AP Precalculus: Rates of Change (Drill 1)

Question 1. The table below shows the height h (in feet) of a balloon at selected times t (in minutes).

t (min)	0	2	4	6
h (ft)	50	74	98	122

What is the average rate of change of h on the interval [2, 6], and what does it represent in context?

• A) 24 feet per minute; the balloon rises 24 feet each minute on average between t = 2 and t = 6.
• B) 48 feet per minute; the total change in height over the interval.
• C) 12 feet per minute; the balloon rises 12 feet each minute on average between t = 2 and t = 6. ✓
• D) 12 feet; the height gained at t = 6 minus the height at t = 2.

Explanation: Choice C is correct. Average rate of change = (h(6) − h(2)) / (6 − 2) = (122 − 74) / 4 = 48 / 4 = 12 feet per minute. This means the balloon rises an average of 12 feet for each minute elapsed between t = 2 and t = 6. Choice A is incorrect because 24 feet per minute results from dividing the total change (48) by 2, the length of one sub-interval, rather than by 4, the full interval length. Choice B is incorrect because 48 feet per minute confuses total change in height with average rate of change; the total change must be divided by the interval length to obtain a rate. Choice D is incorrect because 12 feet is not a rate; it omits "per minute" and misidentifies the result as a height rather than a rate of change.

Question 2. The table below gives values of a function f at equally spaced values of x.

x	0	1	2	3	4
f(x)	3	6	12	24	48

Which of the following claims and supporting statements best describes the data?

• A) f is best modeled by a linear function, because the first differences are constant.
• B) f is best modeled by a quadratic function, because the second differences are constant.
• C) f is best modeled by a quadratic function, because the first differences are increasing.
• D) f is best modeled by an exponential function, because the ratio of consecutive output values is constant. ✓

Explanation: Choice D is correct. The ratios of consecutive outputs are 6/3 = 2, 12/6 = 2, 24/12 = 2, 48/24 = 2, a constant ratio at equally spaced x-values is the defining feature of an exponential function. Choice A is incorrect because the first differences (3, 6, 12, 24) are not constant; they double each step. Choice B is incorrect because constant second differences identify a quadratic model; the second differences here are 3, 6, 12, not constant. Choice C is incorrect because increasing first differences are consistent with both quadratic and exponential models; only the constant ratio distinguishes this data as exponential.

Question 3. The table below gives values of a function g at equally spaced values of x.

x	0	1	2	3	4
g(x)	4	7	12	19	28

Which of the following claim and explanation pairs best describes the data?

• A) g is best modeled by a linear function, because the output values increase by a constant amount each step for the data in the table.
• B) g is best modeled by an exponential function, because the output values are always increasing.
• C) g is best modeled by a quadratic function, because the second differences in the output values are constant. ✓
• D) g is best modeled by an exponential function, because the first differences are increasing.

Explanation: Choice C is correct. First differences: 7−4 = 3, 12−7 = 5, 19−12 = 7, 28−19 = 9. These are not constant, ruling out a linear model. Second differences: 5−3 = 2, 7−5 = 2, 9−7 = 2. The second differences are constant (all equal 2), which is the defining feature of a quadratic model. Choice A is incorrect because the first differences (3, 5, 7, 9) are not constant, the output increases by a different amount each step. Choice B is incorrect because always increasing describes linear, quadratic, and exponential functions alike; it does not identify the model type. Choice D is incorrect because increasing first differences are consistent with both quadratic and exponential models; only constant second differences confirm a quadratic model specifically.

Question 4. A cyclist begins a ride and her speed increases throughout the first 10 minutes. The table below shows her speed s (in miles per hour) at selected times t (in minutes).

t (min)	0	2	4	6	8	10
s (mph)	0	6	10	13	15	16

Which of the following best describes the rate at which the cyclist's speed is changing over the interval [0, 10]?

• A) The cyclist's speed is increasing at a decreasing rate, because the average rate of change of s over successive 2-minute intervals is getting smaller. ✓
• B) The cyclist's speed is increasing at an increasing rate, because the speed values are always getting larger.
• C) The cyclist's speed is increasing at a constant rate, because the speed is increasing throughout the interval.
• D) The cyclist's speed is increasing at a decreasing rate, because the speed never exceeds 16 mph.

Explanation: Choice A is correct. The average rate of change of s over each 2-minute interval: (6−0)/2 = 3, (10−6)/2 = 2, (13−10)/2 = 1.5, (15−13)/2 = 1, (16−15)/2 = 0.5. These values decrease (3, 2, 1.5, 1, 0.5), meaning each additional 2 minutes produces a smaller gain in speed, the speed is increasing at a decreasing rate (concave down). Choice B is incorrect because speed values getting larger only confirms speed is increasing, not the rate at which it is increasing; a student must examine successive rates of change, not output values alone. Choice C is incorrect because a constant rate of increase would require equal speed gains in equal time intervals, but the gains (6, 4, 3, 2, 1) are shrinking. Choice D is incorrect because the maximum speed reached has no bearing on whether the rate of increase is increasing or decreasing.

Question 5. The function f is given by f(x) = 2x² + 3x. What is the average rate of change of f on the interval [0, 3]?

• A) 27
• B) 3
• C) 9 ✓
• D) 5

Explanation: Choice C is correct. Compute f(0) and f(3): f(0) = 2(0)² + 3(0) = 0. f(3) = 2(3)² + 3(3) = 2(9) + 9 = 18 + 9 = 27. Average rate of change = (f(3) − f(0)) / (3 − 0) = (27 − 0) / 3 = 9. Choice A is incorrect because 27 is the value of f(3) itself, the student correctly computed the output but forgot to subtract f(0) and divide by the interval length to find the average rate of change. Choice B is incorrect because 3 results from treating the denominator as 3² = 9 rather than the interval length 3 − 0 = 3, giving 27/9 = 3, a sign that the student confused squaring the endpoint with computing interval length. Choice D is incorrect because 5 results from computing f(3) as 2(3) + 3(3) = 6 + 9 = 15, using 2(3) instead of 2(3²) = 18, and then dividing by 3: 15/3 = 5, which is the classic error of forgetting to apply the exponent before multiplying.