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AP ExamUC A-G · Section CUC Honors · +1.0 GPAMay 5, 2026

AP Calculus AB
The Mathematics of Change

Limits · Derivatives · Integrals · Mastered

The most comprehensive agentic AP Calculus AB course available. Master all 8 units, nail every FRQ type, and score a 5 — powered by Prof. Sofia Euler and SofAI. ~300,000 students take AP Calculus AB annually — one of the largest AP exams.

Start with Prof. Sofia

Enroll in This Course

Duration1 Year

Credits10

DeliveryRemote / Hybrid / In-Person

UC SectionC – Mathematics

Honors GPA+1.0 weighted point

No commitment until admissions confirms your placement.

AP Resources

Score Target

Quick LinksCollegeBoard AP Calculus AB VRS AP Resources AP Seminar Exemplar ↗

Exam: May 5, 2026

Exam Blueprint

Four Sections · MC + FRQ

🔢

MC No Calculator

Section I · Part A (No Calculator)

~33%60 min30 questions

› Pure mathematical reasoning — no technology shortcuts
› Tests limits, derivatives (all rules), and basic integration analytically
› Speed matters: ~2 min per question; skip and return if stuck

Score 5 Tip: Master derivative rules until they're automatic. Product rule, chain rule, and implicit differentiation appear in 30% of no-calc questions. Write out every step — the process reveals the answer.

🧮

MC With Calculator

Section I · Part B (Calculator)

~17%45 min15 questions

› Use calculator for: numerical derivatives, definite integrals, solve equations, find intersections
› Questions often involve context (position/velocity/acceleration, area between curves)
› Know HOW to use nDeriv() and fnInt() on your TI-84 or Nspire

Score 5 Tip: Don't use the calculator as a crutch — know what you're computing before you press buttons. The calculator computes; YOU must interpret the result and connect it to the question.

📐

FRQ With Calculator

Section II · Part A (Calculator)

~17%30 min2 FRQs

› Usually involves area/volume, accumulation, or particle motion with calculator-found values
› Show ALL setup work before using calculator: write the integral, set up the equation
› Round final answers to 3 decimal places unless the problem specifies otherwise

Score 5 Tip: In FRQ Part A, always SHOW the integral or derivative you're computing before using the calculator. The setup earns points. Just writing a numerical answer earns ZERO points even if correct.

✏️

FRQ No Calculator

Section II · Part B (No Calculator)

~33%60 min4 FRQs

› Tests analytical skills: using graphs, tables, and symbolic manipulation
› FRQ types: graph analysis (reading f' and f'' to determine f behavior), related rates, differential equations, accumulation problems
› Show every step — intermediate work earns partial credit

Score 5 Tip: For graph-based FRQs: always connect sign changes of f'(x) to increasing/decreasing of f(x), and sign changes of f''(x) to concavity. A sign chart for f' and f'' is your best friend.

Score Distribution (2024)

Where Students Land

~300,000 students take AP Calculus AB annually — one of the largest AP exams. 19% score a 5. Strong conceptual reasoning + FRQ precision separates the top scorers.

Extremely Qualified

← Your target19%

Well Qualified

17%

Qualified

19%

Possibly Qualified

23%

No Recommendation

22%

What It Takes to Score a 5

Precision + conceptual depth

✍️

FRQ excellence — show every step, justify every claim, earn partial credit on every part

🔢

Derivative rules from memory: power, product, quotient, chain, implicit — no hesitation

🎯

90%+ accuracy on no-calculator MC — this is 30% of the exam and fully analytical

∫

FTC mastery — both parts, including d/dx of integrals with variable upper limits

📊

Graph analysis fluency — read f, f', f'' relationships instantly from any graph

🧮

Calculator proficiency: nDeriv(), fnInt(), intersection finding on TI-84 or Nspire

CollegeBoard CED Aligned

Eight Calculus Units

📉

UNIT 110–12%

Limits and Continuity

Expand ›

Key Topics

Intuitive definition of limits
Limit laws
One-sided limits
Infinite limits and asymptotes
Continuity (definition, types of discontinuity)
IVT
Squeeze Theorem

Key Terms & Formulas

limit

the value a function approaches as x approaches c

continuity

function is continuous at c if f(c) = lim(x→c) f(x)

removable discontinuity

hole in graph (limit exists but ≠ f(c))

vertical asymptote

where lim = ±∞

horizontal asymptote

limit as x→±∞

squeeze theorem

if g(x)≤f(x)≤h(x) and lim g = lim h = L, then lim f = L

FRQ Practice Prompt

Evaluate: lim(x→3) (x²-9)/(x-3). Now find lim(x→0) sin(x)/x using the Squeeze Theorem approach. Finally, describe the difference between lim(x→2) f(x) and f(2) — when can they be different?

Practice with Prof. Sofia →

Curated Video Resources

3Blue1Brown "Essence of Calculus Chapter 1"

Khan Academy Limits

PatrickJMT limits series

📈

UNIT 210–12%

Derivatives — Basics

Expand ›

Key Topics

Derivative as rate of change
Derivative from definition (difference quotient)
Power rule
Sum/difference rule
Constant multiple
Derivatives of sin, cos, eˣ, ln x
Tangent line equations

Key Terms & Formulas

derivative

instantaneous rate of change at a point

difference quotient

(f(x+h)-f(x))/h

power rule

d/dx(xⁿ) = nxⁿ⁻¹

tangent line

line touching curve at one point with slope = f'(c)

differentiability

function is differentiable at c if f'(c) exists

secant line

line through two points on a curve

FRQ Practice Prompt

Find dy/dx for: y = 3x⁴ - 2x² + 5x - 1. Find the equation of the tangent line to f(x) = x³ at x = 2. Finally, explain in plain English what f'(3) = -2 means in context of a position function.

Practice with Prof. Sofia →

Curated Video Resources

Krista King "Power Rule"

Khan Academy derivatives

PatrickJMT basic derivatives

🔗

UNIT 39–13%

Derivatives — Advanced Rules

Expand ›

Key Topics

Product rule
Quotient rule
Chain rule
Implicit differentiation
Derivatives of inverse functions
Derivatives of inverse trig (arcsin, arccos, arctan)

Key Terms & Formulas

chain rule

d/dx[f(g(x))] = f'(g(x))·g'(x)

product rule

d/dx[uv] = u'v + uv'

implicit differentiation

differentiating both sides with respect to x when y is defined implicitly

inverse function derivative

(f⁻¹)'(b) = 1/f'(f⁻¹(b))

composite function

f(g(x)) — function applied inside another

related rates

rates of change of related quantities

FRQ Practice Prompt

Differentiate: y = (x²+1)³·sin(x). Then find dy/dx for x² + y² = 25 (circle, implicit). Finally find dy/dx for y = arctan(2x).

Practice with Prof. Sofia →

Curated Video Resources

Krista King chain rule

3Blue1Brown implicit differentiation

PatrickJMT product/quotient rule

🚗

UNIT 410–15%

Contextual Applications of Derivatives

Expand ›

Key Topics

Position/velocity/acceleration (s(t), v(t)=s'(t), a(t)=v'(t))
Rates of change in context
Related rates problems (ladder, shadow, cone, sphere)
L'Hôpital's Rule (0/0 or ∞/∞ forms)
Linear approximation (tangent line approximation)

Key Terms & Formulas

velocity

instantaneous rate of change of position: v(t) = s'(t)

acceleration

rate of change of velocity: a(t) = v'(t) = s''(t)

related rates

when multiple quantities change together, differentiate with respect to time

L'Hôpital's Rule

lim f(x)/g(x) = lim f'(x)/g'(x) when form is 0/0 or ∞/∞

speeding up

when v(t) and a(t) have the same sign

slowing down

when v(t) and a(t) have opposite signs

FRQ Practice Prompt

A ladder 10 feet long leans against a wall. The bottom slides away at 2 ft/sec. How fast is the top sliding down when the bottom is 6 feet from the wall? Show complete related rates setup.

Practice with Prof. Sofia →

Curated Video Resources

Krista King related rates

Khan Academy L'Hôpital's Rule

PatrickJMT particle motion

📊

UNIT 515–18%

Analytical Applications of Derivatives

Expand ›

Key Topics

Mean Value Theorem
Extreme Value Theorem
Critical points (f'=0 or undefined)
First Derivative Test (increasing/decreasing → local extrema)
Second Derivative Test (concavity → local extrema)
Absolute extrema on closed intervals
Inflection points (f'' changes sign)
Optimization problems

Key Terms & Formulas

critical point

where f'(c) = 0 or f'(c) is undefined

relative maximum

f(c) ≥ f(x) for x near c

inflection point

where f changes concavity (f'' changes sign)

Mean Value Theorem

f'(c) = [f(b)-f(a)]/(b-a) for some c

concave up

f''>0 (smile shape)

Extreme Value Theorem

continuous f on [a,b] attains absolute max and min

FRQ Practice Prompt

For f(x) = x³ - 6x² + 9x on [-1, 5]: (a) find all critical points; (b) identify local max/min using first derivative test; (c) find absolute maximum and minimum on the interval; (d) find all inflection points and describe concavity.

Practice with Prof. Sofia →

Curated Video Resources

Khan Academy first/second derivative test

Krista King optimization

PatrickJMT MVT

∫

UNIT 617–20%

Integration and Accumulation of Change

Expand ›

Key Topics

Riemann sums (left, right, midpoint, trapezoidal approximation)
Definite integral as limit of Riemann sums
Fundamental Theorem of Calculus (Parts 1 and 2)
Antiderivatives
Indefinite integrals
Integration rules (power, trig, exponential)
U-substitution

Key Terms & Formulas

antiderivative

F(x) such that F'(x) = f(x)

definite integral

∫ₐᵇ f(x)dx represents signed area under f from a to b

FTC Part 1

d/dx ∫ₐˣ f(t)dt = f(x)

FTC Part 2

∫ₐᵇ f(x)dx = F(b)-F(a)

Riemann sum

approximation of integral using rectangles

u-substitution

substitution technique: let u = g(x), then du = g'(x)dx

FRQ Practice Prompt

Evaluate: ∫(2x³ - 4x + 3)dx (indefinite). Then evaluate ∫₀² (3x² + 1)dx using FTC Part 2. Finally, use u-substitution to evaluate ∫ 2x(x²+1)⁴ dx.

Practice with Prof. Sofia →

Curated Video Resources

3Blue1Brown "Integration and the FTC"

Khan Academy FTC

Krista King u-substitution

🌀

UNIT 76–12%

Differential Equations

Expand ›

Key Topics

Differential equations (dy/dx = f(x,y))
Slope fields (drawing and interpreting)
Separation of variables
Exponential growth and decay (y = Ce^(kt))
Euler's method (numerical approximation)

Key Terms & Formulas

differential equation

equation relating a function to its derivatives (dy/dx = ky)

separation of variables

technique: move all y terms to one side, x terms to other, then integrate

slope field

visual representation of solutions (tiny tangent segments at each point)

exponential growth

y = y₀e^(kt) when k > 0

exponential decay

y = y₀e^(kt) when k < 0

Euler's method

numerical approximation: y_next ≈ y_now + (dy/dx)·Δx

FRQ Practice Prompt

Solve the differential equation dy/dx = 2xy with initial condition y(0) = 3. (a) Separate variables and integrate. (b) Apply initial condition to find the constant C. (c) Write the particular solution. (d) Interpret: what does this model in a real-world context?

Practice with Prof. Sofia →

Curated Video Resources

Khan Academy slope fields

Krista King differential equations

PatrickJMT separation of variables

📐

UNIT 810–15%

Applications of Integration

Expand ›

Key Topics

Area between two curves (∫[f(x)-g(x)]dx)
Volume of solids of revolution (Disk/Washer method: π∫[R²-r²]dx)
Volume using cross sections (known shapes: square, rectangle, triangle)
Accumulation problems (net change theorem)
Average value of a function

Key Terms & Formulas

area between curves

∫ₐᵇ [f(x)-g(x)]dx where f≥g on [a,b]

disk method

π∫ₐᵇ [f(x)]²dx — volume when cross sections are disks

washer method

π∫ₐᵇ [R(x)²-r(x)²]dx — volume with hole

net change

∫ₐᵇ f'(x)dx = f(b)-f(a)

average value

1/(b-a) ∫ₐᵇ f(x)dx

accumulation

total change from rate of change

FRQ Practice Prompt

Find the area enclosed between f(x) = x² and g(x) = 2x. Then find the volume generated by rotating the region about the x-axis using the disk/washer method. Show complete setup including the limits of integration.

Practice with Prof. Sofia →

Curated Video Resources

Khan Academy area between curves

Krista King disk/washer method

PatrickJMT volumes

50% of Total Score

FRQ Mastery Suite

AP Calculus AB's FRQ section tests both calculator-active and no-calculator skills. Showing your work and justifying answers is the biggest differentiator between a 3 and a 5.

FRQ Coach →

∫~17% of total score

Section II Part A

Accumulation/Area FRQ

Calculator-Active FRQ · ~15 min

Calculator-active FRQ involving area between curves, accumulation functions, or particle motion. Calculator results must be clearly set up first.

Scoring Criteria

· Setup: integral or derivative written before calculator use

· Correct limits of integration and integrand

· Proper rounding: 3 decimal places unless otherwise specified

· Interpretation: what the numerical result means in context

Score 5 Strategy

Write the complete integral expression BEFORE entering anything in the calculator

Identify intersection points using the calculator's solve or graph feature

Label your setup clearly: 'Area = ∫[a to b] [f(x) - g(x)] dx'

Show the numerical result with proper units and round to 3 decimal places

Connect back to the problem: 'therefore the total distance is 4.832 feet'

Model Opener

The area of the region enclosed by f and g is given by ∫[a to b] [f(x) − g(x)] dx. Using a calculator, the intersection points are x = [value] and x = [value], so the area = [value].

📊~17% of total score

Section II Part B

Analytical Graph FRQ

No-Calc Graph Analysis FRQ · ~15 min

Given a graph of f'(x) or f''(x), answer questions about f(x): increasing/decreasing, concavity, local extrema, and inflection points. No calculator allowed.

Scoring Criteria

· Correct identification of intervals using sign of f'(x)

· Correct concavity using sign of f''(x)

· Proper justification — must cite the derivative, not just the graph

· Correct location and type of local extrema with full reasoning

Score 5 Strategy

Build a sign chart for f'(x) immediately — mark +/- on each interval

f is increasing where f'(x) > 0; decreasing where f'(x) < 0

f has a local max where f' changes from + to -; local min where f' changes from - to +

f''(x) > 0 means concave up; f''(x) < 0 means concave down

Inflection points occur where f''(x) changes sign (not just equals zero)

Always JUSTIFY: write 'because f'(x) changes from positive to negative at x = c'

Model Opener

Since f'(x) changes from positive to negative at x = c, f has a local maximum at x = c by the First Derivative Test.

🔧~8% of total score

Section II Part B

Related Rates / Optimization

Applied Derivatives FRQ · ~15 min

Multi-part problem requiring setup of a related rates equation or optimization problem. No calculator — must show complete algebraic work.

Scoring Criteria

· Correct equation relating the variables (e.g., Pythagorean theorem, volume formula)

· Proper implicit differentiation with respect to time

· Substitution of known values AFTER differentiating

· Final answer with correct units and sign

Score 5 Strategy

Draw and label a diagram — always. Assign variables to all changing quantities

Write the equation relating the variables BEFORE differentiating

Differentiate BOTH sides with respect to t (implicit differentiation)

Substitute known values for rates and positions only after differentiating

State your answer as a complete sentence with units: 'The height is decreasing at 3 ft/sec'

For optimization: find critical points with f'(c) = 0 and verify with second derivative test

Model Opener

Let x = the distance from the wall and y = the height of the ladder. By the Pythagorean theorem, x² + y² = 100. Differentiating both sides with respect to t: 2x(dx/dt) + 2y(dy/dt) = 0.

🌀~8% of total score

Section II Part B

Differential Equations FRQ

Slope Field / DE FRQ · ~15 min

Problem involving slope fields, separation of variables, and/or Euler's method. May include drawing a slope field segment or verifying a solution.

Scoring Criteria

· Correct separation of variables (y terms one side, x terms other)

· Proper integration of both sides with constant of integration +C

· Application of initial condition to find C

· Correct particular solution written explicitly

Score 5 Strategy

Separate variables completely: all y and dy on one side, all x and dx on the other

Integrate both sides — include +C on one side only

Apply initial condition immediately to solve for C

Write the particular solution clearly and check by substituting back

For slope fields: at each labeled point, draw a short segment with slope = dy/dx at that point

For Euler's method: y_new = y_old + (dy/dx)·Δx — show each step

Model Opener

Separating variables: (1/y) dy = 2x dx. Integrating both sides: ln|y| = x² + C. Applying the initial condition y(0) = 3: ln(3) = C. Therefore y = 3e^(x²).

Curated for Score 5

Practice Tests & Resources

🏛

OFFICIALFREE

CollegeBoard AP Calculus AB

Official CED, sample questions, and exam format from CollegeBoard.

Open resource

📂

OFFICIALFREE

Past AP Calc AB FRQs (2014–2024)

Actual past exam free-response questions with scoring guidelines.

Open resource

🎥

CONCEPTUALFREE

3Blue1Brown Essence of Calculus

The most beautiful calculus series ever made. Build DEEP conceptual understanding. Watch before the course begins.

Open resource

📺

FRQ STRATEGYFREE

Krista King Math

Systematic, clear AP Calculus tutorials for every topic. Best for worked examples and FRQ walkthroughs.

Open resource

🎯

FREE PRACTICEFREE

Khan Academy AP Calculus AB

Complete practice problems and exercises for every unit. Use alongside video resources.

Open resource

🔢

WORKED EXAMPLESFREE

PatrickJMT

Thousands of calculus worked examples. Search any topic and Patrick has a clear step-by-step video.

Open resource

📚

COMPREHENSIVEFREE

Fiveable AP Calculus AB

Complete course review, FRQ practice, and live cram sessions aligned to the AP CED.

Open resource

📘

EXAM PREP

Barron's AP Calculus AB Prep

Best prep book for practice exams. Includes 4 full-length practice tests.

Open resource

AI-Powered Progress

16-Week Score 5 Study Plan

Weeks 1–4

Phase 1: Limits and Differentiation (Units 1-3) — foundations

Master limit laws, one-sided limits, and continuity definitions
Learn all derivative rules: power, product, quotient, chain rule
Implicit differentiation and inverse function derivatives
Daily practice: 10 derivative problems from PatrickJMT or Krista King

Weeks 5–8

Phase 2: Applications and Integration (Units 4-6) — conceptual depth

Related rates and particle motion (Units 4–5)
Critical points, first/second derivative tests, optimization
Introduction to integration: Riemann sums, FTC Parts 1 & 2
U-substitution practice — 10 problems per day

Weeks 9–12

Phase 3: Differential equations, Integration apps, FRQ mastery

Slope fields, separation of variables, Euler's method (Unit 7)
Area between curves, disk/washer volume, average value (Unit 8)
Complete 2 full timed FRQ sets per week with self-scoring
Use past AP FRQs (2014–2024) from CollegeBoard — timed 15 min each

Weeks 13–16

Phase 4: Full timed exams — 1 per week (105 min MC + 90 min FRQ)

Complete 1 full AP practice exam per week (both sections timed)
Review every missed question with Prof. Sofia (SofAI chat)
Drill calculator techniques: nDeriv(), fnInt(), intersection finding
Final formula review: all derivative and integral rules from memory

Official & Curated

AP Resources Hub

🏛

Official Source

CollegeBoard AP Calculus AB

Official course description, exam format, sample questions, and scoring guidelines.

Visit AP Central →

📚

The VR School

VRS AP Resources Center

All VR School AP course resources, study guides, and score submission guidance.

Open AP Resources →

⭐

Student Exemplar

AP Seminar Exemplar by Jiang

See the standard every VRS student aspires to — and the path to getting there.

View Exemplar →

Agentic AI Tutoring

Your Score 5 AI Tutor

Prof. Sofia Euler is your AP Calculus AB expert — every FRQ type, scoring rubric, and exam strategy. SofAIconnects calculus to every other subject you're studying.

🔗 Walk me through solving a related rates problem step by step ∫ Explain the Fundamental Theorem of Calculus — both parts — with examples 📐 Give me a timed FRQ on area between curves and check my work 📊 Help me master the difference between f, f', and f'' reading from a graph

🌟 Next Level

Your Calculus Skills Are a Research Superpower — Use Them in AP Seminar

AP Calculus AB builds exactly the analytical thinking AP Seminar demands: rigorous reasoning, evidence-based argumentation, and quantitative problem-solving. See how Jiang combined STEM disciplines to build an outstanding portfolio recognized at the national level.

View AP Seminar Exemplar Explore AP Seminar →

🎓

📐

Ready to Score a 5 in Calculus?

Enroll in the most comprehensive, AI-powered AP Calculus AB course available. WASC accredited. UC A-G Section C approved. Exam: May 5, 2026.

Enroll in This Course

Duration1 Year

Credits10

DeliveryRemote / Hybrid / In-Person

UC SectionC – Mathematics

Honors GPA+1.0 weighted point

No commitment until admissions confirms your placement.

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AP ExamUC A-G · Section CUC Honors · +1.0 GPAMay 5, 2026

AP Calculus AB
The Mathematics of Change

Limits · Derivatives · Integrals · Mastered

Start with Prof. Sofia

Enroll in This Course

Duration1 Year

Credits10

DeliveryRemote / Hybrid / In-Person

UC SectionC – Mathematics

Honors GPA+1.0 weighted point

No commitment until admissions confirms your placement.

AP Resources

Score Target

Quick LinksCollegeBoard AP Calculus AB VRS AP Resources AP Seminar Exemplar ↗

Exam: May 5, 2026

Exam Blueprint

Four Sections · MC + FRQ

🔢

MC No Calculator

Section I · Part A (No Calculator)

~33%60 min30 questions

› Pure mathematical reasoning — no technology shortcuts
› Tests limits, derivatives (all rules), and basic integration analytically
› Speed matters: ~2 min per question; skip and return if stuck

🧮

MC With Calculator

Section I · Part B (Calculator)

~17%45 min15 questions

› Use calculator for: numerical derivatives, definite integrals, solve equations, find intersections
› Questions often involve context (position/velocity/acceleration, area between curves)
› Know HOW to use nDeriv() and fnInt() on your TI-84 or Nspire

Score 5 Tip: Don't use the calculator as a crutch — know what you're computing before you press buttons. The calculator computes; YOU must interpret the result and connect it to the question.

📐

FRQ With Calculator

Section II · Part A (Calculator)

~17%30 min2 FRQs

› Usually involves area/volume, accumulation, or particle motion with calculator-found values
› Show ALL setup work before using calculator: write the integral, set up the equation
› Round final answers to 3 decimal places unless the problem specifies otherwise

✏️

FRQ No Calculator

Section II · Part B (No Calculator)

~33%60 min4 FRQs

› Tests analytical skills: using graphs, tables, and symbolic manipulation
› FRQ types: graph analysis (reading f' and f'' to determine f behavior), related rates, differential equations, accumulation problems
› Show every step — intermediate work earns partial credit

Score 5 Tip: For graph-based FRQs: always connect sign changes of f'(x) to increasing/decreasing of f(x), and sign changes of f''(x) to concavity. A sign chart for f' and f'' is your best friend.

Score Distribution (2024)

Where Students Land

~300,000 students take AP Calculus AB annually — one of the largest AP exams. 19% score a 5. Strong conceptual reasoning + FRQ precision separates the top scorers.

Extremely Qualified

← Your target19%

Well Qualified

17%

Qualified

19%

Possibly Qualified

23%

No Recommendation

22%

What It Takes to Score a 5

Precision + conceptual depth

✍️

FRQ excellence — show every step, justify every claim, earn partial credit on every part

🔢

Derivative rules from memory: power, product, quotient, chain, implicit — no hesitation

🎯

90%+ accuracy on no-calculator MC — this is 30% of the exam and fully analytical

∫

FTC mastery — both parts, including d/dx of integrals with variable upper limits

📊

Graph analysis fluency — read f, f', f'' relationships instantly from any graph

🧮

Calculator proficiency: nDeriv(), fnInt(), intersection finding on TI-84 or Nspire

CollegeBoard CED Aligned

Eight Calculus Units

📉

UNIT 110–12%

Limits and Continuity

Expand ›

Key Topics

Intuitive definition of limits
Limit laws
One-sided limits
Infinite limits and asymptotes
Continuity (definition, types of discontinuity)
IVT
Squeeze Theorem

Key Terms & Formulas

limit

the value a function approaches as x approaches c

continuity

function is continuous at c if f(c) = lim(x→c) f(x)

removable discontinuity

hole in graph (limit exists but ≠ f(c))

vertical asymptote

where lim = ±∞

horizontal asymptote

limit as x→±∞

squeeze theorem

if g(x)≤f(x)≤h(x) and lim g = lim h = L, then lim f = L

FRQ Practice Prompt

Evaluate: lim(x→3) (x²-9)/(x-3). Now find lim(x→0) sin(x)/x using the Squeeze Theorem approach. Finally, describe the difference between lim(x→2) f(x) and f(2) — when can they be different?

Practice with Prof. Sofia →

Curated Video Resources

3Blue1Brown "Essence of Calculus Chapter 1"

Khan Academy Limits

PatrickJMT limits series

📈

UNIT 210–12%

Derivatives — Basics

Expand ›

Key Topics

Derivative as rate of change
Derivative from definition (difference quotient)
Power rule
Sum/difference rule
Constant multiple
Derivatives of sin, cos, eˣ, ln x
Tangent line equations

Key Terms & Formulas

derivative

instantaneous rate of change at a point

difference quotient

(f(x+h)-f(x))/h

power rule

d/dx(xⁿ) = nxⁿ⁻¹

tangent line

line touching curve at one point with slope = f'(c)

differentiability

function is differentiable at c if f'(c) exists

secant line

line through two points on a curve

FRQ Practice Prompt

Find dy/dx for: y = 3x⁴ - 2x² + 5x - 1. Find the equation of the tangent line to f(x) = x³ at x = 2. Finally, explain in plain English what f'(3) = -2 means in context of a position function.

Practice with Prof. Sofia →

Curated Video Resources

Krista King "Power Rule"

Khan Academy derivatives

PatrickJMT basic derivatives

🔗

UNIT 39–13%

Derivatives — Advanced Rules

Expand ›

Key Topics

Product rule
Quotient rule
Chain rule
Implicit differentiation
Derivatives of inverse functions
Derivatives of inverse trig (arcsin, arccos, arctan)

Key Terms & Formulas

chain rule

d/dx[f(g(x))] = f'(g(x))·g'(x)

product rule

d/dx[uv] = u'v + uv'

implicit differentiation

differentiating both sides with respect to x when y is defined implicitly

inverse function derivative

(f⁻¹)'(b) = 1/f'(f⁻¹(b))

composite function

f(g(x)) — function applied inside another

related rates

rates of change of related quantities

FRQ Practice Prompt

Differentiate: y = (x²+1)³·sin(x). Then find dy/dx for x² + y² = 25 (circle, implicit). Finally find dy/dx for y = arctan(2x).

Practice with Prof. Sofia →

Curated Video Resources

Krista King chain rule

3Blue1Brown implicit differentiation

PatrickJMT product/quotient rule

🚗

UNIT 410–15%

Contextual Applications of Derivatives

Expand ›

Key Topics

Position/velocity/acceleration (s(t), v(t)=s'(t), a(t)=v'(t))
Rates of change in context
Related rates problems (ladder, shadow, cone, sphere)
L'Hôpital's Rule (0/0 or ∞/∞ forms)
Linear approximation (tangent line approximation)

Key Terms & Formulas

velocity

instantaneous rate of change of position: v(t) = s'(t)

acceleration

rate of change of velocity: a(t) = v'(t) = s''(t)

related rates

when multiple quantities change together, differentiate with respect to time

L'Hôpital's Rule

lim f(x)/g(x) = lim f'(x)/g'(x) when form is 0/0 or ∞/∞

speeding up

when v(t) and a(t) have the same sign

slowing down

when v(t) and a(t) have opposite signs

FRQ Practice Prompt

A ladder 10 feet long leans against a wall. The bottom slides away at 2 ft/sec. How fast is the top sliding down when the bottom is 6 feet from the wall? Show complete related rates setup.

Practice with Prof. Sofia →

Curated Video Resources

Krista King related rates

Khan Academy L'Hôpital's Rule

PatrickJMT particle motion

📊

UNIT 515–18%

Analytical Applications of Derivatives

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Key Topics

Mean Value Theorem
Extreme Value Theorem
Critical points (f'=0 or undefined)
First Derivative Test (increasing/decreasing → local extrema)
Second Derivative Test (concavity → local extrema)
Absolute extrema on closed intervals
Inflection points (f'' changes sign)
Optimization problems

Key Terms & Formulas

critical point

where f'(c) = 0 or f'(c) is undefined

relative maximum

f(c) ≥ f(x) for x near c

inflection point

where f changes concavity (f'' changes sign)

Mean Value Theorem

f'(c) = [f(b)-f(a)]/(b-a) for some c

concave up

f''>0 (smile shape)

Extreme Value Theorem

continuous f on [a,b] attains absolute max and min

FRQ Practice Prompt

Practice with Prof. Sofia →

Curated Video Resources

Khan Academy first/second derivative test

Krista King optimization

PatrickJMT MVT

∫

UNIT 617–20%

Integration and Accumulation of Change

Expand ›

Key Topics

Riemann sums (left, right, midpoint, trapezoidal approximation)
Definite integral as limit of Riemann sums
Fundamental Theorem of Calculus (Parts 1 and 2)
Antiderivatives
Indefinite integrals
Integration rules (power, trig, exponential)
U-substitution

Key Terms & Formulas

antiderivative

F(x) such that F'(x) = f(x)

definite integral

∫ₐᵇ f(x)dx represents signed area under f from a to b

FTC Part 1

d/dx ∫ₐˣ f(t)dt = f(x)

FTC Part 2

∫ₐᵇ f(x)dx = F(b)-F(a)

Riemann sum

approximation of integral using rectangles

u-substitution

substitution technique: let u = g(x), then du = g'(x)dx

FRQ Practice Prompt

Evaluate: ∫(2x³ - 4x + 3)dx (indefinite). Then evaluate ∫₀² (3x² + 1)dx using FTC Part 2. Finally, use u-substitution to evaluate ∫ 2x(x²+1)⁴ dx.

Practice with Prof. Sofia →

Curated Video Resources

3Blue1Brown "Integration and the FTC"

Khan Academy FTC

Krista King u-substitution

🌀

UNIT 76–12%

Differential Equations

Expand ›

Key Topics

Differential equations (dy/dx = f(x,y))
Slope fields (drawing and interpreting)
Separation of variables
Exponential growth and decay (y = Ce^(kt))
Euler's method (numerical approximation)

Key Terms & Formulas

differential equation

equation relating a function to its derivatives (dy/dx = ky)

separation of variables

technique: move all y terms to one side, x terms to other, then integrate

slope field

visual representation of solutions (tiny tangent segments at each point)

exponential growth

y = y₀e^(kt) when k > 0

exponential decay

y = y₀e^(kt) when k < 0

Euler's method

numerical approximation: y_next ≈ y_now + (dy/dx)·Δx

FRQ Practice Prompt

Practice with Prof. Sofia →

Curated Video Resources

Khan Academy slope fields

Krista King differential equations

PatrickJMT separation of variables

📐

UNIT 810–15%

Applications of Integration

Expand ›

Key Topics

Area between two curves (∫[f(x)-g(x)]dx)
Volume of solids of revolution (Disk/Washer method: π∫[R²-r²]dx)
Volume using cross sections (known shapes: square, rectangle, triangle)
Accumulation problems (net change theorem)
Average value of a function

Key Terms & Formulas

area between curves

∫ₐᵇ [f(x)-g(x)]dx where f≥g on [a,b]

disk method

π∫ₐᵇ [f(x)]²dx — volume when cross sections are disks

washer method

π∫ₐᵇ [R(x)²-r(x)²]dx — volume with hole

net change

∫ₐᵇ f'(x)dx = f(b)-f(a)

average value

1/(b-a) ∫ₐᵇ f(x)dx

accumulation

total change from rate of change

FRQ Practice Prompt

Practice with Prof. Sofia →

Curated Video Resources

Khan Academy area between curves

Krista King disk/washer method

PatrickJMT volumes

50% of Total Score

FRQ Mastery Suite

AP Calculus AB's FRQ section tests both calculator-active and no-calculator skills. Showing your work and justifying answers is the biggest differentiator between a 3 and a 5.

FRQ Coach →

∫~17% of total score

Section II Part A

Accumulation/Area FRQ

Calculator-Active FRQ · ~15 min

Calculator-active FRQ involving area between curves, accumulation functions, or particle motion. Calculator results must be clearly set up first.

Scoring Criteria

· Setup: integral or derivative written before calculator use

· Correct limits of integration and integrand

· Proper rounding: 3 decimal places unless otherwise specified

· Interpretation: what the numerical result means in context

Score 5 Strategy

Write the complete integral expression BEFORE entering anything in the calculator

Identify intersection points using the calculator's solve or graph feature

Label your setup clearly: 'Area = ∫[a to b] [f(x) - g(x)] dx'

Show the numerical result with proper units and round to 3 decimal places

Connect back to the problem: 'therefore the total distance is 4.832 feet'

Model Opener

The area of the region enclosed by f and g is given by ∫[a to b] [f(x) − g(x)] dx. Using a calculator, the intersection points are x = [value] and x = [value], so the area = [value].

📊~17% of total score

Section II Part B

Analytical Graph FRQ

No-Calc Graph Analysis FRQ · ~15 min

Given a graph of f'(x) or f''(x), answer questions about f(x): increasing/decreasing, concavity, local extrema, and inflection points. No calculator allowed.

Scoring Criteria

· Correct identification of intervals using sign of f'(x)

· Correct concavity using sign of f''(x)

· Proper justification — must cite the derivative, not just the graph

· Correct location and type of local extrema with full reasoning

Score 5 Strategy

Build a sign chart for f'(x) immediately — mark +/- on each interval

f is increasing where f'(x) > 0; decreasing where f'(x) < 0

f has a local max where f' changes from + to -; local min where f' changes from - to +

f''(x) > 0 means concave up; f''(x) < 0 means concave down

Inflection points occur where f''(x) changes sign (not just equals zero)

Always JUSTIFY: write 'because f'(x) changes from positive to negative at x = c'

Model Opener

Since f'(x) changes from positive to negative at x = c, f has a local maximum at x = c by the First Derivative Test.

🔧~8% of total score

Section II Part B

Related Rates / Optimization

Applied Derivatives FRQ · ~15 min

Multi-part problem requiring setup of a related rates equation or optimization problem. No calculator — must show complete algebraic work.

Scoring Criteria

· Correct equation relating the variables (e.g., Pythagorean theorem, volume formula)

· Proper implicit differentiation with respect to time

· Substitution of known values AFTER differentiating

· Final answer with correct units and sign

Score 5 Strategy

Draw and label a diagram — always. Assign variables to all changing quantities

Write the equation relating the variables BEFORE differentiating

Differentiate BOTH sides with respect to t (implicit differentiation)

Substitute known values for rates and positions only after differentiating

State your answer as a complete sentence with units: 'The height is decreasing at 3 ft/sec'

For optimization: find critical points with f'(c) = 0 and verify with second derivative test

Model Opener

Let x = the distance from the wall and y = the height of the ladder. By the Pythagorean theorem, x² + y² = 100. Differentiating both sides with respect to t: 2x(dx/dt) + 2y(dy/dt) = 0.

🌀~8% of total score

Section II Part B

Differential Equations FRQ

Slope Field / DE FRQ · ~15 min

Problem involving slope fields, separation of variables, and/or Euler's method. May include drawing a slope field segment or verifying a solution.

Scoring Criteria

· Correct separation of variables (y terms one side, x terms other)

· Proper integration of both sides with constant of integration +C

· Application of initial condition to find C

· Correct particular solution written explicitly

Score 5 Strategy

Separate variables completely: all y and dy on one side, all x and dx on the other

Integrate both sides — include +C on one side only

Apply initial condition immediately to solve for C

Write the particular solution clearly and check by substituting back

For slope fields: at each labeled point, draw a short segment with slope = dy/dx at that point

For Euler's method: y_new = y_old + (dy/dx)·Δx — show each step

Model Opener

Separating variables: (1/y) dy = 2x dx. Integrating both sides: ln|y| = x² + C. Applying the initial condition y(0) = 3: ln(3) = C. Therefore y = 3e^(x²).

Curated for Score 5

Practice Tests & Resources

🏛

OFFICIALFREE

CollegeBoard AP Calculus AB

Official CED, sample questions, and exam format from CollegeBoard.

Open resource

📂

OFFICIALFREE

Past AP Calc AB FRQs (2014–2024)

Actual past exam free-response questions with scoring guidelines.

Open resource

🎥

CONCEPTUALFREE

3Blue1Brown Essence of Calculus

The most beautiful calculus series ever made. Build DEEP conceptual understanding. Watch before the course begins.

Open resource

📺

FRQ STRATEGYFREE

Krista King Math

Systematic, clear AP Calculus tutorials for every topic. Best for worked examples and FRQ walkthroughs.

Open resource

🎯

FREE PRACTICEFREE

Khan Academy AP Calculus AB

Complete practice problems and exercises for every unit. Use alongside video resources.

Open resource

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WORKED EXAMPLESFREE

PatrickJMT

Thousands of calculus worked examples. Search any topic and Patrick has a clear step-by-step video.

Open resource

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COMPREHENSIVEFREE

Fiveable AP Calculus AB

Complete course review, FRQ practice, and live cram sessions aligned to the AP CED.

Open resource

📘

EXAM PREP

Barron's AP Calculus AB Prep

Best prep book for practice exams. Includes 4 full-length practice tests.

Open resource

AI-Powered Progress

16-Week Score 5 Study Plan

Weeks 1–4

Phase 1: Limits and Differentiation (Units 1-3) — foundations

Master limit laws, one-sided limits, and continuity definitions
Learn all derivative rules: power, product, quotient, chain rule
Implicit differentiation and inverse function derivatives
Daily practice: 10 derivative problems from PatrickJMT or Krista King

Weeks 5–8

Phase 2: Applications and Integration (Units 4-6) — conceptual depth

Related rates and particle motion (Units 4–5)
Critical points, first/second derivative tests, optimization
Introduction to integration: Riemann sums, FTC Parts 1 & 2
U-substitution practice — 10 problems per day

Weeks 9–12

Phase 3: Differential equations, Integration apps, FRQ mastery

Slope fields, separation of variables, Euler's method (Unit 7)
Area between curves, disk/washer volume, average value (Unit 8)
Complete 2 full timed FRQ sets per week with self-scoring
Use past AP FRQs (2014–2024) from CollegeBoard — timed 15 min each

Weeks 13–16

Phase 4: Full timed exams — 1 per week (105 min MC + 90 min FRQ)

Complete 1 full AP practice exam per week (both sections timed)
Review every missed question with Prof. Sofia (SofAI chat)
Drill calculator techniques: nDeriv(), fnInt(), intersection finding
Final formula review: all derivative and integral rules from memory

Official & Curated

AP Resources Hub

🏛

Official Source

CollegeBoard AP Calculus AB

Official course description, exam format, sample questions, and scoring guidelines.

Visit AP Central →

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The VR School

VRS AP Resources Center

All VR School AP course resources, study guides, and score submission guidance.

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Student Exemplar

AP Seminar Exemplar by Jiang

See the standard every VRS student aspires to — and the path to getting there.

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Agentic AI Tutoring

Your Score 5 AI Tutor

Prof. Sofia Euler is your AP Calculus AB expert — every FRQ type, scoring rubric, and exam strategy. SofAIconnects calculus to every other subject you're studying.

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Credits10

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UC SectionC – Mathematics

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WASC Accredited · UC A-G Approved · CollegeBoard Aligned · Exam: May 5, 2026

AP Calculus ABThe Mathematics of Change

Enroll in This Course

Four Sections · MC + FRQ

MC No Calculator

MC With Calculator

FRQ With Calculator

FRQ No Calculator

Where Students Land

What It Takes to Score a 5

Eight Calculus Units

Limits and Continuity

Key Topics

Key Terms & Formulas

Curated Video Resources

Derivatives — Basics

Key Topics

Key Terms & Formulas

Curated Video Resources

Derivatives — Advanced Rules

Key Topics

Key Terms & Formulas

Curated Video Resources

Contextual Applications of Derivatives

Key Topics

Key Terms & Formulas

Curated Video Resources

Analytical Applications of Derivatives

Key Topics

Key Terms & Formulas

Curated Video Resources

Integration and Accumulation of Change

Key Topics

Key Terms & Formulas

Curated Video Resources

Differential Equations

Key Topics

Key Terms & Formulas

Curated Video Resources

Applications of Integration

Key Topics

Key Terms & Formulas

Curated Video Resources

FRQ Mastery Suite

Accumulation/Area FRQ

Analytical Graph FRQ

Related Rates / Optimization

Differential Equations FRQ

Practice Tests & Resources

CollegeBoard AP Calculus AB

Past AP Calc AB FRQs (2014–2024)

3Blue1Brown Essence of Calculus

Krista King Math

Khan Academy AP Calculus AB

PatrickJMT

Fiveable AP Calculus AB

Barron's AP Calculus AB Prep

16-Week Score 5 Study Plan

Phase 1: Limits and Differentiation (Units 1-3) — foundations

Phase 2: Applications and Integration (Units 4-6) — conceptual depth

Phase 3: Differential equations, Integration apps, FRQ mastery

Phase 4: Full timed exams — 1 per week (105 min MC + 90 min FRQ)

AP Resources Hub

CollegeBoard AP Calculus AB

VRS AP Resources Center

AP Seminar Exemplar by Jiang

Your Score 5 AI Tutor

Your Calculus Skills Are a Research Superpower — Use Them in AP Seminar

Ready to Score a 5 in Calculus?

Enroll in This Course

AP Calculus ABThe Mathematics of Change

Enroll in This Course

Four Sections · MC + FRQ

MC No Calculator

MC With Calculator

FRQ With Calculator

FRQ No Calculator

Where Students Land

What It Takes to Score a 5

Eight Calculus Units

Limits and Continuity

AP Calculus AB
The Mathematics of Change

AP Calculus AB
The Mathematics of Change