AP Calculus AB Study Guide

The course opens with limits and continuity, the foundation that makes everything else rigorous: what it means for ${\lim }_{x\to a}f(x)$ to exist, and why continuity matters for the theorems that come later. From there you build the derivative as a limit of average rates of change, $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$, and learn the rules — power, product, quotient, and chain — that turn that definition into a fast tool.

The back half pivots to the integral. You meet it twice: as an accumulation of change (the area under a curve, ${\int }_a^{b}f(x)dx$) and as the antiderivative. The Fundamental Theorem of Calculus ties those together and is, in a real sense, the whole point of the course. The final units apply integration to differential equations and to geometry — areas between curves, volumes of solids, and average values. If you can see the derivative and the integral as inverse operations, the course stops feeling like a list of techniques.

Not every unit is worth the same on the exam. Roughly, the weighting looks like this:

• Analytical Applications of Differentiation — ~18%
• Integration and Accumulation of Change — ~18%
• Limits and Continuity — ~12%
• Differentiation: Definition and Fundamental Properties — ~12%
• Contextual Applications of Differentiation — ~12%
• Applications of Integration — ~12%
• Differentiation: Composite, Implicit, and Inverse Functions — ~9%
• Differential Equations — ~7%

The takeaway: the two "applications" pillars — analyzing functions with derivatives and accumulating change with integrals — are over a third of the exam by themselves. They also reward the same skills, so time spent there pays off twice. Get the core differentiation and integration mechanics automatic first, then push hard on applying them.

Calculus AB rewards fluency plus interpretation. A routine that works:

1. Automate the algebra and the rules. You should not be thinking about the chain rule or about clearing a fraction while you're trying to set up a related-rates problem. Drill the mechanics until they're free.
2. Always ask what the answer means. The exam constantly attaches units and context — $f^{\prime }(t)$ is a rate, ${\int }_a^b{f}^{\prime }(t)dt$ is a net change. Practice translating between the math and the sentence.
3. Learn the calculator-active and no-calculator styles separately. One section expects clean by-hand work; the other expects you to evaluate definite integrals and find roots numerically. Know which habits belong where.
4. Work in mixed sets and review with full solutions. Studying one unit at a time hides the real difficulty — choosing the right approach. Reading a worked explanation for a problem you missed, including why each wrong choice was tempting, is worth more than three problems you already get right.

• Forgetting the chain rule in implicit differentiation or related rates — every $y$ that depends on $x$ picks up a $\frac{dy}{dx}$.
• Dropping the constant of integration $+C$ on indefinite integrals, or forgetting to solve for it using an initial condition.
• Confusing a function with its rate of change — reading a graph of $f^{\prime }$ as if it were $f$, or vice versa.
• Misusing the Fundamental Theorem, especially with a variable upper limit: $\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))\cdot g^{\prime }(x)$ requires its own chain rule.
• Skipping justification on free-response. Saying a function has a maximum isn't enough; you must point to where $f^{\prime }$ changes from positive to negative.

Every unit below has a focused practice set with full written explanations and a rationale for every wrong choice, plus a worked-solutions page you can read straight through. Start with a unit you're shaky on, then take a mixed set across the whole subject to pressure-test whether you can pick the right tool under time. It's free and needs no account.