Calculus Summary Notes
1. Limits and Continuity
- Definition: The limit of f(x) as x → a is L if f(x) approaches L as x approaches a.
- Notation: lim■→■ f(x) = L.
- Techniques: Direct substitution, factoring, rationalization, and L'Hôpital's Rule.
- Continuity: f is continuous at a if lim■→■ f(x) = f(a).
- Common limits: lim■→0 (sin x)/x = 1, lim■→∞ (1 + 1/x)■ = e.
2. Derivatives
- Definition: f'(x) = lim■→0 [f(x+h) − f(x)] / h.
- Basic rules: Power rule, product rule, quotient rule, chain rule.
- Common derivatives: (sin x)' = cos x, (cos x)' = −sin x, (e■)' = e■, (ln x)' = 1/x.
- Applications: Tangent lines, increasing/decreasing functions, maxima/minima, optimization,
related rates.
3. Integrals
- Definition: ∫ f(x) dx represents the area under the curve of f(x).
- Fundamental Theorem of Calculus: ∫■■ f'(x) dx = f(b) − f(a).
- Techniques: Substitution, integration by parts, partial fractions, trigonometric substitution.
- Common integrals: ∫ e■ dx = e■ + C, ∫ 1/x dx = ln|x| + C, ∫ sin x dx = −cos x + C.
- Improper integrals: Check for convergence using limits.
4. Series and Sequences
- Convergence tests: n-th term test, ratio test, root test, integral test, alternating series test.
- Power series: ∑ a■(x − c)■, converges on an interval around c.
- Taylor/Maclaurin Series: f(x) = ∑ (f■■■(a)/n!) (x − a)■.
- Common expansions: e■ = ∑ x■/n!, sin x = ∑ (−1)■x²■■¹/(2n+1)!.
5. Multivariable Calculus
- Partial derivatives: ∂f/∂x, ∂f/∂y measure rate of change in one direction.
- Gradient vector: ∇f = (∂f/∂x, ∂f/∂y), points in direction of greatest increase.
- Double integrals: ■ f(x, y) dA compute volume under surface.
- Applications: Area, volume, optimization with constraints (Lagrange multipliers).
- Line integrals: ∫■C■ f(x, y) ds measure accumulation along a path.
6. Key Graphical Ideas
- Limit → approaching a point (smooth or jump).
- Derivative → slope of tangent line.
- Integral → area under curve.
- Gradient → arrow pointing to steepest rise.
- Divergence and curl appear in vector fields.