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100+ Free AP Calculus BC Practice Questions

Pass your AP Calculus BC (Advanced Placement Calculus BC) exam on the first try — instant access, no signup required.

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What is the Maclaurin series for sin(x)?

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Key Facts: AP Calculus BC Exam

45

multiple-choice questions (30 no-calculator + 15 calculator)

College Board

6

free-response questions in 1 hour 30 minutes

College Board

1-5

score scale, with 3 or higher generally earning credit

College Board

50%

weight of each section in the composite score

College Board

17-20%

exam weight of Integration and Accumulation of Change

College Board CED

~30%

of the exam is BC-only parametric/polar/vector plus series content

College Board CED

$99

approximate US exam fee for 2025-26

College Board

AP Calculus BC has 45 multiple-choice questions (30 no-calculator in 60 minutes and 15 calculator-active in 45 minutes) plus 6 free-response questions in 90 minutes, for about 3 hours 15 minutes total. The two sections are weighted equally at 50% each, and the exam is scored 1 to 5. It covers all AP Calculus AB content plus parametric, polar, and vector-valued functions and infinite sequences and series, which together account for nearly 30% of the exam. A 3 or higher typically earns college credit (source: College Board, apcentral.collegeboard.org).

Sample AP Calculus BC Practice Questions

Try these sample questions to test your AP Calculus BC exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Evaluate the limit as x approaches 3 of (x^2 - 9)/(x - 3).
A.0
B.3
C.6
D.The limit does not exist
Explanation: Factor the numerator: (x^2 - 9) = (x - 3)(x + 3). Cancel the (x - 3) factor to get x + 3, then substitute x = 3 to obtain 6. This removes the 0/0 indeterminate form.
2The function f is continuous on [a, b] with f(a) = -2 and f(b) = 5. Which theorem guarantees f has a zero on (a, b)?
A.Mean Value Theorem
B.Intermediate Value Theorem
C.Extreme Value Theorem
D.Squeeze Theorem
Explanation: The Intermediate Value Theorem states that a function continuous on a closed interval takes every value between f(a) and f(b). Since 0 lies between -2 and 5, there must be some c in (a, b) where f(c) = 0.
3Evaluate the limit as x approaches infinity of (3x^2 + 2x)/(5x^2 - 1).
A.0
B.3/5
C.1
D.Infinity
Explanation: When the numerator and denominator have the same degree, the limit at infinity equals the ratio of the leading coefficients. Here that ratio is 3/5, so the horizontal asymptote is y = 3/5.
4Evaluate the limit as x approaches 0 of (sin 5x)/x.
A.0
B.1
C.5
D.Does not exist
Explanation: Rewrite (sin 5x)/x as 5 times (sin 5x)/(5x). As x approaches 0, (sin 5x)/(5x) approaches 1, so the limit is 5 times 1, which equals 5.
5For what value of k is f continuous at x = 2, where f(x) = (x^2 - 4)/(x - 2) for x not equal to 2 and f(2) = k?
A.0
B.2
C.4
D.No value of k works
Explanation: For x not equal to 2, simplify (x^2 - 4)/(x - 2) to x + 2. The limit as x approaches 2 is 4, so setting k = 4 makes f continuous at x = 2.
6Given that -x^2 <= g(x) <= x^2 for all x, what is the limit of g(x) as x approaches 0?
A.0
B.1
C.-1
D.Cannot be determined
Explanation: By the Squeeze Theorem, since both bounding functions -x^2 and x^2 approach 0 as x approaches 0, g(x) is forced to the same limit of 0.
7Using the limit definition of the derivative, f'(x) is the limit as h approaches 0 of which expression?
A.(f(x) - f(x - h))/h only
B.(f(x + h) - f(x))/h
C.(f(x + h) + f(x))/h
D.(f(x + h) - f(x))/x
Explanation: The derivative is defined as the limit as h approaches 0 of (f(x + h) - f(x))/h. This represents the slope of the tangent line as the secant interval shrinks to zero.
8If f(x) = x^4 - 3x^2 + 7, what is f'(x)?
A.4x^3 - 6x
B.4x^3 - 6x + 7
C.x^5 - x^3 + 7x
D.4x^3 - 3x
Explanation: Apply the power rule to each term: the derivative of x^4 is 4x^3, the derivative of -3x^2 is -6x, and the derivative of the constant 7 is 0. So f'(x) = 4x^3 - 6x.
9If f(x) = x^2 * sin(x), what is f'(x)?
A.2x cos(x)
B.2x sin(x) + x^2 cos(x)
C.2x sin(x) - x^2 cos(x)
D.x^2 cos(x)
Explanation: Use the product rule: f'(x) = (derivative of x^2)(sin x) + (x^2)(derivative of sin x) = 2x sin(x) + x^2 cos(x).
10If f(x) = (2x + 1)/(x - 3), what is f'(x)?
A.2
B.-7/(x - 3)^2
C.7/(x - 3)^2
D.2/(x - 3)
Explanation: By the quotient rule, f'(x) = [(2)(x - 3) - (2x + 1)(1)]/(x - 3)^2 = (2x - 6 - 2x - 1)/(x - 3)^2 = -7/(x - 3)^2.

About the AP Calculus BC Exam

AP Calculus BC is a College Board Advanced Placement course covering all AP Calculus AB topics plus parametric, polar, and vector-valued functions and infinite sequences and series. The exam has two sections: Section I is 45 multiple-choice questions (Part A, 30 questions in 60 minutes with no calculator; Part B, 15 questions in 45 minutes with a graphing calculator) and Section II is 6 free-response questions in 90 minutes. Each section is worth 50% of the score, which ranges from 1 to 5.

Questions

45 scored questions

Time Limit

3 hours 15 minutes

Passing Score

Scored 1-5; a 3 or higher typically earns college credit

Exam Fee

About $99 per exam (US, 2025-26) (College Board)

AP Calculus BC Exam Content Outline

17-20%

Integration and Accumulation of Change

Riemann sums, the definite integral, the Fundamental Theorem of Calculus, u-substitution, integration by parts, and partial fractions.

17-18%

Infinite Sequences and Series

Convergence tests, geometric and p-series, Taylor and Maclaurin series, power series, and the Lagrange error bound.

11-12%

Parametric, Polar, and Vector-Valued Functions

Derivatives and arc length of parametric and vector functions, planar motion, and area and slope in polar coordinates.

8-11%

Analytical Applications of Differentiation

The Mean Value Theorem, extrema, concavity, the candidates test, and optimization.

6-9%

Contextual Applications of Differentiation

Related rates, linearization, motion, and L'Hospital's Rule.

6-9%

Differential Equations

Slope fields, separation of variables, exponential and logistic models, and Euler's method.

6-9%

Applications of Integration

Average value, area between curves, volumes, and arc length.

4-7%

Limits and Continuity

Evaluating limits, continuity, the Squeeze Theorem, and the Intermediate Value Theorem.

4-7%

Differentiation: Definition and Fundamental Properties

The limit definition of the derivative, power, product, and quotient rules.

4-7%

Differentiation: Composite, Implicit, and Inverse Functions

The chain rule, implicit differentiation, and derivatives of inverse functions.

How to Pass the AP Calculus BC Exam

What You Need to Know

  • Passing score: Scored 1-5; a 3 or higher typically earns college credit
  • Exam length: 45 questions
  • Time limit: 3 hours 15 minutes
  • Exam fee: About $99 per exam (US, 2025-26)

Keys to Passing

  • Complete 500+ practice questions
  • Score 80%+ consistently before scheduling
  • Focus on highest-weighted sections
  • Use our AI tutor for tough concepts

AP Calculus BC Study Tips from Top Performers

1Master the convergence tests for series early — they appear in nearly every BC exam and are the most common source of lost points.
2Practice the no-calculator Part A under strict timing; you average two minutes per question and must do all algebra and antiderivatives by hand.
3Memorize the common Maclaurin series (e^x, sin x, cos x, 1/(1-x)) so you can build new series quickly by substitution and differentiation.
4For parametric and polar problems, keep the formulas for dy/dx, arc length, and polar area on a quick-reference sheet and drill them.
5Always justify answers in free-response using calculus reasoning (sign of the derivative, theorem conditions) rather than just stating numerical results.

Frequently Asked Questions

How many questions are on the AP Calculus BC exam and how long is it?

Section I has 45 multiple-choice questions in 1 hour 45 minutes (30 with no calculator in 60 minutes, then 15 with a graphing calculator in 45 minutes). Section II has 6 free-response questions in 1 hour 30 minutes. The total testing time is about 3 hours 15 minutes.

How is AP Calculus BC scored?

The exam is scored on a 1 to 5 scale, with the multiple-choice and free-response sections each contributing 50% of the composite score. A score of 3 or higher is generally considered passing and often earns college credit.

What is the difference between AP Calculus AB and BC?

AP Calculus BC covers everything in AP Calculus AB and adds parametric, polar, and vector-valued functions and infinite sequences and series. BC exam takers also receive an AB subscore reflecting performance on the shared AB content.

Can I use a calculator on the AP Calculus BC exam?

Part A of the multiple-choice section (30 questions) and Part A of the free-response section do not allow a calculator. Part B of the multiple-choice section (15 questions) and Part B of the free-response section require a graphing calculator.

Which units carry the most weight on the AP Calculus BC exam?

Integration and Accumulation of Change (17-20%) and Infinite Sequences and Series (17-18%) are the most heavily weighted, followed by Parametric, Polar, and Vector-Valued Functions (11-12%).

How much does the AP Calculus BC exam cost?

The standard AP exam fee in the US is about $99 per exam for 2025-26. Fee reductions are available for eligible students, and some schools cover part of the cost.