Calculus Syllabus

Calculus is a dynamic, problem-solving course designed for students who have completed Pre-Calculus. Aligned with the Common Core Standards for Calculus, this course introduces foundational concepts like limits, continuity, derivatives and their applications, as well as definite and indefinite integrals. Students will also explore infinite sequences and series, plane curves, polar coordinates, and basic differential equations. Ideal for students already familiar with trigonometric functions, polar and exponential coordinate systems, and vector and matrix operations up to 3x3, Calculus combines rigorous theory with engaging problem-solving exercises, logic puzzles, and competition-style challenges to inspire mastery and confidence in advanced mathematics. It is aligned with AP Calculus A/B.
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Required Textbook: Art of Problem Solving: Calculus

Concepts, skills, and learning tools students see in this course include, but are not limited to:

• Functions, Continuity and Limits
• Derivatives
• Integrals
• Problem-solving skills, logic puzzles, algebraic and geometric thinking, competition-style problems

Upon completing the course, students will be able to:

• Foundations of Calculus: Explore set theory, set notation, and their applications to functions, as well as properties of functions, continuity, and limits.
• Derivatives and Their Applications: Learn to calculate derivatives of common functions, apply the Chain Rule, and use derivatives to solve real-life problems.
• Integrals and Techniques of Integration: Perform antiderivatives and use methods like integration by parts, substitution, and partial fractions to solve real-life applications, including areas, volumes, and curve lengths.
• Advanced Calculus Concepts: Study limits involving infinity, use L'Hopital's Rule for indeterminate forms, explore improper integrals, and understand differential equations, including second-order linear equations.
• AP Calculus A/B Exam: Students who engage with all of the classes and homework will be well-prepared to take the AP Calculus exam.

Students registering for this course should be comfortable with the following Math:‍

• Introductory Geometry: Foundations in trigonometric functions and identities.
• Algebra 2 and Precalculus Concepts:
  • • Graphing, solving equations, and systems of equations with three unknowns.
    • Logarithms and exponential functions.
    • Polynomials: factoring, finding roots, and analyzing graph behavior.
    • Sequences and series.
    • Understanding radians, polar coordinates, and complex numbers.
    • Geometry and Measurement: Understand triangle and circle properties, apply the Pythagorean Theorem, and calculate volumes of 3D shapes with straight edges.
    • Algebra and Equations: Solve equations and systems with up to 3 unknowns, work with logarithms, exponents, and polynomials (factoring, roots, and graph behavior).
    • Graphing and Complex Numbers: Graph functions on the Cartesian Plane and perform arithmetic and simplification with complex numbers.
• Completion of Pre-Calculus course or equivalent.

• Foundations of Functions and Limits (6 weeks): Explore set and interval notation, function basics, graphing, trigonometric identities, and the concepts of limits and continuity.
• Derivatives and Applications (11 weeks): Understand the fundamentals of derivatives, rules (product, quotient, chain), and their real-life applications such as optimization, Newton’s Method, and rates of change.
• Integrals and Techniques (10 weeks): Study integration basics, the Fundamental Theorem of Calculus, and advanced techniques like integration by parts, substitution, and partial fractions. Apply integrals to calculate areas, volumes, curve lengths, and more.
• Infinity, Indeterminate Forms, and Improper Integrals (5 weeks): Delve into limits at infinity, L'Hopital’s Rule, and improper integrals.
• Differential Equations and AP Review (6 weeks): Learn basic and second-order differential equations, slope fields, and review key concepts in preparation for AP Calculus AB, culminating with a mock AP exam.