Learning with AI (Calculus)
Revisiting A’ Level Maths – Calculus
🔢 What is Calculus?
Calculus is the branch of mathematics that studies change. It has two big halves:
1. Differential Calculus
- Focus: Rates of change, slopes of curves
- Main tool: Derivatives
- Examples:
- “How fast is something changing right now?”
- “What’s the steepness (gradient) at this exact point?”
2. Integral Calculus
- Focus: Accumulation, area under curves
- Main tool: Integrals
- Examples:
- “How much total distance was covered?”
- “What’s the area under this curve?”
- “Add up infinitely small parts”
Bonus: They’re Connected!
They’re two sides of the same coin — and the Fundamental Theorem of Calculus links them.
It basically says:
Differentiation undoes integration, and
Integration undoes differentiation.
But in real life? Here’s where stuff like that can pop up:
✅ 1. Area/Volume Problems
✅ 2. Cost or Revenue Models
Again, if you want rate of change of revenue, take the derivative.
✅ 3. Motion with Variable Forces
Boom — product rule.
Bottom Line:
A lot of textbook examples are engineered for practice — but in real-world modeling, product rule shows up naturally when two varying things are multiplied.
Example
Answer:
Notatation:
Calculus with code uses regular ASCII to represent derivatives and integrals. For derivatives, functions like f(x) = x**2 can be differentiated manually or symbolically using tools like SymPy in Python: diff(f(x), x).
For numerical differentiation, we approximate slopes using: (f(x+h) - f(x)) / h. Integrals can be written as ∫ f(x) dx, but in code we use integrate(f(x), x) for symbolic or use numerical methods like Riemann sums or Simpson’s rule.
ASCII syntax like ** for powers and lambda x: for anonymous functions makes coding calculus readable and practical without needing special math symbols.
