Introduction to Calculus
Understanding differential & integral calculus β core ideas, examples, and applications.
Essentials
Course Outline
Differential Calculus
Limits β’ Derivatives β’ Applications (tangent lines, rates, optimization)
Integral Calculus
Indefinite/Definite Integrals β’ Techniques β’ Area & accumulation problems
Foundations
Limits, continuity, algebra of functions
Applications
Physics (velocity/acceleration), engineering, economics & area computations
What is Calculus?
Calculus is the study of continuous change. Historically developed by Newton and Leibniz, it focuses on two complementary ideas:
Differential calculus studies rates of change (derivatives). The derivative f'(x) = dy/dx measures how the function y = f(x) changes as x changes. It arises from the limit of a quotient: the slope of the secant line approaches the slope of the tangent line.
Integral calculus reverses differentiation: integration accumulates small pieces to get a whole. Indefinite integrals include an arbitrary constant (C); definite integrals compute accumulated values like area under a curve.
Limits describe the behavior of a function as the input approaches a certain value. Continuity means the limit equals the function value. Limits are the foundation on which both derivatives and integrals are built.
Interactive Demo β Secant β Tangent (Derivative)
Use the slider to move the second point (h). The slope of the secant line approaches the tangent slope as h β 0 for f(x) = xΒ² at x = 1.
Secant slope: 2.6
Tangent (derivative) at x: 2
Secant
Tangent
Key Formulas & Notes
Derivative: f'(x) = limββ0 (f(x+h) - f(x))/h
Indefinite Integral: β« f(x) dx = F(x) + C
Definite Integral: β«βα΅ f(x) dx = F(b) - F(a)
Applications: Motion, Area, Optimization
Tools: Analytical techniques, substitution, parts
Prereqs: Functions, algebra, exponents