AP PreCalculus Practice Quiz: Rates of Change
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Test your understanding with short quizzes. This quiz has 15 questions to check your progress.
Question 1 of 15
All Questions (15)
A) The rate of change at the midpoint of the interval.
B) The constant rate of change that would produce the same total change in the output values over that interval.
C) The maximum rate of change that occurs within the interval.
D) The rate at which the function's output is changing at the specific point a.
Correct Answer: B
This directly matches the definition provided in Content Point 3: 'The average rate of change... is the constant rate of change that yields the same change in the output values as the function yielded on that interval.'
A) As the input increases, the output decreases.
B) The output value is always a positive number.
C) As the input increases, the output also increases.
D) The rate of change itself is increasing.
Correct Answer: C
Content Point 6 states: 'A positive rate of change indicates that as one quantity increases or decreases, the other quantity does the same.' Therefore, if the input increases, the output also increases.
A) The rate of change at x=2 is approximately equal to the rate of change at x=4.
B) The rate of change at x=4 is likely greater than the rate of change at x=2.
C) The rate of change at x=2 is likely greater than the rate of change at x=4.
D) The function has a negative rate of change at both points.
Correct Answer: B
According to Content Point 5, we can compare rates of change at two points using average rate of change approximations over small intervals. Since the average rate of change near x=4 (which is 8) is greater than the average rate of change near x=2 (which is 5), it is reasonable to conclude the rate of change at x=4 is greater.
A) The total change in the function's output over its entire domain.
B) The average of all possible rates of change across the function's domain.
C) A quantification of how output values would change if the input were to change at that specific point.
D) The constant slope of a line connecting the start and end points of the function.
Correct Answer: C
This is a direct paraphrase of Content Point 4: 'The rate of change of a function at a point quantifies the rate at which output values would change were the input values to change at that point.'
A) The population increased by exactly 300 people each year between 2015 and 2020.
B) In 2015, the population was increasing at a rate of 300 people per year.
C) The total increase in population between 2015 and 2020 was 300 people.
D) The same total population change between 2015 and 2020 would have occurred if the population had grown by a constant 300 people per year.
Correct Answer: D
The average rate of change is 0.3 thousand people (or 300 people) per year. Content Point 3 explains this is the constant rate that yields the same total change. It does not mean the rate was constant every year (eliminating A) or that this was the rate at a specific point (eliminating B). The total change is 0.3 * (10-5) = 1.5 thousand, or 1500 people, not 300 (eliminating C).
A) As x increases, f(x) increases, and the rate of increase is growing.
B) As x increases, f(x) decreases, and the rate of decrease is slowing down (becoming less negative).
C) As x increases, f(x) decreases, and the rate of decrease is speeding up (becoming more negative).
D) As x increases, f(x) increases, and the rate of increase is slowing down.
Correct Answer: C
The rates of change are negative, so as x increases, f(x) decreases. The rate changes from -10.5 to -10.8, which is a larger magnitude of decrease (more negative). This indicates that the function is decreasing at a faster rate. This aligns with Content Point 2, which involves describing how quantities vary together.
A) Calculate the average rate of change over the function's entire domain.
B) Find the value of the function at point c.
C) Calculate the average rate of change over a sufficiently small interval containing c.
D) Find the average of the function's maximum and minimum values.
Correct Answer: C
Content Point 5 explicitly states: 'The rates of change at two points can be compared using average rate of change approximations over sufficiently small intervals containing each point...' This principle applies to finding the rate at a single point as well.
A) The particle moves in the positive direction on both intervals, but its average speed is slower on the second interval.
B) The particle moves in the positive direction on the first interval and in the negative direction on the second.
C) The particle moves faster on average during the interval [3, 6] than on [1, 3].
D) The particle is slowing down at the exact instant t=3.
Correct Answer: A
Both average rates of change (7 m/s and 2 m/s) are positive, indicating that as time (the input) increases, position (the output) also increases on both intervals (Content Point 6). This means the particle is moving in the positive direction. Since 7 > 2, the average rate of change (average velocity) is greater on the first interval than the second. This describes how the quantities vary together over different intervals (Content Point 2).
A) Because small intervals are easier to calculate.
B) Because over a small interval, the function's behavior is more likely to be nearly constant, similar to the constant rate of the average.
C) Because this method guarantees the exact instantaneous rate of change.
D) Because large intervals often result in a rate of change of zero.
Correct Answer: B
The concept of using a small interval (Content Point 5) is based on the idea that a function's curve can be approximated by a straight line over a very short distance. The average rate of change (Content Point 3) is a constant rate. Therefore, over a small interval, the function's change is closely modeled by this constant rate, providing a good approximation of the rate at a specific point (Content Point 4).
A) The company's revenue is higher than its costs.
B) The company's revenue in March is greater than its revenue in January.
C) The company's revenue increased by the same amount each month.
D) The company made a profit during this period.
Correct Answer: B
A positive rate of change means that as the input (time) increases, the output (revenue) also increases (Content Point 6). Therefore, the revenue at the end of the interval (March) must be greater than the revenue at the start (January).
A) The function f is a straight line with a positive slope.
B) The function f is always positive.
C) The function f is an increasing function.
D) The rate of change of f is constant.
Correct Answer: C
A positive rate of change indicates that as the input increases, the output also increases (Content Point 6). Since the average rate of change is positive over every small interval, it means the function's output is always increasing as the input increases. This is the definition of an increasing function.
A) The function has a value of -2 when x=5.
B) If x were to increase by a very small amount from 5, g(x) would decrease by approximately twice that amount.
C) The average rate of change of g(x) on any interval containing 5 will be -2.
D) The function is decreasing, and its graph is a straight line with a slope of -2.
Correct Answer: B
Content Point 4 defines the rate of change at a point as quantifying how output values would change. A rate of -2 means that for a small change in x (let's call it Δx), the change in g(x) would be approximately -2 * Δx. So, if x increases by a small amount, g(x) decreases by twice that amount. Option A is a common misconception. Option C is not necessarily true, as this is the instantaneous rate, not the average. Option D is too strong; we only know the rate at one point, not everywhere.
A) The rate of change at point c is always greater than k.
B) The rate of change at point c is equal to k.
C) The rate of change at point c is unrelated to k.
D) The rate of change at point c is k only if c is the midpoint of the interval.
Correct Answer: B
If the average rate of change is constant (k) for every possible interval, it implies the function itself must be changing at that constant rate everywhere. The only function for which this is true is a linear function, f(x) = kx + C. Therefore, the rate of change at any single point (the instantaneous rate) must also be k. This connects the definition of average rate of change (Content Point 3) with the rate of change at a point (Content Point 4).
A) As the number of hours spent studying increases, the number of mistakes on a test decreases.
B) As a car brakes, its speed decreases over time.
C) As the altitude of a hiker increases, the air temperature decreases.
D) As the number of items produced in a factory increases, the total production cost increases.
Correct Answer: D
Content Point 6 states that a positive rate of change occurs when as one quantity increases, the other does the same. In option D, the number of items (input) increases, and the total cost (output) also increases. The other options all describe one quantity increasing while the other decreases, which corresponds to a negative rate of change.
A) 5
B) 3
C) 15
D) Cannot be determined from the given information.
Correct Answer: C
The average rate of change is defined as the total change in output divided by the total change in input. From Content Point 3, we know it's the constant rate that yields the same change. So, (Change in Output) / (Change in Input) = Average Rate. (Change in Output) / (4 - 1) = 5. Therefore, the Change in Output is 5 * 3 = 15.