Discover the mathematics of change, motion, and rates with limits, derivatives, and integrals
Calculus is the mathematical study of continuous change. It provides powerful tools for understanding how quantities change and accumulate, forming the foundation of physics, engineering, economics, and many other fields.
Essential for physics, engineering, and all quantitative sciences
Critical for understanding rates of change in business and finance
Develops sophisticated mathematical reasoning and problem-solving
Solves complex problems in medicine, climate science, and technology
Understand the foundation of calculus with limits and continuous functions
Practice LimitsLinks differentiation and integration
Part 1: \frac{d}{dx}\int_a^x f(t) \, dt = f(x)
Part 2: \int_a^b f(x) \, dx = F(b) - F(a)
Guarantees a point where instantaneous rate equals average rate
f'(c) = \frac{f(b) - f(a)}{b - a} for some c in (a,b)
Continuous functions take every value between f(a) and f(b)
If f(a) < k < f(b), then āc in (a,b) such that f(c) = k
Represents functions as infinite polynomials
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n
Calculus describes velocity, acceleration, and projectile motion
Marginal analysis and optimization problems
Growth rates and population modeling
Algorithms, graphics, and machine learning
Eudoxus and Archimedes developed early concepts of limits and areas
Isaac Newton and Gottfried Leibniz independently developed calculus
Calculus powers modern science, technology, and engineering
Draw graphs and diagrams. Calculus is about understanding change visually.
Calculus requires consistent practice. Work problems daily.
Understand how limits, derivatives, and integrals are related.
Strong algebra skills are essential for calculus success.
Don't just memorize formulas - understand the concepts behind them.
Show all steps clearly. Calculus problems often have multiple steps.