Illinois High School Calculus: Topics and Skills
Calculus is one of the most important math courses Illinois high school students take. It builds directly on algebra, trigonometry, and precalculus to introduce three major areas: limits, derivatives, and integrals. Each area connects to real-world applications in physics, engineering, economics, and data science.
Limits and Continuity
The course begins with understanding limits — what a function approaches as the input gets close to a value. Students learn to evaluate limits graphically and numerically, then algebraically using substitution. From there, they study continuity at a point, identify types of discontinuities, and analyze end behavior using limits at infinity.
Derivatives
The derivative section covers the definition of a derivative as a rate of change and slope of a tangent line. Students apply differentiation rules including the power rule, product rule, quotient rule, and chain rule. They differentiate trigonometric, exponential, and logarithmic functions, and find derivatives of implicitly defined functions. Applications include tangent line equations, linear approximation, optimization, curve sketching, and related rates problems.
- Power, product, quotient, and chain rules
- Derivatives of trig, exponential, and logarithmic functions
- Implicit differentiation
- Critical points, local maxima and minima
- Optimization and related rates
- Velocity, acceleration, and real-world rate problems
Integrals
The integration unit introduces antiderivatives and the concept of accumulation. Students approximate definite integrals using left, right, and midpoint Riemann sums, then use the Fundamental Theorem of Calculus to evaluate integrals exactly. They apply substitution to handle more complex integrands, find area under and between curves, and use integrals to analyze motion — including displacement, distance, and average value of a function.
- Antiderivatives with initial conditions
- Left, right, and midpoint Riemann sums
- Fundamental Theorem of Calculus (Parts 1 and 2)
- U-substitution
- Area under and between curves
- Displacement and distance from velocity functions
- Average value of a function over an interval
How StudyPug Supports Illinois Calculus Students
StudyPug provides video lessons and practice problems for every topic listed above. Each lesson is broken into short segments students can pause and replay. After watching, students practice with problems that mirror what they see in class. Content is aligned to Illinois Learning Standards for Math and covers the core topics found in AP Calculus AB and BC courses.
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