Calculus Fundamentals: A Comprehensive
Study Guide
Overview
Calculus is the mathematical study of continuous change, in the same way that geometry is the
study of shape and algebra is the study of generalizations of arithmetic operations. This guide
explores the core foundational pillars of introductory calculus: Limits, Derivatives, and Integrals.
1. Limits and Continuity
The concept of a limit is the foundational baseline for all of calculus. It describes the behavior of a
function as the input approaches a specific value, rather than evaluating the function at that exact
value itself.
Definition: If f(x) becomes arbitrarily close to a unique number L as x approaches c from both
sides, then the limit of f(x) as x approaches c is L.
Notation: limx → c f(x) = L
2. The Derivative and Rates of Change
The derivative measures the instantaneous rate of change of a function with respect to one of its
variables. Geometrically, the derivative at a given point represents the slope of the tangent line to
the graph of the function at that specific point.
The Limit Definition of a Derivative:
f'(x) = limh → 0 [f(x + h) - f(x)] / h
Key Differentiation Rules:
• Power Rule: d/dx [xn] = n · xn-1
• Product Rule: d/dx [u · v] = 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)
3. Integrals and Accumulation
Integration is the inverse operation of differentiation. While differentiation breaks a curve down into
instantaneous slopes, integration accumulates individual pieces to determine total areas, volumes,
and displacements.
Definite vs. Indefinite Integrals: An indefinite integral represents a family of functions (the
antiderivative) and always includes a constant of integration (+ C). A definite integral calculates the
precise net signed area under a curve between two specific boundaries, a and b.
Fundamental Theorem of Calculus (FTC):
If f is continuous on [a, b] and F is the antiderivative of f, then:
∫ab f(x) dx = F(b) - F(a)
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