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AP PreCalculus Practice Quiz: Rates of Change in Polar Functions

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 11 questions to check your progress.

Question 1 of 11

For a polar function r = f(θ), if f(θ) is positive and increasing for θ in the interval (a, b), which statement describes the graph on this interval?

All Questions (11)

For a polar function r = f(θ), if f(θ) is positive and increasing for θ in the interval (a, b), which statement describes the graph on this interval?

A) The points on the graph are moving farther from the origin.

B) The points on the graph are moving closer to the origin.

C) The points on the graph are at a constant distance from the origin.

D) The points on the graph are crossing the origin.

Correct Answer: A

According to the provided content, if a polar function r = f(θ) is positive and increasing, then the distance between f(θ) and the origin is increasing.

A particle moves along the path of a polar function r = f(θ). On the interval π/4 < θ < π/2, the function is negative and decreasing. How is the particle's distance from the origin changing on this interval?

A) It is increasing.

B) It is decreasing.

C) It is constant.

D) It cannot be determined without knowing the specific function.

Correct Answer: A

According to the provided content, if a polar function is negative and decreasing, then the distance between the point and the origin is increasing.

For a polar function r = f(θ), under which of the following conditions is the distance between a point on the curve and the origin decreasing?

A) r > 0 and r is increasing.

B) r < 0 and r is decreasing.

C) r > 0 and r is decreasing.

D) r is constant and positive.

Correct Answer: C

The provided content states that if a polar function is positive and decreasing or negative and increasing, the distance to the origin is decreasing. Option C matches the first of these conditions.

The distance between a point on the graph of a polar function r = f(θ) and the origin is increasing. Which of the following must be true about the function r and its rate of change with respect to θ?

A) The function r = f(θ) must be positive.

B) The function r = f(θ) must be increasing.

C) The value of r and its rate of change with respect to θ have the same sign.

D) The value of r and its rate of change with respect to θ have opposite signs.

Correct Answer: C

The content states the distance from the origin is increasing if (r is positive and increasing) or (r is negative and decreasing). In the first case, r > 0 and dr/dθ > 0. In the second case, r < 0 and dr/dθ < 0. In both scenarios, r and its rate of change have the same sign.

For a polar function r = f(θ), the function changes from increasing to decreasing at θ = c. What does this change signify for the graph of the function?

A) The graph has a vertical tangent.

B) The graph has a horizontal tangent.

C) The graph has a point that is a relative extremum with respect to its distance from the origin.

D) The graph crosses the polar axis.

Correct Answer: C

According to the provided content, if the function changes from increasing to decreasing or vice versa, it has a relative extremum corresponding to a point relatively closest to or farthest from the origin.

What is the average rate of change of the polar function r = 4θ - 2 on the interval [0, π/2]?

A) 2

B) 4

C) 2π - 2

D) π

Correct Answer: B

The average rate of change is the change in r divided by the change in θ. At θ = π/2, r = 4(π/2) - 2 = 2π - 2. At θ = 0, r = 4(0) - 2 = -2. The average rate of change is ((2π - 2) - (-2)) / (π/2 - 0) = (2π) / (π/2) = 4.

A point moves along a polar curve r = f(θ) such that r < 0 for a < θ < b. If the point is moving closer to the origin on this interval, which statement about the function f(θ) must be true?

A) The function is positive.

B) The function is negative.

C) The function is increasing.

D) The function is decreasing.

Correct Answer: C

According to the provided content, for the distance to the origin to be decreasing, the function must be (positive and decreasing) or (negative and increasing). Since we are given that r < 0, the function must be increasing.

The average rate of change of r with respect to θ over an interval [θ₁, θ₂] is defined as:

A) The change in the angle divided by the change in the radius.

B) The instantaneous rate of change of the radius at the midpoint of the interval.

C) The ratio of the change in the radius values to the change in θ over the interval.

D) The total distance traveled by the point divided by the change in θ.

Correct Answer: C

This is the definition provided in the content: 'The average rate of change of r with respect to θ over an interval of θ is the ratio of the change in the radius values to the change in θ over an interval of θ.'

For a polar function r = f(θ) that is always positive, a point on its graph is determined to be relatively closest to the origin on an interval. What must be true about the function r = f(θ) at this point?

A) The function changes from increasing to decreasing.

B) The function changes from decreasing to increasing.

C) The function is equal to zero.

D) The rate of change of the function is zero, but the function's behavior does not change.

Correct Answer: B

A point relatively closest to the origin is a relative minimum for the distance. Since r is always positive, this corresponds to a relative minimum for r itself. A relative minimum occurs when the function's rate of change goes from negative to positive, meaning the function changes from decreasing to increasing.

Consider a polar function r = f(θ). If r > 0 and the derivative dr/dθ < 0 on an interval, which of the following describes the motion of a point on the curve?

A) Moving away from the origin.

B) Moving towards the origin.

C) Moving in a circle of constant radius.

D) The direction of motion relative to the origin cannot be determined.

Correct Answer: B

The condition dr/dθ < 0 means the function r is decreasing. The content states that if a polar function is positive and decreasing, the distance between the point and the origin is decreasing. Therefore, the point is moving towards the origin.

If a polar function r = f(θ) has a relative extremum on an interval, what does this correspond to on its graph?

A) A point where the graph intersects the polar axis.

B) A point of maximum or minimum curvature.

C) A cusp or sharp turn in the graph.

D) A point that is relatively closest to or farthest from the origin.

Correct Answer: D

This is a direct application of the provided content, which states: '...the function has a relative extremum on the interval corresponding to a point relatively closest to or farthest from the origin.'