AP Calculus AB common mistakes
AP Calculus AB · Understand the traps and how to avoid them.
Evaluating a limit by direct substitution and reporting the resulting 0/0 (or k/0) as the answer instead of recognizing it as indeterminate or nonexistent.
Substitution is the first tool taught, so students stop there and never check whether the expression is defined at the point or whether the left- and right-hand behaviors differ.
What to do instead
Before substituting, classify the form. If 0/0, factor/cancel, rationalize, or use a trig identity; if k/0, check one-sided signs to decide +∞, −∞, or DNE; if the function is piecewise or has an absolute value, evaluate both one-sided limits explicitly and require them to agree.
Treating a removable discontinuity (a hole) as making the limit nonexistent, or claiming a function is continuous at a point just because the limit exists there.
Students conflate the three continuity conditions and forget that the limit depends only on nearby values, not on the function's value at the point.
What to do instead
State the three conditions separately each time: f(a) is defined, the two-sided limit exists, and the limit equals f(a). For hole problems, compute the limit by cancellation, then note the value is undefined (or redefined), so continuity fails while the limit still exists.
Using the power rule on expressions like (2x+1)^3 or x^x, or differentiating a product/quotient by differentiating each factor and multiplying/dividing the results.
The basic rules are overlearned and pattern-matched, and students do not pause to identify the structure (composite vs. product vs. quotient) before differentiating.
What to do instead
Before differentiating, label the structure out loud: outer function and inner function, or u and v. Then apply chain rule (outer derivative times inner derivative) or the product/quotient formulas in full. For x^x, rewrite as e^(x ln x) rather than forcing a rule.
Differentiating implicitly and then solving for dy/dx while dropping the chain-rule factor, e.g. writing d/dx(y^2)=2y instead of 2y·dy/dx.
Students treat y as a constant or as an independent variable because it looks like a single symbol rather than a function of x.
What to do instead
Annotate every y-term with '·dy/dx' as you differentiate, before doing any algebra. After differentiating, isolate all dy/dx terms on one side, factor out dy/dx, then divide. Verify by substituting a known point.
Setting a derivative equal to zero and calling every solution a maximum or minimum without checking whether the derivative actually changes sign there.
Students memorize 'f'=0 means extremum' and skip the sign analysis, so inflection points with horizontal tangents get misclassified.
What to do instead
Build a sign chart for f' around each critical point and require a sign change for an extremum (+ to − max, − to + min). If no sign change, it is neither. Also confirm the point lies in the domain of f.
On related-rates and optimization word problems, substituting the instantaneous numerical values into the equation before differentiating, which makes all derivatives vanish.
Students plug in every number they see as soon as possible, not distinguishing constants from quantities that vary with time or with the decision variable.
What to do instead
Write the relating equation in symbols only. Differentiate with respect to time (or the decision variable) first, then substitute the given instantaneous values into the differentiated equation. Keep a labeled list of 'varies' vs. 'constant at the instant'.
Applying the Mean Value Theorem or Intermediate Value Theorem without verifying its hypotheses, or citing MVT when the problem only guarantees IVT (or vice versa).
The theorems are memorized as names without their conditions, and both are invoked whenever a problem mentions an interval.
What to do instead
Before citing a theorem, write its hypotheses as a checklist: MVT needs continuity on [a,b] and differentiability on (a,b); IVT needs only continuity on [a,b]. Test each condition against the given function, and state the conclusion in the theorem's own terms (existence of c with f'(c)= the average rate).
Computing a definite integral with the Fundamental Theorem but ignoring the constant of integration incorrectly in indefinite integrals, or forgetting that an antiderivative of 1/x is ln|x| and of a constant multiple needs the reciprocal factor.
Students blur indefinite and definite procedures and memorize a short antiderivative table without the absolute value and linear-substitution adjustments.
What to do instead
Separate the two tasks: indefinite integrals always end with '+C'; definite integrals never carry a C. For 1/x write ln|x|. For linear inner functions, divide by the inner slope (∫f(ax+b)dx = F(ax+b)/a), and check every antiderivative by differentiating it back.
Solving a separable differential equation but omitting the constant of integration until after solving for y, or misapplying the initial condition to the wrong side.
Students treat the constant as an afterthought and are unsure where it belongs once variables are separated.
What to do instead
Add +C immediately after integrating both sides, then use the initial condition to solve for that single C before isolating y. If the problem gives an initial value, substitute it into the general solution, not into the derivative.
Setting up area-between-curves or volume integrals with the wrong integrand order or forgetting to subtract the lower curve, and using disk/washer formulas without adjusting the radius when the axis of revolution is not the x-axis.
Students focus on computing the antiderivative and rush the setup, and they memorize volume formulas with r=x rather than r=distance to the axis.
What to do instead
Sketch the region, identify top minus bottom (or right minus left) for area, and for volumes write radius as distance from the typical slice to the axis (e.g. r = x − (−2) = x+2). Find intersection points first to set limits, and sanity-check that the result is positive and has plausible magnitude.