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Prepare for the timed AP Calculus BC test.

Your answers are held until submission. The final review identifies the units that need another pass.

AP Calculus BC coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

01

Limits and Continuity

Limits, continuity, asymptotic behavior

Goal: recognise a limits and continuity problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Differentiation: Definition and Properties

Core derivative concepts and rules

Goal: recognise a differentiation: definition and properties problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Composite, Implicit, and Inverse Differentiation

Chain rule and advanced derivatives

Goal: recognise a composite, implicit, and inverse differentiation problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Contextual Applications of Differentiation

Rates and motion

Goal: recognise a contextual applications of differentiation problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Analytical Applications of Differentiation

Function analysis and optimization

Goal: recognise a analytical applications of differentiation problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Integration and Accumulation of Change

FTC, methods, improper integrals

Goal: recognise a integration and accumulation of change problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Differential Equations

Models, slope fields, Euler's method

Goal: recognise a differential equations problem from its wording, carry out the governing method, and check that the result is reasonable.

08

Applications of Integration

Area, volume, arc length

Goal: recognise a applications of integration problem from its wording, carry out the governing method, and check that the result is reasonable.

09

Parametric, Polar, and Vector-Valued Functions

Motion, slopes, area, arc length

Goal: recognise a parametric, polar, and vector-valued functions problem from its wording, carry out the governing method, and check that the result is reasonable.

10

Infinite Sequences and Series

Convergence tests, power and Taylor series

Goal: recognise a infinite sequences and series problem from its wording, carry out the governing method, and check that the result is reasonable.

AP Calculus BC questions with worked answers

These 8 questions are printed in full on this page and are drawn across 8 of the 10 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: Find the limit of (e^(2x) - 1) / x as x approaches 0.

AP Calculus BC · Limits and Continuity

  1. 2
  2. 1
  3. 0
  4. infinity
Show the worked answer
  1. Substituting 0 gives 0/0, so use another method.
  2. The Maclaurin series is e^(2x) = 1 + 2x + 2x^2 + ... .
  3. Subtracting 1 and dividing by x leaves 2 + 2x + ... .
  4. As x approaches 0 this tends to 2. L'Hospital's rule gives the same answer.

Answer: 2

Question 2: What is the derivative of arctan(3x) at x = 1/3?

AP Calculus BC · Composite, Implicit, and Inverse Differentiation

  1. 3/2
  2. 3
  3. 1/2
  4. 3/10
Show the worked answer
  1. The derivative of arctan(u) is u' / (1 + u^2).
  2. With u = 3x, u' = 3, so the derivative is 3 / (1 + 9x^2).
  3. At x = 1/3, 9x^2 = 9(1/9) = 1.
  4. So the value is 3 / (1 + 1) = 3/2.

Answer: 3/2

Question 3: A spherical balloon has radius growing at 0.2 cm/s. How fast is its volume growing when the radius is 3 cm?

AP Calculus BC · Contextual Applications of Differentiation

  1. 7.2 pi cubic cm/s
  2. 3.6 pi cubic cm/s
  3. 36 pi cubic cm/s
  4. 2.4 pi cubic cm/s
Show the worked answer
  1. The volume is V = (4/3) pi r^3.
  2. Differentiate with respect to time: dV/dt = 4 pi r^2 times dr/dt.
  3. At r = 3, 4 pi r^2 = 36 pi.
  4. 36 pi x 0.2 = 7.2 pi cubic cm per second.

Answer: 7.2 pi cubic cm/s

Question 4: Evaluate the integral of x / (x^2 + 1) from 0 to 1.

AP Calculus BC · Integration and Accumulation of Change

  1. half of ln 2
  2. ln 2
  3. pi / 4
  4. 1/2
Show the worked answer
  1. Substitute u = x^2 + 1, so du = 2x dx and x dx = du / 2.
  2. The limits become u = 1 and u = 2.
  3. The integral turns into half the integral of 1/u from 1 to 2.
  4. That is half of (ln 2 - ln 1) = half of ln 2, about 0.347.

Answer: half of ln 2

Question 5: For the logistic model dP/dt = 0.05P(1 - P/2000), what is the carrying capacity and at what population does P grow fastest?

AP Calculus BC · Differential Equations

  1. carrying capacity 2000, fastest at P = 1000
  2. carrying capacity 2000, fastest at P = 2000
  3. carrying capacity 1000, fastest at P = 500
  4. carrying capacity 0.05, fastest at P = 1000
Show the worked answer
  1. Growth stops when the bracket is zero, so P = 2000 is the carrying capacity.
  2. The growth rate as a function of P is a downward parabola through P = 0 and P = 2000.
  3. A parabola peaks halfway between its zeros.
  4. Halfway is P = 1000, which is half the carrying capacity.

Answer: carrying capacity 2000, fastest at P = 1000

Question 6: What is the average value of f(x) = x^2 on the interval from 0 to 3?

AP Calculus BC · Applications of Integration

  1. 3
  2. 9
  3. 4.5
  4. 1
Show the worked answer
  1. Average value is the integral divided by the length of the interval.
  2. The integral of x^2 from 0 to 3 is x^3 / 3 = 27 / 3 = 9.
  3. The interval has length 3 - 0 = 3.
  4. 9 / 3 = 3.

Answer: 3

Question 7: A curve is given by x = t^2 and y = t^3 - 3t. What is dy/dx at t = 2?

AP Calculus BC · Parametric, Polar, and Vector-Valued Functions

  1. 2.25
  2. 9
  3. 4
  4. 0.75
Show the worked answer
  1. For a parametric curve, dy/dx = (dy/dt) divided by (dx/dt).
  2. dy/dt = 3t^2 - 3 and dx/dt = 2t.
  3. At t = 2: dy/dt = 12 - 3 = 9 and dx/dt = 4.
  4. 9 / 4 = 2.25.

Answer: 2.25

Question 8: Use the third-degree Maclaurin polynomial for e^x to approximate e^0.5, to four decimal places.

AP Calculus BC · Infinite Sequences and Series

  1. 1.6458
  2. 1.6487
  3. 1.6250
  4. 1.5000
Show the worked answer
  1. The polynomial is 1 + x + x^2 / 2 + x^3 / 6.
  2. At x = 0.5: x^2 / 2 = 0.25 / 2 = 0.125.
  3. x^3 / 6 = 0.125 / 6 = 0.0208333.
  4. 1 + 0.5 + 0.125 + 0.0208333 = 1.6458, close to the true value 1.6487.

Answer: 1.6458

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed AP Calculus BC questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Limits and ContinuityFind the limit of (e^(2x) - 1) / x as x approaches 0.2
2Composite, Implicit, and Inverse DifferentiationWhat is the derivative of arctan(3x) at x = 1/3?3/2
3Contextual Applications of DifferentiationA spherical balloon has radius growing at 0.2 cm/s. How fast is its volume growing when the radius is 3 cm?7.2 pi cubic cm/s
4Integration and Accumulation of ChangeEvaluate the integral of x / (x^2 + 1) from 0 to 1.half of ln 2
5Differential EquationsFor the logistic model dP/dt = 0.05P(1 - P/2000), what is the carrying capacity and at what population does P grow fastest?carrying capacity 2000, fastest at P = 1000
6Applications of IntegrationWhat is the average value of f(x) = x^2 on the interval from 0 to 3?3
7Parametric, Polar, and Vector-Valued FunctionsA curve is given by x = t^2 and y = t^3 - 3t. What is dy/dx at t = 2?2.25
8Infinite Sequences and SeriesUse the third-degree Maclaurin polynomial for e^x to approximate e^0.5, to four decimal places.1.6458

How the generated AP Calculus BC sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this AP Calculus BC course test

What does this page cover?

It covers all 10 AP Calculus BC units listed above.

When can I see correct answers and explanations?

Submit the full test first. The result screen then shows every response, the correct answer, and the explanation so you can review each miss.

How long is each section of the AP Calculus BC exam?

Multiple choice: 1 hour 40 minutes, 42 questions. Free response: 1 hour 30 minutes, 6 questions.

Which AP Calculus BC sections allow a graphing calculator?

Rules change from section to section. Multiple choice, Part A: no calculator. Multiple choice, Part B: graphing calculator required. Free response, Part A: graphing calculator required. Free response, Part B: no calculator. Practise each section under its own rule.

Which AP Calculus BC units does this test cover?

All ten: Limits and Continuity, Differentiation: Definition and Properties, Composite, Implicit, and Inverse Differentiation, Contextual Applications of Differentiation, Analytical Applications of Differentiation, Integration and Accumulation of Change, Differential Equations, Applications of Integration, Parametric, Polar, and Vector-Valued Functions, Infinite Sequences and Series. Study first: Integration and Accumulation of Change (17–20%) carries the largest published weighting.

What should I check first when I review a missed AP Calculus BC question?

Polar area needs the one-half out front and theta limits where the curve is actually traced. Sketch first, then integrate.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.