Single Variable Calculus

Single variable calculus is the standard one-year (or two-semester) introduction to calculus. It restricts itself to functions of one input variable, so all the geometry lives in the x-y plane. Topics typically include limits and continuity, differentiation and its applications (related rates, optimization, linearization), Riemann sums and the definite integral, techniques of integration (u-substitution, integration by parts, partial fractions, trig substitution), improper integrals, and infinite sequences and series including Taylor series.

It corresponds to AP Calculus AB plus BC at the high school level, or Calc 1 and Calc 2 in college. It's a prerequisite for multivariable calculus, differential equations, linear algebra in many programs, and most upper-level physics and engineering. The bar to clear is fluency, not just exposure — engineering and physics courses assume students can differentiate and integrate without thinking about it.

Single variable calculus is mostly a fluency problem. The ideas (rate of change, accumulation) aren't hard to grasp; the problem is that every problem demands clean algebra, correct trig, and careful bookkeeping all at once. Students who try to learn by watching videos almost always underestimate how much problem-solving practice is required.

Practical approach:

• Pair a textbook with Paul's Online Math Notes and 3Blue1Brown's Essence of Calculus. Read, watch, then work problems — in that order.
• Keep a running list of integration techniques and when to use each one. The hard part of Calc 2 is recognizing which tool fits, not executing it.
• Do old AP free-response problems or MIT OCW problem sets. Both are free and harder than most textbooks.

Gut-check: give your student an unfamiliar integral and ask which technique to try first and why. If they can reason about the structure of the integrand, they're ready for what comes next. If they want to look it up, they need more reps.