Lesson 1: What is Calculus? Limits Intuition

1. What is Calculus?

Calculus is the mathematics of **change**.

It has two main parts:

• Differential Calculus: how things change at an instant (slopes, rates, speed)
• Integral Calculus: how change accumulates (areas, total distance, volume)

Used in:

• Machine learning: gradients, optimization
• Physics: velocity, acceleration
• Economics: marginal cost, growth rates

2. The Big Idea: Limits

A limit asks: “What value does a function approach as x gets very close to a number?”

Example: What happens to \(f(x) = x^{2}\) when \(x\) is very close to 3?

\(f(2.9) = 8.41, f(2.99) = 8.9401, f(3.01) = 9.0601, f(3.001) = 9.006001\) → approaches 9

Summary – Lesson 1

• Calculus studies change (derivatives) and accumulation (integrals)
• Limits are the foundation: “what value is approached?”
• Many limits can be found by direct substitution

Next: more on limits, including tricky cases.