Basics

Limits: the value that a function $f(x)$ approaches as the input $x$ approaches some value $c$. Understanding limits is crucial for defining continuity and derivatives.

• Notation: $\lim_{x \to c} f(x) = L$, which means as $x$ approaches $c$, $f(x)$ approaches $L$.

• Techniques for evaluating limits include direct substitution, factoring, rationalizing, and using L'Hôpital's Rule.

Derivatives: the instantaneous rate of change of a function, representing the slope of the tangent line at a given point. It measures how a function's output changes with respect to its input.

• Definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Basic rules: Power Rule, Product Rule, Quotient Rule, and Chain Rule.

• Applications: finding critical points, optimization problems, related rates, and curve sketching.

Integrals: the area under a curve/function. Integration is the reverse process of differentiation and is used to find the area, volume, and other accumulation-related quantities.

• Types: Definite integrals (with limits of integration) and indefinite integrals (antiderivatives).

• Fundamental Theorem of Calculus: connects differentiation and integration, stating that the derivative of the integral of a function is the original function itself.

• Techniques: substitution, integration by parts, trigonometric substitution, and partial fractions.

• Applications: finding areas, volumes, average values, and solving differential equations.