AP CALCULUS AB –QUESTIONS AND CORRECT ANSWERS (VERIFIED ANSWERS) PLUS RATIONALES 2026
Q&A | INSTANT DOWNLOAD PDF.
*Core Domains*
*Limits and Continuity*
*Differentiation: Definition and Fundamental Properties*
*Differentiation: Composite, Implicit, and Inverse Functions*
*Contextual Applications of Differentiation*
*Analytical Applications of Differentiation*
*Integration and Accumulation of Change*
*Differential Equations*
*Applications of Integration*
*Introduction*
*This assessment is designed to rigorously evaluate a candidate's mastery of essential
SECTION ONE: QUESTIONS 1–100
1. What is the limit of (x^2 - 1) / (x - 1) as x approaches 1?
A. 0
B. 1
🟢 C. 2
D. Does not exist
🔴 RATIONALE: By factoring the numerator as (x-1)(x+1), the (x-1) terms cancel, leaving x+1, which evaluates
to 2 at x=1.
,2. If f(x) = 3x^2 + 5x, what is f'(x)?
A. 3x + 5
🟢 B. 6x + 5
C. 6x + 5x
D. 6x^2 + 5
🔴 RATIONALE: Applying the power rule to each term individually results in 6x + 5.
3. Which of the following conditions must be met for a function to be continuous at a point c?
A. The derivative must exist at c.
🟢 B. The limit as x approaches c must equal f(c).
C. The function must be differentiable on an open interval.
D. The slope must be zero at c.
🔴 RATIONALE: Continuity requires the limit from both sides to exist and equal the function value at that point.
4. Find the derivative of sin(x) with respect to x.
A. -cos(x)
B. sin(x)
🟢 C. cos(x)
D. -sin(x)
🔴 RATIONALE: The derivative of the sine function is defined as the cosine function.
5. What is the integral of 2x dx?
A. x + C
B. x^2 + C
🟢 C. x^2 + C
D. 2x^2 + C
🔴 RATIONALE: Using the power rule for integration, the integral of x^n is (x^(n+1))/(n+1), which for x^1
becomes x^2.
6. If a particle's position is given by s(t), what does s'(t) represent?
A. Acceleration
🟢 B. Velocity
, C. Jerk
D. Displacement
🔴 RATIONALE: The velocity is defined as the instantaneous rate of change of position with respect to time.
7. What is the slope of the tangent line to y = x^3 at x = 2?
A. 4
B. 6
🟢 C. 12
D. 8
🔴 RATIONALE: The derivative y' = 3x^2. Evaluating at x=2 gives 3(2^2) = 12.
8. Which rule is used to differentiate a product of two functions?
A. Chain Rule
B. Quotient Rule
🟢 C. Product Rule
D. Power Rule
🔴 RATIONALE: The product rule states (uv)' = u'v + uv'.
9. The Mean Value Theorem requires a function to be:
A. Differentiable only.
B. Continuous only.
🟢 C. Continuous and differentiable.
D. Constant.
🔴 RATIONALE: The MVT requires continuity on [a, b] and differentiability on (a, b).
10. What is the derivative of e^x?
🟢 A. e^x
B. x*e^(x-1)
C. e
D. ln(x)
🔴 RATIONALE: The exponential function e^x is unique because it is its own derivative.
Q&A | INSTANT DOWNLOAD PDF.
*Core Domains*
*Limits and Continuity*
*Differentiation: Definition and Fundamental Properties*
*Differentiation: Composite, Implicit, and Inverse Functions*
*Contextual Applications of Differentiation*
*Analytical Applications of Differentiation*
*Integration and Accumulation of Change*
*Differential Equations*
*Applications of Integration*
*Introduction*
*This assessment is designed to rigorously evaluate a candidate's mastery of essential
SECTION ONE: QUESTIONS 1–100
1. What is the limit of (x^2 - 1) / (x - 1) as x approaches 1?
A. 0
B. 1
🟢 C. 2
D. Does not exist
🔴 RATIONALE: By factoring the numerator as (x-1)(x+1), the (x-1) terms cancel, leaving x+1, which evaluates
to 2 at x=1.
,2. If f(x) = 3x^2 + 5x, what is f'(x)?
A. 3x + 5
🟢 B. 6x + 5
C. 6x + 5x
D. 6x^2 + 5
🔴 RATIONALE: Applying the power rule to each term individually results in 6x + 5.
3. Which of the following conditions must be met for a function to be continuous at a point c?
A. The derivative must exist at c.
🟢 B. The limit as x approaches c must equal f(c).
C. The function must be differentiable on an open interval.
D. The slope must be zero at c.
🔴 RATIONALE: Continuity requires the limit from both sides to exist and equal the function value at that point.
4. Find the derivative of sin(x) with respect to x.
A. -cos(x)
B. sin(x)
🟢 C. cos(x)
D. -sin(x)
🔴 RATIONALE: The derivative of the sine function is defined as the cosine function.
5. What is the integral of 2x dx?
A. x + C
B. x^2 + C
🟢 C. x^2 + C
D. 2x^2 + C
🔴 RATIONALE: Using the power rule for integration, the integral of x^n is (x^(n+1))/(n+1), which for x^1
becomes x^2.
6. If a particle's position is given by s(t), what does s'(t) represent?
A. Acceleration
🟢 B. Velocity
, C. Jerk
D. Displacement
🔴 RATIONALE: The velocity is defined as the instantaneous rate of change of position with respect to time.
7. What is the slope of the tangent line to y = x^3 at x = 2?
A. 4
B. 6
🟢 C. 12
D. 8
🔴 RATIONALE: The derivative y' = 3x^2. Evaluating at x=2 gives 3(2^2) = 12.
8. Which rule is used to differentiate a product of two functions?
A. Chain Rule
B. Quotient Rule
🟢 C. Product Rule
D. Power Rule
🔴 RATIONALE: The product rule states (uv)' = u'v + uv'.
9. The Mean Value Theorem requires a function to be:
A. Differentiable only.
B. Continuous only.
🟢 C. Continuous and differentiable.
D. Constant.
🔴 RATIONALE: The MVT requires continuity on [a, b] and differentiability on (a, b).
10. What is the derivative of e^x?
🟢 A. e^x
B. x*e^(x-1)
C. e
D. ln(x)
🔴 RATIONALE: The exponential function e^x is unique because it is its own derivative.
