AP Calculus AB Practice

AP Calculus AB — Practice

16 MCQ practice problems. Select an answer, submit, and review the explanation.

Limits and Continuity

Limits are the foundation of calculus. Before you can differentiate or integrate, you need to understand what happens to a function as the input approaches a value, even if the function is undefined there. The key intuition is that limits describe behaviour near a point, not behaviour at the point.

Common limit patterns to recognise:

$\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ is the definition of $f'(a)$
$\lim_{x \to 0} \frac{\sin x}{x} = 1$ (the most important trigonometric limit)
$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$ (the definition of $e$ )

When L’Hopital’s rule applies: Only when the limit produces $\frac{0}{0}$ or $\frac{\infty}{\infty}$ . Always verify the indeterminate form first.

Common Mistakes:

Applying L’Hopital’s rule without verifying the indeterminate form first
Forgetting that limits describe behaviour near a point, not at the point
Confusing $\lim_{x \to a} f(x) = L$ with $f(a) = L$

Derivatives

The derivative measures the instantaneous rate of change of a function. Geometrically, it is the slope of the tangent line. The power rule, product rule, quotient rule, and chain rule are your computational tools, but understanding why the chain rule works (composition of functions) matters more than memorising the formula.

The chain rule in plain English: If $y$ depends on $u$ , and $u$ depends on $x$ , then the rate of change of $y$ with respect to $x$ is the product of the individual rates: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ .

Common Mistakes:

Forgetting the chain rule when differentiating composite functions
Confusing the product rule $(fg)' = f'g + fg'$ with the quotient rule
Applying the power rule to functions like $e^x$ or $\sin x$

Integrals

Integration is the reverse of differentiation, but its deeper meaning is accumulation. The definite integral $\int_a^b f(x)\,dx$ sums up infinitesimal contributions of $f(x)$ over the interval $[a, b]$ . The Fundamental Theorem of Calculus connects the two branches: differentiation and integration are inverse operations.

Common integration strategies:

u-substitution: Use when you see a function and its derivative together
Integration by parts: Use for products of different types of functions (polynomial × exponential, polynomial × trig)
Recognise special forms: $\int \frac{1}{x}\,dx = \ln|x| + C$ , $\int e^x\,dx = e^x + C$

Common Mistakes:

Forgetting the constant of integration $+C$ on indefinite integrals
Incorrectly applying u-substitution (wrong $du$ or wrong limits)
Confusing definite and indefinite integrals

Differential Equations

Differential equations describe how quantities change in relation to each other. The most common type on the AP exam is separation of variables: rearrange the equation so all $y$ terms are on one side and all $x$ terms on the other, then integrate both sides. Exponential growth and decay ( $\frac{dy}{dt} = ky$ ) and logistic growth ( $\frac{dy}{dt} = ky(1 - y/L)$ ) are the models you must know.

Common Mistakes:

Forgetting to separate variables before integrating
Mixing up exponential growth ( $k > 0$ ) and decay ( $k < 0$ )
Incorrectly applying initial conditions to find the constant $C$

Cross-References

Limits and Continuity: The foundation of all calculus, limits, continuity, and L’Hopital’s rule.
Derivatives: Differentiation rules, chain rule, and applications.
Integrals: FTC, u-substitution, and area under curves.

AP Calculus AB Practice