AP Calculus BC Exam, AP Calculus BC Actual Exam Testbank - 3 Versions with Verified Answers latest update guaranteed pass ()

11



AP Calculus BC Exam, AP Calculus BC
Actual Exam Testbank - 3 Versions with
Verified Answers latest update
guaranteed pass (2026-2027)
Intermediate Value Theorem — Given that a function 𝑓 is continuous on [1, 6] with 𝑓(1) =
−4 and 𝑓(6) = 9, what conclusion can be drawn?
A) 𝑓(𝑥) has a relative maximum at some point in (1, 6)
B) There is at least one value 𝑐 in (1, 6) such that 𝑓 ′ (𝑐) = 0
C) There must be at least one 𝑥-value between 1 and 6 where 𝑓(𝑥) = 0
D) The average rate of change of 𝑓 on [1, 6] is 13/5

Correct Answer: C
Rationale: The Intermediate Value Theorem states that for a continuous function on a closed
interval, it takes on every value between 𝑓(𝑎) and 𝑓(𝑏). Since 𝑓(1) = −4 and 𝑓(6) = 9, and 0
is between -4 and 9, there must be a 𝑐 in (1, 6) such that 𝑓(𝑐) = 0 (the x-axis).



Question 2

Average Rate of Change — The slope of the secant line between two points on a function is the
definition of which of the following?
A) Instantaneous rate of change
B) Average rate of change
C) Derivative at a point
D) Limit of the difference quotient

Correct Answer: B
Rationale: The average rate of change of a function over an interval [𝑎, 𝑏] is calculated
𝑓(𝑏)−𝑓(𝑎)
as , which geometrically represents the slope of the secant line connecting the
𝑏−𝑎
points (𝑎, 𝑓(𝑎)) and (𝑏, 𝑓(𝑏)).



Question 3



11

,11


Instantaneous Rate of Change — The slope of the tangent line at a point on a curve represents
the:
A) Average rate of change
B) Secant line slope
C) Instantaneous rate of change
D) Arc length

Correct Answer: C
Rationale: The instantaneous rate of change of a function at a point is the slope of the tangent
line at that point. It is the value of the derivative at that specific point.



Question 4

Formal Definition of Derivative — Which of the following is the formal definition of the
derivative of 𝑓(𝑥)?
𝑓(𝑥+ℎ)−𝑓(𝑥)
A) lim⁡ℎ→0 ℎ
𝑓(𝑏)−𝑓(𝑎)
B) 𝑏−𝑎
𝑏
C) ∫𝑎 𝑓(𝑥) 𝑑𝑥
𝑓(𝑥)−𝑓(𝑎)
D) lim⁡𝑥→𝑎
𝑥−𝑎

Correct Answer: A
𝑓(𝑥+ℎ)−𝑓(𝑥)
Rationale: The formal definition of the derivative is 𝑓 ′ (𝑥) = lim⁡ℎ→0 . Option D is the
ℎ
alternate definition using a different limit notation.



Question 5

Alternate Definition of Derivative — Which limit expression represents the derivative
of 𝑓(𝑥) at 𝑥 = 𝑎?
𝑓(𝑎+ℎ)−𝑓(𝑎)
A) lim⁡ℎ→0 ℎ
𝑓(𝑥)−𝑓(𝑎)
B) lim⁡𝑥→𝑎 𝑥−𝑎
C) Both A and B
D) Neither A nor B

Correct Answer: C
Rationale: Both expressions are equivalent definitions of the derivative. Option A uses ℎ as the
increment, and Option B uses 𝑥 approaching 𝑎.


11

,11




Question 6

Positive Derivative — If 𝑓 ′ (𝑥) > 0 for all 𝑥 in an interval, then 𝑓(𝑥) is:
A) Decreasing
B) Constant
C) Increasing
D) Concave up

Correct Answer: C
Rationale: A positive derivative indicates that the function's slope is positive, meaning the
function is increasing over that interval.



Question 7

Negative Derivative — If 𝑓 ′ (𝑥) < 0 for all 𝑥 in an interval, then 𝑓(𝑥) is:
A) Increasing
B) Decreasing
C) Constant
D) Concave down

Correct Answer: B
Rationale: A negative derivative indicates that the function's slope is negative, meaning the
function is decreasing over that interval.



Question 8

Relative Minimum (First Derivative Test) — According to the First Derivative Test,
if 𝑓 ′ (𝑥) changes from negative to positive at a critical point 𝑐, then 𝑓 has a:
A) Relative maximum at 𝑐
B) Relative minimum at 𝑐
C) Point of inflection at 𝑐
D) Vertical asymptote at 𝑐

Correct Answer: B
Rationale: A sign change from negative (decreasing) to positive (increasing) indicates that the
function reaches a valley, or relative minimum, at that point.




11

, 11


Question 9

Relative Maximum (First Derivative Test) — According to the First Derivative Test,
if 𝑓 ′ (𝑥) changes from positive to negative at a critical point 𝑐, then 𝑓 has a:
A) Relative maximum at 𝑐
B) Relative minimum at 𝑐
C) Point of inflection at 𝑐
D) Vertical asymptote at 𝑐

Correct Answer: A
Rationale: A sign change from positive (increasing) to negative (decreasing) indicates that the
function reaches a peak, or relative maximum, at that point.



Question 10

Concave Up — A function 𝑓(𝑥) is concave up on an interval if:
A) 𝑓 ′ (𝑥) is decreasing
B) 𝑓 ′ (𝑥) is increasing
C) 𝑓 ′ (𝑥) = 0
D) 𝑓(𝑥) is decreasing

Correct Answer: B
Rationale: Concavity is related to the slope of the derivative. If the derivative 𝑓 ′ (𝑥) is increasing
(i.e., 𝑓 ′′ (𝑥) > 0), the original function is concave up.



Question 11

Concave Down — A function 𝑓(𝑥) is concave down on an interval if:
A) 𝑓 ′ (𝑥) is increasing
B) 𝑓 ′ (𝑥) is decreasing
C) 𝑓 ′ (𝑥) = 0
D) 𝑓(𝑥) is increasing

Correct Answer: B
Rationale: If the derivative 𝑓 ′ (𝑥) is decreasing (i.e., 𝑓 ′′ (𝑥) < 0), the original function is concave
down.



Question 12


11