AP Calculus BC Practice Quiz: Estimating Derivatives of a Function at a Point
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Test your understanding with short quizzes. This quiz has 9 questions to check your progress.
Question 1 of 9
All Questions (9)
A) 5
B) 6
C) 7
D) 12
Correct Answer: B
The best estimate for the derivative at a point from a table is the average rate of change over the smallest symmetric interval containing that point. Here, we use the interval [2, 4] to estimate f'(3). The calculation is (f(4) - f(2)) / (4 - 2) = (19 - 7) / 2 = 12 / 2 = 6.
A) -2
B) -1
C) 1
D) 2
Correct Answer: B
The value of the derivative at an endpoint of a data set can be estimated using the average rate of change over the closest interval. To estimate g'(7), we use the interval [5, 7]. The calculation is (g(7) - g(5)) / (7 - 5) = (20 - 22) / 2 = -1.
A) -3/2
B) -2/3
C) 2/3
D) 3/2
Correct Answer: A
The derivative of a function at a point is the slope of the tangent line at that point. The tangent line shown passes through the point of tangency (2, 3) and another point (4, 0). The slope is calculated as the change in y divided by the change in x: (0 - 3) / (4 - 2) = -3 / 2.
A) a
B) b
C) c
D) d
Correct Answer: A
The derivative g'(x) represents the slope of the tangent line to the graph of g(x). By visually inspecting the graph, the slope is positive at point a, zero at point b (a local maximum), and negative at points c and d. Since a positive value is always greater than zero or any negative value, the derivative is greatest at point a.
A) The value of the function at x=c, f(c).
B) The slope of a secant line through two points on the curve f(x) that are very close to x=c.
C) The area under the curve of f(x) near x=c.
D) The y-intercept of the tangent line at x=c.
Correct Answer: B
Technology estimates the derivative at a point by calculating the slope of a secant line over a very small interval around that point. This is a numerical implementation of the limit definition of the derivative, often using a symmetric difference quotient like (f(c+h) - f(c-h))/(2h) for a very small h. The slope of this secant line provides a very close approximation to the slope of the tangent line.
A) At t=4 seconds, the particle's position is 7.5 meters.
B) At t=4 seconds, the particle's velocity is approximately 7.5 meters per second.
C) Over the first 4 seconds, the particle's average velocity was 7.5 meters per second.
D) At t=4 seconds, the particle's acceleration is approximately 7.5 meters per second squared.
Correct Answer: B
The derivative of a position function with respect to time gives the instantaneous velocity. Therefore, p'(4) represents the velocity of the particle at the specific instant t=4 seconds. The value 7.5 has units of meters per second.
A) (9.61 - 8.41) / (3.1 - 2.9)
B) (9.06 - 9) / (3.01 - 3)
C) (9 - 8.94) / (3 - 2.99)
D) (9.06 - 8.94) / (3.01 - 2.99)
Correct Answer: D
The best estimate for a derivative at a point is found by using the average rate of change over the smallest possible interval surrounding that point. Option D uses the points x=2.99 and x=3.01, which form the narrowest symmetric interval around x=3 provided in the table. This symmetric difference quotient provides the most accurate approximation of the instantaneous rate of change at x=3.
A) x < -2
B) -2 < x < 3
C) x > 3
D) x < -2 and x > 3
Correct Answer: B
The derivative f'(x) is negative on intervals where the function f(x) is decreasing. By observing the graph, the function's slope is negative (the curve goes downwards from left to right) between the local maximum at x=-2 and the local minimum at x=3. Therefore, f'(x) < 0 on the interval -2 < x < 3.
A) The exact velocity of the rocket at t=15 seconds.
B) The average velocity of the rocket over the interval from t=10 to t=20 seconds.
C) The acceleration of the rocket at t=15 seconds.
D) The total height gained by the rocket in the first 15 seconds.
Correct Answer: B
The expression (h(b) - h(a)) / (b - a) represents the average rate of change of the function h(t) on the interval [a, b]. Since h(t) is height (a position), its rate of change is velocity. Therefore, the calculation gives the average velocity over the time interval [10, 20]. This value is often used to estimate the instantaneous velocity at the midpoint, t=15, but it is fundamentally an average velocity.