Overview & units
Unit Title Exam % Key Topics U1 Limits & Continuity 10–12% Limit definition, one/two-sided, squeeze theorem, continuity, IVT U2 Differentiation — Definition & Rules 10–12% Avg vs. instantaneous rate, limit definition of derivative, power/trig/exp rules U3 Composite, Implicit & Inverse 9–13% Chain rule, implicit differentiation, derivatives of inverse trig U4 Contextual Applications 10–15% Related rates, straight-line motion, L'Hôpital's rule U5 Analytical Applications 15–18% MVT, EVT, increasing/decreasing, concavity, optimisation, 1st & 2nd derivative tests U6 Integration & Accumulation 17–20% Riemann sums, FTC Parts 1 & 2, u-substitution, definite integrals U7 Differential Equations 6–12% Slope fields, separation of variables, exponential growth/decay U8 Applications of Integration 10–15% Area between curves, cross-section & disk/washer volumes, average value
Full per-unit lessons for all eight units are below, followed by exam orientation, worked free-response questions, and a 50-question practice exam. For interactive visual intuition on any concept, the Calculus subject carries matching explorers you can drag and explore.
AP Calculus AB covers single-variable differential and integral calculus across eight units, assessed by a multiple-choice and free-response exam.
Unit 1 · Limits & Continuity
📌 AP Calc AB Unit 1 — Limits & Continuity 10–12% of exam · Limit laws · Squeeze theorem · Continuity & types of discontinuity · IVT
Calculus begins with the limit — the value a function approaches near a point — and with continuity, which guarantees no breaks in a graph.
Limit laws apply term by term. A function is continuous at \( a \) when \( \lim_{x\to a}f(x)=f(a) \). Discontinuities are removable, jump, or infinite. The Intermediate Value Theorem: a function continuous on \( [a,b] \) attains every value between \( f(a) \) and \( f(b) \).
Evaluate \( \displaystyle\lim_{x\to2}\frac{x^2-4}{x-2} \).
Use the Intermediate Value Theorem to show \(f(x)=x^3-x-1\) has a root on \([1,2]\). \(f(1)=1-1-1=-1<0\) and \(f(2)=8-2-1=5>0\). Since \(f\) is continuous on \([1,2]\) and changes sign, by IVT there exists \(c\in(1,2)\) with \(f(c)=0\).
Unit 2 · Differentiation — Definition & Basic Rules
📌 AP Calc AB Unit 2 — Differentiation 10–12% of exam · Limit definition of derivative · Power/trig/exp/ln rules · Higher-order derivatives
The derivative is the instantaneous rate of change: the slope of the tangent line, defined as a limit of average rates.
Below, \( f(x)=\tfrac{x^{3}}{3}-x \) is in indigo and its derivative \( f'(x)=x^{2}-1 \) in amber. Slide \( x \): the height of the amber curve equals the slope of the indigo one.
| x | f(x) | f′(x) |
|---|
Where \( f \) is flat (\( f'=0 \)) the amber curve crosses zero — those are the max and min of \( f \).
\( f'(x)=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}{h} \). From it follow the power, constant-multiple, sum, product, and quotient rules.
Differentiate \( f(x)=3x^{4}-2x \).
Find \(f'(x)\) for \(f(x)=x^2-3x\) using the limit definition.
Unit 3 · Composite, Implicit & Inverse Functions
📌 AP Calc AB Unit 3 — Composite, Implicit & Inverse 9–13% of exam · Chain rule · Implicit differentiation · Inverse trig derivatives
The chain rule differentiates compositions; implicit differentiation handles curves not solved for \( y \); and inverse-function derivatives complete the toolkit.
\( \dfrac{d}{dx}f(g(x))=f'(g(x))\,g'(x) \). Differentiate relations implicitly term by term. For inverses, \( (f^{-1})'(b)=\dfrac{1}{f'(a)} \) where \( f(a)=b \).
Differentiate \( \sin(x^{2}) \).
Find \(\dfrac{dy}{dx}\) for \(x^2+y^2=25\) and the equation of the tangent at \((3,4)\). At \((3,4)\): slope \(=-\tfrac{3}{4}\). Tangent: \(y-4=-\tfrac{3}{4}(x-3)\).
Unit 4 · Contextual Applications of Differentiation
📌 AP Calc AB Unit 4 — Contextual Applications 10–15% of exam · Related rates · Straight-line motion · L'Hôpital's rule
Derivatives model the real world: velocity and acceleration, related rates, and local linear approximation.
For motion, \( v(t)=s'(t) \) and \( a(t)=v'(t) \). Related rates differentiate a relationship with respect to time. Linearization: \( L(x)=f(a)+f'(a)(x-a) \). L'Hôpital's rule resolves \( \tfrac00 \) and \( \tfrac{\infty}{\infty} \).
A particle has position \( s(t)=t^{3}-6t^{2} \). When is it at rest?
A spherical balloon is inflated so its volume increases at 10 cm³/s. How fast is the radius increasing when \(r=5\) cm?
Unit 5 · Analytical Applications of Differentiation
📌 AP Calc AB Unit 5 — Analytical Applications 15–18% of exam · MVT & EVT · 1st & 2nd derivative tests · Optimisation · Concavity
Derivatives reveal where a function rises, falls and curves, and pin down its maxima and minima.
The Mean Value Theorem made visible: the amber secant joins the endpoints, and somewhere between them a tangent (dashed) runs exactly parallel. Slide the right endpoint \( b \).
| x | f(x) |
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There is a \( c \) in \( (a,b) \) with \( f'(c)=\dfrac{f(b)-f(a)}{b-a} \) — the ringed point.
Mean Value Theorem: some \( c \) has \( f'(c)=\dfrac{f(b)-f(a)}{b-a} \). \( f \) increases where \( f'>0 \) and is concave up where \( f''>0 \). The first- and second-derivative tests classify critical points; optimization finds extreme values.
Classify the critical points of \( f(x)=x^{3}-3x^{2} \). \( f'(x)=3x^{2}-6x=3x(x-2) \), so \( x=0,2 \). With \( f''(x)=6x-6 \): \( f''(0)<0 \), \( f''(2)>0 \).
A farmer has 200 m of fencing to enclose a rectangular field with one side along a river (no fence needed there). What dimensions maximise the area? Let width \(=x\), length \(=y\). Constraint: \(2x+y=200\Rightarrow y=200-2x\).
Unit 6 · Integration & Accumulation of Change
📌 AP Calc AB Unit 6 — Integration & Accumulation 17–20% of exam · Riemann sums · FTC Part 1 & 2 · u-substitution · Definite integrals
Integration reverses differentiation and accumulates change — the second great engine of calculus.
A definite integral is a limit of Riemann sums. Add rectangles and switch the sample point — the amber estimate marches toward the exact area.
| i | [x, x+Δx] | sample x* | f(x*) | area |
|---|
As \( n\to\infty \) the estimate converges to \( \displaystyle\int_0^4(0.4x^{2}+1)\,dx \).
Part 1: \( \dfrac{d}{dx}\displaystyle\int_a^x f(t)\,dt=f(x) \). Part 2: \( \displaystyle\int_a^b f=F(b)-F(a) \). Use \( u \)-substitution to reverse the chain rule, and Riemann sums to approximate area.
Evaluate \( \displaystyle\int_0^2 3x^{2}\,dx \).
If \(g(x)=\int_0^x\sin(t^2)\,dt\), find \(g'(x)\) and evaluate \(\int_1^3(2x-x^2)\,dx\). FTC Part 1: \(g'(x)=\sin(x^2)\).
Unit 7 · Differential Equations
📌 AP Calc AB Unit 7 — Differential Equations 6–12% of exam · Slope fields · Separation of variables · Exponential growth & decay
A differential equation links a function to its derivatives; solving one means recovering the family of curves that fit.
Slope fields sketch solution curves. A separable equation \( \dfrac{dy}{dx}=g(x)h(y) \) solves as \( \displaystyle\int\frac{dy}{h(y)}=\int g(x)\,dx \). Exponential change obeys \( y=y_0e^{kt} \).
Solve \( \dfrac{dy}{dx}=2y \) with \( y(0)=3 \).
Solve \(\dfrac{dy}{dx}=2xy\) with \(y(0)=3\). Initial condition: \(3=Ae^0=A\). Solution: \(y=3e^{x^2}\).
Unit 8 · Applications of Integration
📌 AP Calc AB Unit 8 — Applications of Integration 10–15% of exam · Area between curves · Disk/washer volumes · Average value of a function
Definite integrals measure areas between curves, volumes of solids, and average values.
Area between curves: \( \displaystyle\int(\text{top}-\text{bottom})\,dx \). Volumes by disks/washers about an axis. Average value: \( \dfrac{1}{b-a}\displaystyle\int_a^b f \). Solids with known cross-sections integrate the section area.
Find the area between \( y=x \) and \( y=x^{2} \) on \( [0,1] \).
Find the volume of the solid formed by revolving \(y=\sqrt{x}\) about the \(x\)-axis on \([0,4]\).
Exam format
45 questions, 1 hr 45 min total. Part A: 30 questions, 60 min, no calculator. Part B: 15 questions, 45 min, graphing calculator required.
6 questions, 1 hr 30 min total. Part A: 2 questions, 30 min, calculator. Part B: 4 questions, 60 min, no calculator.
The composite score is reported on the 1–5 scale. Free-response questions reward clear setup and justification, not just the final number — show the integral or derivative you evaluate.
MCQ: 45 points raw (1 pt each, no penalty). FRQ: 54 points (6 problems × 9 pts). Total 108 pts scaled to 1–5. Score of 3+ is generally considered passing.
Worked FRQs
A particle moves along a line with velocity \( v(t)=t^2-4t+3 \) for \( t\geq 0 \). (a) When is the particle at rest? \( v(t)=(t-1)(t-3)=0 \Rightarrow t=1,\ 3 \). (b) On \( [0,3] \), when is it moving left? \( v<0 \) on \( (1,3) \), so it moves left there (and right on \( [0,1) \)). (c) Total distance travelled on \( [0,3] \):
Let \( R \) be the region bounded by \( y=\sqrt{x} \) and \( y=x \). (a) Area of \( R \) (here \( \sqrt{x}\geq x \) on \( [0,1] \)): (b) Volume when \( R \) is revolved about the \( x \)-axis (washers):
Exam practice
Differentiate \( f(x)=x^2\sin x \).
MCQProduct rule:
\[ f'(x)=2x\sin x+x^2\cos x. \]Evaluate \( \displaystyle\int_1^{2} 3x^2\,dx \).
MCQAt which \( x \) does \( f(x)=x^3-3x^2+2 \) have a local maximum?
FRQ\( f'(x)=3x^2-6x=3x(x-2) \Rightarrow x=0,2 \). With \( f''(x)=6x-6 \): \( f''(0)=-6<0 \).
\[ \text{Local maximum at } x=0. \]Evaluate \( \displaystyle\lim_{x\to 3}\frac{x^2-9}{x-3} \).
MCQFactor and cancel: \( \tfrac{(x-3)(x+3)}{x-3} \).
\[ \lim_{x\to 3}(x+3)=6. \]Evaluate \( \displaystyle\lim_{x\to 0}\frac{\sin 5x}{x} \).
MCQEvaluate \( \displaystyle\lim_{x\to\infty}\frac{3x^2+2}{x^2-1} \).
MCQRatio of leading coefficients:
\[ 3. \]Evaluate \( \displaystyle\lim_{x\to 0}\frac{\sqrt{x+9}-3}{x} \).
FRQConjugate \( \sqrt{x+9}+3 \):
\[ \lim_{x\to 0}\frac{1}{\sqrt{x+9}+3}=\frac16. \]Evaluate \( \displaystyle\lim_{x\to 0}\frac{1-\cos x}{x} \).
MCQForm \( \tfrac00 \); L'Hôpital gives \( \tfrac{\sin x}{1}\to 0 \):
\[ 0. \]Differentiate \( f(x)=4x^5-2x^2 \).
MCQDifferentiate \( y=\sin 2x \).
MCQDifferentiate \( y=e^{3x} \).
MCQDifferentiate \( y=\ln\!\big(x^2+1\big) \).
MCQDifferentiate \( y=\big(x^2+1\big)^4 \).
MCQChain rule:
\[ y'=8x\big(x^2+1\big)^3. \]Differentiate \( y=x\sin x \).
MCQDifferentiate \( y=\dfrac{x-1}{x+1} \).
MCQQuotient rule:
\[ y'=\frac{2}{(x+1)^2}. \]Differentiate \( y=\sqrt{3x+1} \).
MCQDifferentiate \( y=\tan x \).
MCQFind \( \dfrac{dy}{dx} \) if \( x^2+xy=4 \).
FRQImplicit: \( 2x+\big(y+x\,y'\big)=0 \).
\[ \frac{dy}{dx}=-\frac{2x+y}{x}. \]Find \( f''(x) \) for \( f(x)=x^4 \).
MCQ\( f'=4x^3 \), \( f''=12x^2 \).
\[ 12x^2. \]Differentiate \( y=e^{x}\cos x \).
MCQFind the tangent line to \( y=x^3 \) at \( x=1 \).
FRQPoint \( (1,1) \); slope \( y'=3x^2=3 \):
\[ y=3x-2. \]Find the critical points of \( f(x)=x^3-6x^2+9x \).
FRQ\( f'(x)=3x^2-12x+9=3(x-1)(x-3) \):
\[ x=1,\ 3. \]At what \( x \) does \( f(x)=x^2-4x+1 \) attain its minimum?
MCQ\( f'(x)=2x-4=0 \):
\[ x=2 \ (\text{min value } -3). \]On what interval is \( f(x)=x^2-6x \) increasing?
MCQ\( f'(x)=2x-6>0 \):
\[ x>3. \]A rectangle has perimeter 40. What is the maximum possible area?
FRQ\( l+w=20 \), \( A=l(20-l) \), maximised at \( l=10 \):
\[ 10\times 10=100. \]A circle's radius grows at 3 cm/s. How fast is its area changing when \( r=4 \) cm?
FRQ\( \dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}=2\pi(4)(3) \):
\[ 24\pi\ \text{cm}^2/\text{s}. \]Evaluate \( \displaystyle\lim_{x\to 0}\frac{e^{2x}-1}{x} \).
MCQL'Hôpital: \( \tfrac{2e^{2x}}{1}\to 2 \):
\[ 2. \]For what \( x \) is \( f(x)=x^3 \) concave up?
MCQ\( f''(x)=6x>0 \):
\[ x>0. \]Find the inflection point of \( f(x)=x^3-3x \).
FRQ\( f''(x)=6x=0\Rightarrow x=0 \), and \( f(0)=0 \):
\[ (0,0). \]A cube's edge grows at 2 cm/s. How fast is the volume changing when the edge is 5 cm?
FRQ\( V=s^3 \), \( \dfrac{dV}{dt}=3s^2\dfrac{ds}{dt}=3(25)(2) \):
\[ 150\ \text{cm}^3/\text{s}. \]Evaluate \( \displaystyle\int\big(8x^3-3x^2\big)\,dx \).
MCQEvaluate \( \displaystyle\int_0^{2} x^3\,dx \).
MCQEvaluate \( \displaystyle\int_1^{3} 2x\,dx \).
MCQEvaluate \( \displaystyle\int_0^{\pi}\sin x\,dx \).
MCQEvaluate \( \displaystyle\int e^{2x}\,dx \).
MCQEvaluate \( \displaystyle\int_1^{e^2}\frac{1}{x}\,dx \).
MCQEvaluate \( \displaystyle\int_0^{\pi/4}\sec^2 x\,dx \).
MCQFind \( \dfrac{d}{dx}\displaystyle\int_0^{x}\sin\!\big(t^2\big)\,dt \).
FRQBy the Fundamental Theorem (Part I):
\[ \sin\!\big(x^2\big). \]Evaluate \( \displaystyle\int_0^{2}\big(3x^2+1\big)\,dx \).
MCQFind the average value of \( f(x)=2x \) on \( [0,4] \).
MCQEvaluate \( \displaystyle\int 3x^2\big(x^3+1\big)^5\,dx \).
MCQLet \( u=x^3+1 \), \( du=3x^2\,dx \):
\[ \frac{\big(x^3+1\big)^6}{6}+C. \]Evaluate \( \displaystyle\int x\,e^{x}\,dx \).
FRQBy parts:
\[ e^{x}(x-1)+C. \]Evaluate \( \displaystyle\int 2x\cos\!\big(x^2\big)\,dx \).
MCQLet \( u=x^2 \):
\[ \sin\!\big(x^2\big)+C. \]Evaluate \( \displaystyle\int x\ln x\,dx \).
FRQBy parts, \( u=\ln x,\ dv=x\,dx \):
\[ \frac{x^2}{2}\ln x-\frac{x^2}{4}+C. \]Find the area between \( y=4x \) and \( y=x^2 \) on \( [0,4] \).
FRQOn \( [0,4] \), \( 4x\ge x^2 \):
\[ \int_0^{4}\big(4x-x^2\big)\,dx=\Big[2x^2-\tfrac{x^3}{3}\Big]_0^{4}=\frac{32}{3}. \]Find the area under \( y=3x^2 \) from \( 0 \) to \( 2 \).
MCQThe region under \( y=\sqrt{x} \) on \( [0,9] \) is revolved about the \( x \)-axis. Find the volume.
FRQDisks with \( R^2=x \):
\[ \pi\int_0^{9}x\,dx=\pi\Big[\tfrac{x^2}{2}\Big]_0^{9}=\frac{81\pi}{2}. \]Find the average value of \( f(x)=x^3 \) on \( [0,2] \).
MCQSolve \( \dfrac{dy}{dx}=4y \) with \( y(0)=2 \).
FRQExponential model with \( k=4 \), \( y_0=2 \):
\[ y=2e^{4x}. \]Find the general solution of \( \dfrac{dy}{dx}=x^2 \).
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Full official exam paper
Mirrors the College Board AP Calculus AB exam: 45 MC (Part A no-calc / Part B calc) + 6 FRQs. 5 answer choices. Scored 1–5.
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AB lessons, FRQs & practice
All eight units, the worked free-response questions, and the full practice exam with solutions.
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Past Papers & Official Tests
Officially released exams and mark schemes from the examining body.
Worksheet Library
15 original ClipSAT worksheets covering AP Calculus AB Units 1–8, each with a matching answer key — printable PDFs, free to download.