New York High School Calculus: Topics and Skills
Calculus is one of the most important courses New York high school students take on the path to college and STEM careers. StudyPug covers every major Calculus topic aligned to the NYS Next Generation Mathematics Learning Standards, giving students the support they need to succeed in class, on exams, and beyond.
Limits and Continuity
Students begin Calculus by exploring limits — understanding how functions behave as inputs approach a value. Topics include evaluating limits graphically and numerically, using substitution, identifying types of discontinuities, and analyzing end behavior using limits at infinity. These foundational concepts underpin everything that follows in Calculus.
Derivatives and Differentiation Rules
Derivatives measure how functions change. New York Calculus students learn to:
- Interpret the derivative as rate of change and slope of a tangent line
- Apply the power rule, product rule, quotient rule, and chain rule
- Differentiate trigonometric, exponential, and logarithmic functions
- Use implicit differentiation for implicitly defined functions
- Write equations of tangent lines and apply linear approximation
Applications of Derivatives
Once students can find derivatives, they use them to solve real problems. Topics include finding critical points, local maxima and minima, and solving optimization problems. Students also analyze increasing and decreasing behavior, concavity, and learn to sketch curves using derivative information. Related rates problems and applications to velocity and acceleration connect Calculus to the physical world.
Integrals and the Fundamental Theorem of Calculus
The second half of Calculus focuses on integration. Students learn to:
- Find antiderivatives of basic functions using initial conditions
- Approximate definite integrals using left, right, and midpoint Riemann sums
- Apply the Fundamental Theorem of Calculus to evaluate definite integrals
- Use substitution to evaluate more complex integrals
Applications of Integrals
Integration has powerful real-world uses. New York Calculus students apply integrals to find area under curves and between curves, calculate displacement and distance from velocity functions, and determine the average value of a function over an interval. These skills are essential for AP Calculus and college-level mathematics.
How StudyPug Helps New York Calculus Students
StudyPug provides video lessons and practice problems for every topic listed above. Each lesson is 5–15 minutes long, broken into short segments students can pause and replay. Whether your child is preparing for an AP Calculus exam, catching up after missing class, or working ahead, StudyPug has the right lesson ready to watch on any device.
New York