Calculus 1 and 2 Topics: A Complete Course Overview

By Pankaj Kumar|Last Updated: July 12, 2026|

Key Takeaways

Calculus is divided into Calculus 1, Calculus 2, and Calculus 3 (Multivariable).
Calculus 1 covers limits, derivatives, integrals, and differential equations.
Calculus 2 extends integration techniques, series, and parametric equations.
Both courses share foundational integral and differential equation topics.
Wikipedia maintains a complete list of calculus topics across all levels.

Calculus is basically divided into Calculus 1 and Calculus 2. Calculus 3 is three-dimensional calculus and also called Multivariable Calculus (we have not covered Calculus 3 topics in this post). If you need a calculus tutor, the topics below outline exactly what each course covers.

Calculus 1 Topics

Calculus 1 builds the foundation of the subject, beginning with limits and continuity before moving into derivatives and their applications. Students working through AP Calculus will recognise most of these topics as core curriculum. The full topic list for Calculus 1 is below.

Limits and Continuity

Limits and continuity
Estimating limits from tables
Estimating limits from graphs
Formal definition of limits (epsilon-delta)
Properties of limits
Limits using algebraic manipulation
Limits by direct substitution
Strategy in finding limits
Squeeze theorem
Continuity at a point
Continuity over an interval
Types of discontinuities
Removing discontinuities
Limits at infinity
Infinite limits
Intermediate value theorem

Derivatives: Definition and Basic Rules

Average vs. instantaneous rate of change
Secant lines
Estimating derivatives
Differentiability
Power rule
Combining the power rule with other derivative rules
Derivatives of cos(x), sin(x), eˣ, and ln(x)
Product rule
Quotient rule

Derivatives: Chain Rule and Advanced Topics

Implicit differentiation
Differentiating inverse functions
Derivatives of inverse trigonometric functions
Strategy in differentiating functions
Differentiation using multiple rules
Second derivatives
Disguised derivatives
Logarithmic differentiation

Applications of Derivatives

Meaning of the derivative in context
Straight-line motion
Non-motion applications of derivatives
Introduction to related rates
Solving related rates problems
Approximation with local linearity
L’Hôpital’s rule
L’Hôpital’s rule: composite exponential functions

Analyzing Functions

Mean value theorem
Extreme value theorem and critical points
Relative (local) extrema
Absolute (global) extrema
Concavity and inflection points intro
Analyzing concavity and inflection points
Second derivative test
Sketching curves
Connecting f, f’, and f”
Solving optimization problems
Analyzing implicit relations

Integrals

For students also studying statistics alongside calculus, this guide to mastering statistical thinking covers complementary mathematical reasoning skills.

Accumulations of change
Approximation with Riemann sums
Summation notation review
Riemann sums in summation notation
Defining integrals with Riemann sums
Fundamental theorem of calculus and accumulation functions
Interpreting the behavior of accumulation functions
Properties of definite integrals
Fundamental theorem of calculus and definite integrals
Reverse power rules
Indefinite integrals of common functions
Definite integrals of common functions
Integrating with u-substitution
Integrating using long division and completing the square
Integrating using trigonometric identities

Differential Equations (Calculus 1)

Verifying solutions for differential equations
Sketching slope fields
Reasoning using slope fields
Separation of variables
Particular solutions to differential equations
Exponential models

Applications of Integrals (Calculus 1)

Average value of a function
Straight-line motion
Non-motion applications of integrals
Area: vertical area between curves
Area: horizontal area between curves
Area: curves that intersect at more than two points
Volume: triangles and semicircles cross-sections
Volume: squares and rectangles cross-sections
Volume: disc method (revolving around x- and y-axes)
Volume: disc method (revolving around other axes)
Volume: washer method (revolving around x- and y-axes)
Volume: washer method (revolving around other axes)

Engineering students will find the engineering mathematics survival kit a useful companion resource alongside these Calculus 1 topics.

Calculus 2 Topics

Calculus 2 extends the integral and differential equation work from Calculus 1 and introduces series, parametric equations, and polar coordinates. Students preparing for AP Calculus BC will cover the majority of these topics. The full topic list for Calculus 2 is below.

Integrals Review

Accumulations of change
Approximation with Riemann sums
Summation notation
Riemann sums in summation notation
Defining integrals with Riemann sums
Fundamental theorem of calculus and accumulation functions
Interpreting the behavior of accumulation functions
Properties of definite integrals
Fundamental theorem of calculus and definite integrals
Reverse power rule
Indefinite integrals of common functions
Definite integrals of common functions

Integration Techniques

Integrating with u-substitution
Integrating using long division and completing the square
Integrating using trigonometric identities
Trigonometric substitution
Integration by parts
Integrating using linear partial fractions
Improper integrals

Differential Equations (Calculus 2)

Verifying solutions for differential equations
Sketching slope fields
Reasoning using slope fields
Approximation with Euler’s method
Separation of variables
Particular solutions to differential equations
Exponential models
Logistic models

Applications of Integrals (Calculus 2)

Students who find probability distributions challenging alongside integral applications may benefit from this statistics guide for engineers on choosing the right probability distribution.

Average value of a function
Straight-line motion
Non-motion applications of integrals
Area: vertical area between curves
Area: horizontal area between curves
Area: curves that intersect at more than two points
Volume: squares and rectangles cross-sections
Volume: triangles and semicircles cross-sections
Volume: disc method (revolving around x- and y-axes)
Volume: disc method (revolving around other axes)
Volume: washer method (revolving around x- and y-axes)
Volume: washer method (revolving around other axes)
Arc length

Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Parametric equations
Second derivatives of parametric equations
Arc length: parametric curves
Vector-valued functions
Planar motion
Polar functions
Area: polar regions (single curve)
Area: polar regions (two curves)
Arc length: polar curves

Series

Students working through series convergence tests often encounter hypothesis-testing logic in their statistics courses as well; this step-by-step guide to hypothesis testing covers that parallel reasoning in detail. Those preparing specifically for AP Calculus AB should note that series topics are generally not part of the AB curriculum.

Convergent and divergent infinite series
Infinite geometric series
nth-term test
Integral test
Harmonic series and p-series
Comparison tests
Alternating series test
Ratio test
Absolute and conditional convergence
Alternating series error bound
Taylor and Maclaurin polynomials
Error bound
Power series
Function as a geometric series
Maclaurin series of eˣ, sin(x), and cos(x)
Representing functions as power series

Check the Wikipedia page on calculus topics to see the complete list of topics in Calculus 1, 2 and more.

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