AP Calculus AB

What this covers

• Evaluating limits algebraically and from graphs or tables, including one-sided limits, infinite limits, limits at infinity, and using continuity conditions to solve for unknown constants in piecewise functions
• Differentiation fluency: power, product, quotient, and chain rules; implicit differentiation; derivatives of inverse and inverse trigonometric functions; and the limit definition when a question demands it
• Applications of the derivative: related rates, optimization, linear approximation, particle motion along a line (position, velocity, speed, acceleration), and curve analysis with first and second derivative sign charts
• The Fundamental Theorem of Calculus in both forms, including differentiating accumulation functions like g(x) = ∫ from a to x of f(t) dt and analysing g from a graph of f
• Definite and indefinite integration by u-substitution, Riemann sums and trapezoidal estimates from tables, average value, and net versus total distance travelled
• Area between curves, volumes by disc, washer, and known cross-sections, plus separable differential equations, slope fields, and exponential growth models
• Writing exam-style justifications: units, interval notation, and the specific wording that earns a point on free-response scoring

Where learners get stuck

Treating the derivative as a number attached to a function rather than a function itself, so f'(3) and f(3) get mixed up in curve-sketching and particle motion questions

Early practice is dominated by 'find the derivative' tasks with no interpretation, so students never build the habit of asking what the output at a point actually measures. It shows up first when velocity is negative but speed is increasing, and again when reading f' graphs to describe f.

Forgetting the chain rule in implicit differentiation and related rates — writing d/dt of y² as 2y instead of 2y(dy/dt)

In ordinary differentiation the variable of differentiation matches the variable in the expression, so the chain factor is invisible. Once everything is a function of time, that factor becomes the whole point, and students carry over the old pattern.

Changing the limits of integration incorrectly (or not at all) after a u-substitution, and adding +C to a definite integral

Indefinite and definite integrals are taught with the same mechanical steps, so the substitution feels like an algebra trick rather than a change of variable that transforms the interval as well as the integrand.

Assuming a continuous function is differentiable, or that a derivative of zero means a maximum

Most textbook functions are smooth, so counterexamples like |x| at zero or x³ at zero rarely come up until a theorem-hypothesis question asks whether the Mean Value Theorem applies.

What a session looks like

Sessions run as a spoken back-and-forth. You bring a problem set, a past free-response question, or a topic you are stuck on, and Evelyn works through it by asking you to say the next step and the reason for it. Because justification wording matters on this exam, you will often be asked to state a conclusion out loud in full sentences — naming the theorem, the interval, and the units — before moving on. Graphs and tables are described verbally, so having the figure in front of you helps. Typical sessions mix one worked problem, one you attempt with prompting, and a short review of the error pattern that showed up.

Helpful to know first

• Algebra 2 fluency: factoring, rational expressions, solving equations, and manipulating exponents and logarithms without a calculator
• Precalculus function work: domain and range, piecewise and composite functions, transformations, and behaviour of polynomial, rational, exponential, and logarithmic graphs
• Unit circle values, basic trigonometric identities, and comfort with radian measure
• Graphing calculator basics: finding zeros, numerical derivatives, and definite integrals, since part of the exam allows one

Questions

What is the difference between AP Calculus AB and BC?

AB covers limits, derivatives, integrals, and their applications through separable differential equations and volumes. BC includes all of that plus parametric and polar calculus, vector-valued functions, integration by parts and partial fractions, improper integrals, Euler's method, logistic growth, and infinite series including Taylor and Maclaurin polynomials. AB is the smaller syllabus taught over the same year.

My child can differentiate fine but keeps losing points on free response. What's going on?

Almost always justification and communication rather than calculus. Points are awarded for stating which theorem applies and why its hypotheses hold, for including units, for setting up an integral before evaluating it, and for answering the question that was asked. Sessions target that gap by making the student say the reasoning aloud.

How much time before the exam should we start?

For steady support, weekly sessions alongside the school course work best because the material is cumulative — weak limits make the definition of the derivative harder, and weak differentiation makes related rates and optimization harder. For targeted review, six to eight weeks focused on free-response patterns and the accumulation-function questions is a common pattern.

Does the tutoring cover calculator versus non-calculator sections?

Yes. Some parts of the exam allow a graphing calculator and some do not, and the expected method differs — a non-calculator question wants exact values and algebraic simplification, while a calculator question often wants a definite integral set up and evaluated numerically to three decimal places. Sessions flag which mode a problem belongs to.