Calculus Syllabus – Complete Unit-Wise Course Outline
This Calculus syllabus presents a structured, unit-wise overview of topics
commonly covered in high school AP Calculus and introductory college-level calculus.
Students progress from limits and derivatives to integrals and real-world applications.
Who This Calculus Syllabus Is For
This syllabus is suitable for high school students, AP Calculus learners,
college freshmen reviewing fundamentals, homeschool students, and anyone
seeking a clear roadmap of calculus concepts.
Calculus Syllabus (Expandable Units)
Unit 1: Precalculus Review and Functions
- Function notation, domain and range
- Polynomial, rational, exponential, logarithmic functions
- Trigonometric functions and identities
- Graph interpretation and modeling
Unit 2: Limits and Continuity
- Concept of limits (graphical and numerical)
- Limit laws and algebraic evaluation
- One-sided and infinite limits
- Continuity and types of discontinuities
- Intermediate Value Theorem
Unit 3: Derivatives – Foundations
- Derivative as rate of change
- Derivative from first principles
- Basic derivative rules
- Derivatives of exponential and trigonometric functions
- Higher-order derivatives
Unit 4: Derivative Rules and Techniques
- Product and quotient rules
- Chain rule
- Implicit differentiation
- Related rates
- Linear approximation and differentials
Unit 5: Applications of Derivatives
- Increasing and decreasing functions
- First and second derivative tests
- Concavity and curve sketching
- Optimization problems
- Mean Value Theorem
Unit 6: Integrals – Foundations
- Antiderivatives and indefinite integrals
- Riemann sums
- Definite integrals
- Fundamental Theorem of Calculus
- Integrals as net change
Unit 7: Integration Techniques
- Substitution (u-substitution)
- Integration by parts (advanced)
- Trigonometric integrals (advanced)
- Partial fractions (advanced)
Unit 8: Applications of Integrals
- Area between curves
- Volume using disk and washer methods
- Average value of a function
- Motion along a line
- Intro to differential equations
Unit 9: Sequences, Series, and Advanced Topics
- Sequences and convergence
- Infinite series overview
- Taylor and Maclaurin series (advanced)
- Parametric and polar equations (advanced)
Learning Outcomes
Students completing this calculus syllabus will be able to evaluate limits,
compute derivatives and integrals, interpret results in real-world contexts,
and solve application-based problems involving change and accumulation.
Need Help With Calculus?
For topic-wise guidance, practice support, or syllabus-based tutoring,
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