AP Calculus AB/BC · Premium Problem Set

📐 About This Problem Set

All 20 questions are written in the style of real AP Calculus free-response and multiple-choice exams. Each question tests a single key concept with precision. Instant feedback and full solutions are provided after each selection.

FormatMultiple Choice (A–D)

CoverageAB + BC Topics

DifficultyMedium → Hard

CalculatorNot Required

Unit 1 · Limits & Continuity

Concept Review

Limits & Continuity

A limit describes the value a function approaches as the input approaches a given point.

\(\displaystyle\lim_{x\to c} f(x) = L\) means \(f(x)\to L\) as \(x\to c\)

L'Hôpital's Rule (for \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\) forms):

\(\displaystyle\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}\)

∞/∞ → L'Hôpital0/0 → L'HôpitalDNE if oscillates

Quick Example

\(\displaystyle\lim_{x\to 0}\frac{\sin x}{x} = 1\) (standard limit — memorize!)

Answer: 1

Limits — L'Hôpital's Rule

Evaluate \(\displaystyle\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}\).

Medium

Continuity — Removable Discontinuity

Let \(f(x) = \dfrac{x^2 - 4}{x - 2}\). Which value of \(f(2)\) makes \(f\) continuous at \(x = 2\)?

Medium

Unit 2 · Differentiation

Concept Review

Differentiation Rules

Key rules every student must memorize:

\(\dfrac{d}{dx}[f(x)g(x)] = f'g + fg'\) (Product Rule)

\(\dfrac{d}{dx}\left[\dfrac{f}{g}\right] = \dfrac{f'g - fg'}{g^2}\) (Quotient Rule)

\(\dfrac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)\) (Chain Rule)

sin→coscos→−sineˣ→eˣln x→1/x

Quick Example

If \(f(x)=\sin(x^2)\), then \(f'(x)=\cos(x^2)\cdot 2x\).

Chain Rule

If \(y = \ln(\cos x)\), what is \(\dfrac{dy}{dx}\)?

Medium

Implicit Differentiation

Given \(x^2 + y^2 = 25\), find \(\dfrac{dy}{dx}\) at the point \((3, 4)\).

Medium

Unit 3 · Applications of Derivatives

Concept Review

Mean Value Theorem & Curve Analysis

MVT: If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then \(\exists\, c\) s.t. \(f'(c)=\dfrac{f(b)-f(a)}{b-a}\)

f'>0 → increasingf''>0 → concave upf''=0 → inflection pt?

Quick Example

Critical points occur where \(f'(x)=0\) or \(f'(x)\) is undefined.

Mean Value Theorem

Let \(f(x) = x^3\) on \([0, 2]\). The Mean Value Theorem guarantees a value \(c \in (0,2)\) such that \(f'(c)\) equals:

Medium

Optimization — Absolute Extrema

The function \(f(x) = x^3 - 3x\) on \([-2, 2]\) attains its absolute maximum value of:

Hard

Related Rates

A spherical balloon is being inflated. Its radius is increasing at \(2\) cm/s. At the instant when \(r = 3\) cm, how fast is the volume increasing? (\(V = \tfrac{4}{3}\pi r^3\))

Hard

Unit 4 · Integration

Concept Review

Fundamental Theorem of Calculus & Techniques

FTC I: \(\displaystyle\frac{d}{dx}\int_a^x f(t)\,dt = f(x)\)

FTC II: \(\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)\)

u-sub: \(\displaystyle\int f(g(x))g'(x)\,dx = \int f(u)\,du\)

∫eˣdx = eˣ+C∫1/x dx = ln|x|+C∫cos x dx = sin x+C

Quick Example

\(\displaystyle\int_0^1 2x\,dx = \Big[x^2\Big]_0^1 = 1\)

FTC Part I — Derivative of Integral

If \(\displaystyle F(x) = \int_1^{x^2} \sin(t)\,dt\), find \(F'(x)\).

Hard

U-Substitution

Evaluate \(\displaystyle\int x e^{x^2}\,dx\).

Medium

Integration by Parts (BC)

Evaluate \(\displaystyle\int x e^x\,dx\).

Medium

Unit 5 · Differential Equations

Concept Review

Separable Differential Equations

Separate variables: \(\dfrac{dy}{dx} = f(x)g(y)\) → \(\displaystyle\int \frac{dy}{g(y)} = \int f(x)\,dx\)

Separate → Integrate → Solve for yLogistic: dP/dt = kP(1−P/M)

Quick Example

\(\dfrac{dy}{dx}=y\) → \(\displaystyle\int\frac{dy}{y}=\int dx\) → \(\ln|y|=x+C\) → \(y=Ae^x\)

Separable Differential Equation

The solution to \(\dfrac{dy}{dx} = 2xy\) with \(y(0) = 3\) is:

Medium

Logistic Growth (BC)

A population satisfies \(\dfrac{dP}{dt} = 0.1P\!\left(1 - \dfrac{P}{500}\right)\). The population grows fastest when \(P =\):

Hard

Unit 6 · Area, Volume & Accumulation

Concept Review

Area Between Curves & Volumes of Revolution

Area \(= \displaystyle\int_a^b [f(x) - g(x)]\,dx\) where \(f(x)\ge g(x)\)

Disk Method: \(V = \pi\displaystyle\int_a^b [f(x)]^2\,dx\)

Washer Method: \(V = \pi\displaystyle\int_a^b \left([f(x)]^2 - [g(x)]^2\right)dx\)

Shell Method: \(V = 2\pi\displaystyle\int_a^b x\,f(x)\,dx\)

Area Between Curves

The area enclosed by \(y = x^2\) and \(y = x\) is:

Medium

Volume — Disk Method

The region bounded by \(y = \sqrt{x}\), \(x = 4\), and the \(x\)-axis is revolved about the \(x\)-axis. The volume is:

Hard

Unit 7 · Parametric & Polar (BC)

Concept Review

Parametric Equations & Polar Curves

Parametric slope: \(\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}\)

Arc length (parametric): \(L = \displaystyle\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt\)

Polar area: \(A = \dfrac{1}{2}\displaystyle\int_\alpha^\beta [r(\theta)]^2\,d\theta\)

dy/dx = (dy/dt)÷(dx/dt)Polar: A = ½∫r²dθ

Parametric Derivatives (BC)

A curve is defined by \(x(t) = t^2 + 1\) and \(y(t) = t^3 - 3t\). Find \(\dfrac{dy}{dx}\) when \(t = 1\).

Hard

Unit 8 · Infinite Series (BC)

Concept Review

Convergence Tests & Taylor Series

Ratio Test: \(\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = L\). Converges if \(L<1\), diverges if \(L>1\)

Taylor Series: \(f(x) = \displaystyle\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n\)

Key Maclaurin series: \(e^x = \displaystyle\sum_{n=0}^{\infty}\frac{x^n}{n!}\), \quad \sin x = \displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}\)

p-series: Σ1/nᵖ converges iff p>1Geometric: Σarⁿ converges iff |r|<1

Geometric Series (BC)

What is the sum of the infinite series \(\displaystyle\sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n\)?

Medium

Taylor Polynomial (BC)

The third-degree Maclaurin polynomial for \(f(x) = e^x\) is used to approximate \(e^{0.1}\). This approximation equals:

Hard

Ratio Test — Convergence (BC)

Using the Ratio Test, the series \(\displaystyle\sum_{n=1}^{\infty} \frac{n!}{n^n}\):

Hard

Unit 9 · Improper Integrals & Advanced Topics

Concept Review

Improper Integrals & Arc Length

\(\displaystyle\int_1^{\infty}\frac{1}{x^p}\,dx\) converges iff \(p > 1\); value \(= \dfrac{1}{p-1}\)

Arc length: \(L = \displaystyle\int_a^b\sqrt{1+[f'(x)]^2}\,dx\)

∫₁^∞ 1/x dx = diverges (p=1)∫₁^∞ 1/x² dx = 1

Improper Integral (BC)

Evaluate \(\displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx\).

Medium

Euler's Method (BC)

Using Euler's method with step size \(h = 0.5\), starting from \(y(0) = 1\) with \(\dfrac{dy}{dx} = y\), approximate \(y(0.5)\).

Hard

Final Score

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Answer Key & Solutions

Q1 (B) ½ — Apply L'Hôpital twice: first derivative gives (eˣ−1)/2x → (0/0), second gives eˣ/2 → ½.

Q2 (C) 4 — Factor: (x²−4)/(x−2)=(x+2). As x→2, limit=4. Set f(2)=4.

Q3 (B) −tan x — By chain rule: (1/cos x)·(−sin x) = −sin x/cos x = −tan x.

Q4 (B) −3/4 — Differentiate implicitly: 2x+2y(dy/dx)=0 → dy/dx=−x/y=−3/4.

Q5 (B) 4 — MVT: (f(2)−f(0))/(2−0)=(8−0)/2=4.

Q6 (A) 2 — f'(x)=3x²−3=0 → x=±1. Check: f(−2)=−2, f(−1)=2, f(1)=−2, f(2)=2. Max=2.

Q7 (D) 72π cm³/s — dV/dt = 4πr²·(dr/dt) = 4π(3²)(2) = 4π·9·2 = 72π cm³/s.

Q8 (C) 2x sin(x²) — FTC I + Chain Rule: sin(x²)·(2x).

Q9 (B) ½e^(x²)+C — Let u=x², du=2x dx; integral becomes ½∫eᵘdu=½eᵘ+C.

Q10 (B) (x−1)eˣ+C — IBP: u=x, dv=eˣdx → xeˣ−∫eˣdx=xeˣ−eˣ=(x−1)eˣ+C.

Q11 (A) y=3e^(x²) — Separate: dy/y=2x dx → ln|y|=x²+C → y=Ae^(x²). y(0)=A=3.

Q12 (B) 250 — Logistic growth is fastest at P=M/2=500/2=250.

Q13 (A) 1/6 — Intersect: x²=x → x=0,1. ∫₀¹(x−x²)dx=[x²/2−x³/3]₀¹=1/2−1/3=1/6.

Q14 (B) 8π — V=π∫₀⁴(√x)²dx=π∫₀⁴x dx=π[x²/2]₀⁴=8π.

Q15 (A) 0 — dx/dt=2t=2, dy/dt=3t²−3=0 at t=1. dy/dx=0/2=0.

Q16 (B) 3 — Geometric series, a=1, r=2/3. Sum=a/(1−r)=1/(1/3)=3.

Q17 (B) — T₃(x)=1+x+x²/2+x³/6. At x=0.1: 1+0.1+0.005+0.001/6.

Q18 (B) — Ratio test: lim|(n+1)!/(n+1)^(n+1) · n^n/n!|= lim(n/(n+1))^n·1=1/e<1. Converges.

Q19 (C) 1 — ∫₁^∞ x⁻² dx=[−x⁻¹]₁^∞=0−(−1)=1.

Q20 (B) 1.5 — Euler: y(0.5)≈y(0)+h·f(0,y(0))=1+0.5·1=1.5.