Arc Length Formula in Calculus: Derivation and Examples
The arc length of y = f(x) from a to b is ∫√(1 + [f'(x)]²) dx. Derivation, worked examples (x^(3/2), a circle, a catenary) and parametric curves.
Author: Yasir Hafeez
Yasir Hafeez is the founder and editor of Calculus Calc. He writes and checks step-by-step calculus guides and builds the site's interactive calculators.
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Mean Value Theorem: Statement, Meaning and Worked Examples
The Mean Value Theorem guarantees a point where the instantaneous rate equals the average rate. Hypotheses, Rolle’s theorem, and how to find c step by step.
Linear Approximation (Linearization): Formula and Examples
Linear approximation uses the tangent line L(x) = f(a) + f'(a)(x − a) to estimate values like √4.1 and sin 0.1. Includes error, differentials and examples.
Newton’s Method: Formula, Worked Example and When It Fails
Newton’s method finds roots with xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Step-by-step iterations for x³ − 2x − 5, approximating √2, and when the method fails.
How to Find the Equation of a Tangent Line (and Normal Line)
Find the tangent line to y = f(x) at x = a in three steps: point, slope f'(a), point-slope form. Includes normal lines and 4 worked examples.
How to Find Inflection Points and Concavity (with Examples)
An inflection point is where concavity changes. Find candidates with f”(x) = 0 or undefined, then confirm a sign change. Worked examples and pitfalls.
How to Find Critical Points (and Classify Them) Step by Step
Find critical points by solving f'(x) = 0 or where f’ is undefined, then classify them with the first or second derivative test. Worked examples inside.
Fundamental Theorem of Calculus (Parts 1 and 2) Explained
The Fundamental Theorem of Calculus links derivatives and integrals. Both parts explained with intuition, formulas and worked examples.
Improper Integrals: How to Tell If They Converge or Diverge
Improper integrals have infinite limits or infinite integrands. Learn the limit definition, the p-integral test, comparison, and 6 worked examples.
Partial Fraction Decomposition: Steps and Examples
How to integrate rational functions with partial fractions: distinct linear factors, repeated factors and irreducible quadratics, with worked examples.