What is Calculus?
Calculus is the mathematical study of change and motion, focusing on concepts such as derivatives and integrals. Widely used in engineering, physics, and computer science, calculus forms the backbone of scientific modeling and modern mathematical analysis.
Why Learn Calculus?
- Model physical phenomena
- Predict system behavior
- Enable algorithmic computation
Key Pillars
Limits
Define the behavior of functions near points
Derivatives
Measure rates of change and slopes
Integrals
Calculate areas under curves and accumulate totals
Applications
Engineering, finance, biology, etc
Differentiation Explained
Definition
The derivative measures instantaneous rate of change or the slope of a curve.
Rules
- • Power Rule
- • Chain Rule
- • Product Rule
Applications
- • Motion modeling
- • Optimization
- • Curve fitting
Integration Techniques
Fundamental Concept
Integrals calculate total accumulation over a domain. Includes definite and indefinite forms.
∫f(x) dx = F(x) + C
Common Methods
- • Substitution
- • Integration by Parts
- • Partial Fractions
- • Numerical Integration
Real-World Uses
Physics Applications
Calculus models velocity, acceleration, energy transfer, and electromagnetic fields in physical systems.
Financial Modeling
Derivatives are crucial for risk assessment, option pricing, and algorithmic trading.
Try a Practice Problem
Problem: Find the derivative of $f(x) = x^3 \cdot \sin(x)$
f'(x) = ?
Answer (Click to Expand):
Solution