Calculus I

Limits and Continuity
• Concept of a limit
• One-sided limits and infinite limits
• Continuity and types of discontinuities
• Limit laws, squeeze theorem
• Precise (ε-δ) definition of limit (optional in some courses)

Derivatives and Differentiation Rules
• Definition of derivative (as limit of difference quotient)
• Tangent lines and rates of change
• Power, constant, and constant-multiple rules
• Sum/difference and product/quotient rules
• Chain rule
• Derivatives of exponential, logarithmic, trigonometric, and inverse-trig functions
• Higher-order derivatives

Applications of Derivatives
• Curve sketching (increasing/decreasing, concavity, extrema, inflection points)
• Optimization problems (max/min in real-world settings)
• Related rates
• Linear approximation and differentials
• Mean Value Theorem, Rolle’s Theorem

Integrals (Antiderivatives)
• Indefinite integrals and basic integration rules
• Initial value problems
• Definite integrals and the area problem
• Fundamental Theorem of Calculus (Parts I & II)
• Substitution method (u-substitution)

Applications of Integrals (often end of Calc 1)
• Area between curves
• Accumulated change
• Simple physics applications (distance from velocity, work, etc.)