = F(b) - F(a)$	Definite integrals
Double integral	$\iint f(x,y)\,dx\,dy = \int \left(\int f(x,y)\,dx\right)dy$	Area/volume under surfaces

Relationship to Python tasks: Tasks 9, 10, and 17 implement programmatically what these 14 exercises compute by hand:

Task 9 (9-sum_total.py) automates the $\sum i^2$ closed form you work through in Tasks 0–1
Task 10 (10-matisse.py) implements the power rule for derivatives that Tasks 4–6 practice manually
Task 17 (17-integrate.py) implements the reverse power rule and integration constant that Tasks 11–16 build intuition for

The manual work teaches you what the answer should be; the Python code teaches you how to compute it at scale.

---\n\n## Task-by-Task Reference (Python Implementations)

Each task below highlights the unique challenge it posed and the new technique introduced.

Task 9 — Summation Formula (`9-sum_total.py`)

Challenge: Compute $\sum_{i=1}^{n} i^2$ efficiently without a loop — introducing the closed-form mathematical formula.

Approach: Implement the formula $\frac{n(n+1)(2n+1)}{6}$ using integer arithmetic. Validate that $n$ is a positive integer. Use // for floor division since the numerator is always divisible by 6.

New techniques introduced:

Technique	Purpose
$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$	Closed-form sum of squares — O(1) instead of O(n)
`//` integer floor division	Guarantee integer result when numerator is divisible by denominator
Input validation: `isinstance(n, int) and n >= 1`	Reject non-integer and non-positive inputs
`return None` for invalid input	Sentinel pattern for invalid parameters

Key takeaway: Mathematical formulas can replace loops. The sum of squares formula is a classic closed form — knowing it turns an O(n) problem into O(1).

Task 10 — Polynomial Derivative (`10-matisse.py`)

Challenge: Compute the derivative of a polynomial represented as a list of coefficients — implementing the power rule programmatically.

Approach: Given coefficients $[a_0, a_1, a_2, \dots]$ representing $a_0 + a_1x + a_2x^2 + \dots$ , the derivative is $a_1 + 2a_2x + 3a_3x^2 + \dots$ . For each coefficient at index $i$ , multiply by $i$ (the power) and shift left by 1 position. Handle edge cases: empty list → None, constant polynomial → [0].

New techniques introduced:

Technique	Purpose
Power rule: $\frac{d}{dx}(a_i x^i) = i \cdot a_i x^{i-1}$	Derivative of each term is coefficient × power
Coefficient list representation	Index = power, value = coefficient
`poly[i] * i` for derivative coefficient	Multiply by the exponent (index)
`all(x == 0 for x in out)` → return `[0]`	Handle the zero-polynomial edge case

Key takeaway: The derivative of $a_n x^n$ is $n \cdot a_n x^{n-1}$ . In a coefficient list, you multiply each coefficient by its index (the power) and shift left. The constant term (index 0) is dropped.

Task 17 — Polynomial Integral (`17-integrate.py`)

Challenge: Compute the indefinite integral (antiderivative) of a polynomial — implementing the reverse power rule with an integration constant.

Approach: For each coefficient $a_i$ at index $i$ (representing $a_i x^i$ ), the integral term is $\frac{a_i}{i+1} x^{i+1}$ . Store at index $i+1$ in the output. The constant $C$ goes at index 0. Use integer simplification: if the division is exact, store as int; otherwise store as float.

New techniques introduced:

Technique	Purpose
Reverse power rule: $\int a_i x^i \, dx = \frac{a_i}{i+1} x^{i+1} + C$	Integral of each term divides by the new exponent
`C` as the integration constant	Arbitrary constant added to every indefinite integral
Index shift: input[i] → output[i+1]	The integral increases each term's degree by 1
`int(val)` if exact else `float(val)`	Preserve exact integer results, use float for fractions
`C.is_integer()` float check	Handle the case where C is a float that represents a whole number

Key takeaway: Integration is the inverse of differentiation. Each term $a x^n$ becomes $\frac{a}{n+1} x^{n+1}$ . The integration constant $C$ represents the family of all antiderivatives. Don't forget to shift indices — integrating increases each power by 1.

Technique Inventory

Task	New technique summarized	Category
0–8	Manual computation: $\Sigma$ , $\Pi$ , derivatives, partial derivatives	Manual Calculus
9	Closed-form $\sum i^2$ formula, `//` integer division	Summation
10	Polynomial derivative via power rule, coefficient list representation	Derivatives
11–16	Manual computation: indefinite, definite, double integrals	Manual Calculus
17	Polynomial integral via reverse power rule, integration constant $C$	Integration

Math — Calculus

Learning Objectives

Module Structure

Computation Exercises (Tasks 0–8, 11–16)

Computation Exercises Approach

Task 9 — Summation Formula (9-sum_total.py)

Task 10 — Polynomial Derivative (10-matisse.py)

Task 17 — Polynomial Integral (17-integrate.py)

Technique Inventory

Resources

Task 9 — Summation Formula (`9-sum_total.py`)

Task 10 — Polynomial Derivative (`10-matisse.py`)

Task 17 — Polynomial Integral (`17-integrate.py`)