🧮 What is Calculus?
Calculus is a branch of mathematics focused on how things change. It helps us analyze motion, growth, areas, and more — from planetary orbits to machine learning.
It’s divided into two main parts: Differential Calculus (concerned with rates of change) and Integral Calculus (concerned with accumulation and area under curves).
🔗 Limits & Continuity
Limits are the foundation of Calculus. They help us understand what a function is approaching at a certain point.
Continuity means there are no breaks, jumps, or holes in the graph of a function.
Example: lim(x→2) (x² - 4)/(x - 2) simplifies to 4 using limit rules.
📉 Derivatives
A derivative tells you how fast something is changing at any point. It's like the speedometer of math.
🧾 Definition
The derivative of a function f(x) at a point x is:
f'(x) = lim(h→0) [f(x + h) - f(x)] / h📚 Common Derivative Rules
- Power Rule: d/dx [xⁿ] = n·xⁿ⁻¹
- Constant Rule: d/dx [c] = 0
- Sum Rule: d/dx [f(x) + g(x)] = f'(x) + g'(x)
- Product Rule: d/dx [uv] = u'v + uv'
- Quotient Rule: d/dx [u/v] = (u'v - uv') / v²
- Chain Rule: d/dx [f(g(x))] = f'(g(x)) · g'(x)
🧪 Examples
- d/dx [x²] = 2x
- d/dx [3x³ + 2x] = 9x² + 2
- d/dx [sin(x)] = cos(x)
- d/dx [ln(x)] = 1/x
🛠 Applications of Derivatives
- Finding maximum or minimum values
- Understanding acceleration, speed
- Optimizing functions in economics or science
🧮 Integrals
An integral helps you calculate area under a curve, total distance, or total accumulation.
🧾 Definitions
- Indefinite Integral: ∫f(x) dx — represents a family of functions (includes "+ C")
- Definite Integral: ∫ab f(x) dx — gives a number (area between a and b)
📚 Common Rules
- ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, for n ≠ -1
- ∫1/x dx = ln|x| + C
- ∫eˣ dx = eˣ + C
- ∫cos(x) dx = sin(x) + C
🛠 Techniques
- U-Substitution
- Integration by Parts
- Partial Fractions
🧪 Applications
- Finding area under a curve
- Calculating total accumulation of quantities
- Physics: Work, energy, and motion
🧠 Fundamental Theorem of Calculus
This connects derivatives and integrals:
If F is the antiderivative of f, then:
∫ab f(x) dx = F(b) - F(a)📚 More Advanced Topics
- Implicit Differentiation – When functions are not explicitly solved for y
- Differential Equations – Equations involving derivatives
- Parametric & Polar Coordinates
- Series and Sequences (Calculus II)