Revisiting A’ Level Maths – Calculus with the help of AI

🔢 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:

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.

Here's some simple Python code using SymPy, a symbolic math library, to compute the derivative using just ASCII:

from sympy import symbols, diff

# Define the variable
x = symbols('x')

# Define the function
f = x**2

# Compute the derivative
f_prime = diff(f, x)

# Print results
print("f(x) =", f)
print("f'(x) =", f_prime)

f(x) = x**2
f'(x) = 2*x

Next, find the area between 2 parabolas – see if you can do this in code?

Try solve this using code and then come back and check the answer!