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The limit is the foundational concept that underlies all of calculus. Informally, the limit of \(f(x)\) as \(x\) approaches \(c\) is the value \(L\) that \(f(x)\) approaches as \(x\) gets arbitrarily close to \(c\). More precisely, we write \(\lim_{x \to c} f(x) = L\) to mean that \(f(x)\) can be made as close to \(L\)...
The derivative measures instantaneous rate of change. Formally, the derivative of \(f\) at a point \(x\) is defined as \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}, \] provided this limit exists. An equivalent formulation focuses on a specific point \(a\): \[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}. \] Th...
Computing derivatives directly from the limit definition is tedious, so a collection of rules streamlines the process. The Power Rule states that if \(f(x) = x^n\) for any real exponent \(n\), then \(f'(x) = n x^{n-1}\). This rule applies to integer, fractional, and negative exponents alike, making it one of the most f...
Sometimes the relationship between \(x\) and \(y\) is given implicitly by an equation like \(x^2 + y^2 = 25\) rather than explicitly as \(y = f(x)\). In such cases, implicit differentiation provides a way to find \(dy/dx\). One differentiates both sides of the equation with respect to \(x\), treating \(y\) as a functio...
Integration can be viewed as the inverse of differentiation. An indefinite integral \(\int f(x) \, dx\) represents the family of all antiderivatives of \(f\), expressed as \(F(x) + C\), where \(F'(x) = f(x)\) and \(C\) is the constant of integration capturing the vertical shifts in this family. A definite integral \(\i...
For many integrals, no single antiderivative rule applies directly, and one must transform the integral into a more tractable form. The most common such technique is u-substitution, which reverses the Chain Rule. One identifies a portion of the integrand as \(u = g(x)\), computes the differential \(du = g'(x) \, dx\),...
Integration gives a precise language for measuring accumulation, with practical applications in geometry. The area between two curves \(y = f(x)\) and \(y = g(x)\) is given by \(\int_a^b |f(x) - g(x)| \, dx\), where \(a\) and \(b\) are the points of intersection. When \(f(x) \geq g(x)\) throughout, this simplifies to \...
A series \(\sum_{n=1}^{\infty} a_n\) assigns a value to an infinite sum by means of partial sums. If the partial sums \(S_n = a_1 + a_2 + \cdots + a_n\) approach a finite limit as \(n \to \infty\), the series converges; otherwise, it diverges. Several standard series serve as benchmarks. The geometric series \(\sum_{n=...