Single-Variable
Learning Objectives
Learning objectives of the Single-Variable section.
Summary Table
Summary of the Single-Variable section.
| Concept | Description | Notation |
|---|---|---|
| Derivative | Instantaneous rate of change; slope of the tangent line | \(f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\) |
| Constant Rule | Derivative of a constant function | \(\frac{d}{dx}[c] = 0\) |
| Power Rule | Derivative of a power function | \(\frac{d}{dx}[x^n] = nx^{n-1}\) |
| Sum/Difference Rule | Derivative of a sum or difference | \(\frac{d}{dx}[g(x) \pm h(x)] = g'(x) \pm h'(x)\) |
| Product Rule | Derivative of a product | \(\frac{d}{dx}[g(x)h(x)] = g'(x)h(x) + g(x)h'(x)\) |
| Quotient Rule | Derivative of a quotient | \(\frac{d}{dx}\left[\frac{g(x)}{h(x)}\right] = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\) |
| Chain Rule | Derivative of a composite function | \(\frac{d}{dx}[g(h(x))] = g'(h(x)) \cdot h'(x)\) |
| Definite Integral | Net area under a curve from \(a\) to \(b\) | \(\int_a^b f(x)\, dx\) |
| Indefinite Integral | General antiderivative | \(\int f(x)\, dx = F(x) + C\) |
| Constant Rule (Int.) | Integral of a constant | \(\int c\,dx = cx + C\) |
| Power Rule (Int.) | Integral of a power function (when \(n \neq -1\)) | \(\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\) |
| Sum/Difference Rule (Int.) | Integral of a sum or difference | \(\int [g(x) \pm h(x)]\,dx = \int g(x)\,dx \pm \int h(x)\,dx\) |
| Exponential Rule (base e) | Integral of \(e^x\) | \(\int e^x\,dx = e^x + C\) |
| Exponential Rule (base a) | Integral of \(a^x\) | \(\int a^x\,dx = \frac{a^x}{\ln a} + C\) |
| Logarithmic Rule | Integral of reciprocal | \(\int \frac{1}{x}\,dx = \ln|x| + C\) |
| Chain Rule (u-sub) | Substitution technique | \(\int f(g(x))g'(x)\,dx = \int f(u)\,du,\ u = g(x)\) |
| Integration by Parts | Product of functions technique | \(\int u(x)v'(x)\,dx = u(x)v(x) - \int u'(x)v(x)\,dx\) |
| Fundamental Theorem | Derivative of accumulation function | \(F(x) = \int_a^x f(t)\,dt \Rightarrow F'(x) = f(x)\) |
| FTC Evaluation | Total accumulation from \(a\) to \(b\) | \(\int_a^b f(x)\,dx = F(b) - F(a)\) |
| Optimization | Find extrema using derivatives | \(f'(x) = 0\), \(f''(x) \gtrless 0\) |
| First Derivative Test | Sign change test for local extrema | \(f'(x): + \to -\) (max), \(- \to +\) (min), no change = none |
| Second Derivative Test | Concavity test for local extrema | \(f''(x) > 0\): min, \(f''(x) < 0\): max, \(f''(x) = 0\): inconclusive |