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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...
The derivative measures instantaneous rate of change. Formally, the derivative of \(f\) at a point \(x\) is defined as \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}, \] provide...
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 exp...
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 differ...
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) +...
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...
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 \(\i...
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...