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Why This Structure?

Calculus builds systematically. Each course depends on what came before:

graph LR
    subgraph PreCalc["Precalculus"]
        P["Functions<br/>Trig<br/>Algebra"]
    end

    subgraph Calc1["Calculus I"]
        A["Limits"]
        B["Derivatives"]
        C["Integrals"]
    end

    subgraph Calc2["Calculus II"]
        D["Applications"]
        E["Techniques"]
        F["Inverse<br/>Functions"]
    end

    subgraph ODE["Diff Eq"]
        G["ODEs"]
    end

    subgraph Calc3["Calculus III"]
        H["Series"]
        I["Vectors"]
    end

    subgraph Multi["Multivariable"]
        J["Partial<br/>Derivatives"]
        K["Multiple<br/>Integrals"]
        L["Vector<br/>Calculus"]
    end

    P --> A
    A --> B
    B --> C
    C --> D
    D --> E
    E --> F
    C --> G
    F --> H
    F --> I
    I --> J
    J --> K
    K --> L

    style P fill:#e0f2fe
    style A fill:#fef3c7
    style B fill:#fef3c7
    style C fill:#fef3c7
    style D fill:#d1fae5
    style E fill:#d1fae5
    style F fill:#d1fae5
    style G fill:#fce7f3
    style H fill:#e0e7ff
    style I fill:#e0e7ff
    style J fill:#fde68a
    style K fill:#fde68a
    style L fill:#fde68a

This single theorem connects differentiation and integration, the two central operations of calculus.

Precalculus (MATH 141 & 142)

Core Question: What mathematical tools do you need before calculus?

Precalculus builds the function toolkit: recognizing, transforming, and combining different function types.

Mastery Check: Can you solve $\log_2(x+1) + \log_2(x-1) = 3$? If not, review Chapter 4.

Calculus I (MATH 161)

Core Question: What are limits, derivatives, and integrals?

This is where calculus begins. Limits formalize “approaching,” derivatives measure instantaneous change, and integrals accumulate quantities.

Chapter Details

Chapter 1: Functions and Limits

The foundation of calculus. Limits answer: “What does $f(x)$ approach as $x$ approaches $a$?”

Chapter 2: Derivatives

The derivative $f’(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ measures instantaneous rate of change.

Chapter 3: Applications of Differentiation

Use derivatives to analyze functions: find extrema, sketch curves, solve optimization problems.

Chapter 4: Integrals

Integration reverses differentiation. The Fundamental Theorem of Calculus connects these operations.

Calculus II (MATH 162)

Core Question: How do we apply integrals and find more integrals?

Calculus II expands integration: computing areas, volumes, and arc lengths, plus techniques for harder integrals.

Mastery Check: Can you evaluate $\int \frac{x^2}{x^2-1}\,dx$ using partial fractions?

Differential Equations (MATH 347)

Core Question: How do we solve equations involving derivatives?

ODEs model dynamic systems: population growth, radioactive decay, oscillations.

Note: MATH 347 typically uses a dedicated ODE textbook. Chapter 9 provides supplementary coverage of first-order equations.

Calculus III (MATH 163)

Core Question: What happens in new coordinate systems and infinite dimensions?

Calculus III explores parametric curves, polar coordinates, infinite series, and vectors in 3D space.

Chapter Details

Chapter 11: Sequences, Series, and Power Series

Infinite sums: when do they converge, and what do they converge to?

Mastery Check: Does $\sum_{n=1}^{\infty} \frac{n}{2^n}$ converge? What is its value?

Multivariable Calculus (MATH 241)

Core Question: How does calculus extend to functions of several variables?

Multivariable calculus studies functions $f(x,y)$ and $f(x,y,z)$: partial derivatives, multiple integrals, and vector fields.

The Big Theorems of Vector Calculus

These three theorems unify all of vector calculus. They relate integrals over boundaries to integrals over regions.

Before You Begin: Prerequisites

Make sure you have these foundations: