Course Description

This course begins with four major ideas: limits, derivatives, indefinite integrals, and definite integrals. It then moves into their applications and continues through sequences and series, parametric and polar equations, vectors in a plane, and advanced integration techniques.

Students learn to work with functions represented graphically, numerically, analytically, and verbally. The derivative is understood as a limit and rate of change, and the definite integral is understood as a limit of sums and as net accumulation of change.

The course emphasizes understanding the “why” behind the mathematics so that problem-solving skills become clear, usable, and lasting.

Grading

Student Proficiencies

• Work with functions in graphical, numerical, analytical, and verbal forms
• Understand derivatives as rate of change and local linear approximation
• Understand definite integrals as Riemann sums and net accumulation
• Use both parts of the Fundamental Theorem of Calculus
• Communicate mathematics clearly in writing and discussion
• Model real situations with functions, differential equations, or integrals
• Use technology to explore, solve, and verify results
• Judge the reasonableness of solutions
• Develop an appreciation of calculus as a coherent body of knowledge

Course Design Philosophy

This course is intentionally structured to develop:

Students consistently engage in all Mathematical Practices (MP1–MP8) across every unit.

Course Outline

Unit-by-Unit Alignment with CR + MP + Activities

Semester 1

Unit 1: Limits and Continuity

Battaglia: P.4, P.5, 1.1–1.4

Focus: Foundational understanding of limits and continuity across representations

Sample Learning Activities:

• Analyze limit behavior using graphs, tables, and algebraic simplification
• Write explanations comparing one-sided vs. two-sided limits
• Apply the Intermediate Value Theorem to justify existence of solutions

CR Alignment:

• CR1 (Multiple representations)
• CR2 (Conceptual understanding)
• CR4 (Modeling situations with limits)

Mathematical Practices:

• MP1 (Modeling)
• MP2 (Representations)
• MP6 (Communication of reasoning)

Unit 2: Differentiation – Definition and Basic Rules

Battaglia: 2.1–2.5

Focus: Derivative as rate of change and slope

Sample Learning Activities:

• Compute derivatives from first principles
• Interpret derivatives in real-world contexts (velocity, growth)
• Compare symbolic derivatives with graphical slopes

CR Alignment:

• CR1, CR2, CR6

Mathematical Practices:

• MP2 (Connections between representations)
• MP3 (Procedural fluency)
• MP7 (Use of notation and structure)

Unit 3: Composite, Implicit, and Inverse Functions

Battaglia: 2.6–2.9

Focus: Advanced differentiation techniques

Sample Learning Activities:

• Apply chain rule in nested function contexts
• Perform implicit differentiation with interpretation
• Justify relationships between inverse functions

CR Alignment:

• CR2, CR6, CR7

Mathematical Practices:

• MP3, MP6, MP7

Unit 4: Contextual Applications of Differentiation

Battaglia: 3.1–3.3

Focus: Interpreting derivatives in applied settings

Sample Learning Activities:

• Solve related rates problems with clear variable definitions
• Model motion scenarios and interpret units
• Translate verbal descriptions into derivative equations

CR Alignment:

• CR4, CR5

Mathematical Practices:

• MP1 (Modeling)
• MP4 (Interpretation)
• MP6 (Communication)

Unit 5: Analytical Applications of Differentiation

Battaglia: 3.4–3.9

Focus: Using derivatives to analyze function behavior

Sample Learning Activities:

• Justify extrema using first and second derivative tests
• Write full curve analysis summaries
• Solve optimization problems with constraints

CR Alignment:

• CR3, CR5, CR7

Mathematical Practices:

• MP2, MP4, MP6

Unit 6: Integration and Accumulation

Battaglia: 4.1–4.5

Focus: Accumulation and area via integrals

Sample Learning Activities:

• Approximate area using Riemann sums
• Connect antiderivatives to accumulation functions
• Explain the Fundamental Theorem of Calculus conceptually

CR Alignment:

• CR1, CR2, CR6

Mathematical Practices:

• MP2, MP3, MP7

Unit 7: Differential Equations

Battaglia: 5.1–5.4

Focus: Modeling change with differential equations

Sample Learning Activities:

• Sketch and interpret slope fields using technology
• Solve separable differential equations
• Model exponential growth and decay

CR Alignment:

• CR4, CR5, CR8

Mathematical Practices:

• MP1, MP4, MP8 (technology use)

Unit 8: Applications of Integration

Battaglia: 6.1–6.4

Focus: Physical and geometric applications

Sample Learning Activities:

• Set up and evaluate area between curves
• Model volume using disk, washer, and shell methods
• Interpret average value of a function in context

CR Alignment:

• CR4, CR5, CR6

Mathematical Practices:

• MP1, MP4, MP6

Semester 2

Unit 9: Parametric, Polar, and Vector Functions

Battaglia: 7.1–7.4

Focus: Alternative representations of motion and curves

Sample Learning Activities:

• Analyze motion with parametric equations
• Convert between polar and Cartesian forms
• Interpret vector-valued functions

CR Alignment:

• CR1, CR2, CR6

Mathematical Practices:

• MP2, MP3, MP7

Unit 10: Infinite Sequences and Series

Battaglia: 8.1–8.7

Focus: Convergence and approximation

Sample Learning Activities:

• Apply convergence tests with justification
• Construct Taylor and Maclaurin polynomials
• Compare series approximations to exact values

CR Alignment:

• CR3, CR6, CR7

Mathematical Practices:

• MP3, MP6, MP7
• AP Exam Preparation

Technology and Materials

Students begin using a graphing calculator early in the course and continue using it throughout the year to solve problems, test ideas, interpret results, and verify conclusions. Technology supports learning, but it does not replace careful written work.

The syllabus also lists Enhanced WebAssign with the Larson and Battaglia AP Calculus text, plus AP exam review materials, as core course resources.