Numerical Integration: Methods, Formulas & Error
Read ArticleMonotonic Sequences: Increasing, Decreasing, Convergent
Monotonic sequences are sequences that head in one direction only: every term is greater than or equal to the one before (increasing) or less than or equal to the one before (decreasing). You test the direction by checking the sign of a n+1 - aₙ, comparing the ratio (a n+1)/aₙ to 1 for positive terms, or looking at f'(x) when aₙ = f(n). The payoff is the Monotone Convergence Theorem: a sequence that is both monotonic and bounded must converge.
Read moreMethods Of Integration: Techniques & Examples
The methods of integration are the standard techniques for finding an antiderivative: direct standard formulas, substitution (the reverse chain rule), integration by parts (the reverse product rule), partial fractions for rational functions, trigonometric substitution for square-root expressions, and trigonometric integrals handled with identities. Which one to use is decided by what the integrand looks like, and every result can be checked by differentiating the answer back to the integrand.
Read moreMarginal Cost and Marginal Revenue With Calculus
Marginal cost and marginal revenue are the derivatives of the total cost and total revenue functions: MC = C'(x) and MR = R'(x). Each measures the rate of change of a total as one more unit is made or sold, so it closely approximates the cost or revenue of that next unit. Profit P(x) = R(x) - C(x) is largest where marginal profit is zero, which means MR = MC, confirmed by the second-order check P''(x) < 0.
Read moreLogarithmic Differentiation: Formula, Steps & Examples
Logarithmic differentiation is a technique that takes the natural log of both sides of y = f(x), uses log laws to break products, quotients, and powers into sums, differences, and multiples, then differentiates implicitly to get (y')/y on the left. Solving gives y' = y·(the log-expanded derivative). Reach for it when the base and the exponent both vary, as in y = x x, or when a function is a large product or quotient of powers.
Read moreList of Integrals: Standard Integral Table
A List of Integrals is an organised table of standard antiderivatives: for each function you look up the formula whose derivative returns that function, then add the constant of integration +C. This page tabulates the power, exponential, logarithmic, trigonometric, inverse-trigonometric, and hyperbolic forms, plus the common patterns 1/(a²+x²) and 1/(√(a²-x²)), and every entry is checked by differentiating it back to the original function.
Read moreLinear Differential Equations: Integrating Factor Method
Linear differential equations are first-order equations of the form dy/dx + P(x),y = Q(x), where y and its derivative appear only to the first power and never multiplied together. You solve them with the integrating factor μ(x) = e ∫ P(x),dx: multiply through by μ, the left side collapses into d/dx[μ y], and one integration gives y = 1/(μ)∫ μ,Q,dx.
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