Calculus is the branch of mathematics that studies how things change and accumulate. Developed by Newton and Leibniz, calculus is used in almost every STEM (Science, Technology, Engineering and Mathematics) field, from computing the trajectory of a rocket to modeling the spread of a disease.
Calculus has two main parts:
- Differential Calculus: study of slope of curves and rates of change
- Integral Calculus: study of the accumulation of quantities and areas between or under the curves
This guide covers all major calculus topics for high school and college students, including full preparation for AP Calculus AB and AP Calculus BC, as well as first and second semester college calculus (Calc 1 and Calc 2).
Prerequisites: You should be comfortable with algebra, trigonometry and precalculus.
Calculus 1 (AP Calculus AB): Limits, Continuity, and Differentiation
Build a strong foundation in limits, derivatives and the fundamental concepts of calculus.
Limits and Continuity
- Graphs and Limits
- Definition of a Limit
- One-Sided Limits
- Limit Laws
- Limits of Piecewise Functions
- Squeeze Theorem
- Limits and Continuity
- Limits at Infinity
- Intermediate Value Theorem
Derivatives Basics
- Implicit Differentiation
- Average Rate of Change
- Limit Definition of Derivative
- Tangent Line Equations
- Differentiability
- Power Rule
- Product Rule
- Quotient Rule
Special Derivatives
- Derivatives of Trigonometric Functions
- Chain Rule
- Derivatives of Exponential Functions
- Derivatives of Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Inverse Trigonometric Functions
Applications of Derivatives
- Maxima and Minima
- First & Second Derivative Tests
- Extreme Value Examples
- Mean Value Theorem (and Proof)
- Concavity
- Curve Sketching
- Higher Order Derivatives
- Linear Approximation and Differentials
- L'Hôpital's Rule
- Indeterminate forms
- Newton's Method
- Real-Life Applications
Introduction to Integration
Calculus 2 (AP Calculus BC): Integration, Series, and Advanced Topics
Explore advanced integration techniques, infinite series and multivariable concepts.
Integration Techniques
- Substitution
- Integration by Parts
- Trigonometric Integrals
- Trigonometric Substitution
- Partial Fractions
- Improper Integrals
Applications of Integration
Sequences and Series
Advanced Calculus Topics
Practice for Calculus
Strengthen your understanding of calculus concepts through quizzes and practice problems.
- Limits Quiz
- Continuity of Function Quiz
- Maxima and Minima Quiz
- Integration Quiz
- Practice Questions on Calculus
Programs for Calculus
Apply calculus concepts using programming tools such as Python and MATLAB.