Analytic Geometry: Formulas, Lines, Circles & Conics
Read ArticleAlternating Series Test: Leibniz Rule & Examples
The Alternating Series Test, also called the Leibniz Test, says that a series whose signs flip, ∑ (-1)ⁿ bₙ or ∑ (-1)ⁿ⁺¹ bₙ with every bₙ > 0, converges when two things hold: the terms bₙ are eventually decreasing, and lim n→∞ bₙ = 0. When both are true, the sum settles on a finite value, and the error from stopping at the Nth term is never larger than the first term you left out, S - S N ≤ b N+1.
Read moreAlternating Series: Test, Convergence & Error Bound
An alternating series is a sum whose terms flip sign, written ∑ (-1)ⁿ bₙ or ∑ (-1)ⁿ⁺¹ bₙ with bₙ ≥ 0. The Alternating Series Test says such a sum converges when the sizes bₙ decrease and shrink to zero: b n+1 ≤ bₙ and lim n → ∞ bₙ = 0. The alternating harmonic series 1 - 12 + 13 - 14 + … converges to ln 2, and if you stop after N terms the error is no bigger than the first term you dropped, R N ≤ b N+1.
Read moreAbsolute Value Functions: Graph, Derivative & Integral
Absolute Value Functions measure distance from zero, so f(x) = |x| equals x when x ≥ 0 and -x when x < 0. The graph is a V with its vertex at the origin, domain all real numbers, and range y ≥ 0. In calculus the surprise is that |x| is continuous everywhere yet not differentiable at x = 0, where the V has a sharp corner; away from that corner its derivative is sgn(x) and its integral is (x|x|)/2 + C.
Read moreWhat Is Calculus? Definition, Branches & Examples
What is calculus? It is the mathematics of continuous change. It has two branches that mirror each other: differential calculus, which measures how fast something changes (the derivative, the slope of a curve), and integral calculus, which measures how much accumulates (the integral, the area under a curve). Both rest on one idea, the limit, and the Fundamental Theorem of Calculus proves the two branches are inverses of each other.
Read moreTrigonometric Substitution: Rules & Examples
Trigonometric substitution is an integration method that replaces a square root of a quadratic with a trig function, so a Pythagorean identity can clear the root. Match the radical to its substitution: √(a²-x²) takes x=asinθ, √(a²+x²) takes x=atanθ, and √(x²-a²) takes x=asecθ. After integrating in θ, a reference triangle converts the answer back to x.
Read moreTrapezoidal Rule: Formula, Examples & Error
The Trapezoidal Rule estimates a definite integral ∫ₐ b f(x),dx by slicing [a,b] into n strips of width h=(b-a)/n and treating each strip as a trapezoid, giving Tₙ=h/2[f(x₀)+2f(x₁)+2f(x₂)+…+2f(x n-1)+f(xₙ)]. It is the simplest reliable way to find an area under a curve when the integral has no clean antiderivative, and its error shrinks like 1/(n²) as you add strips.
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