AP CollegeBoard Unit 1 Progress Check: MCQ Part A Scoring Guide 16. A new special attraction opened at a museum. The museum management tracked the number of people who visited the attraction each day and created a function model M for the number of people for each day d after the attraction opened. Each day they also calculated the rate of change of the number of people visiting the attraction. They created a function model R for the rate of change, in people per day, for each day d after the attraction opened. The function R is given by R(d) = (-d +35d³ – 411d² + 1845d — 2686.5). At which of the following values of d does the graph of y = M(d) have a point of inflection? = 1 200
The correct answer and explanation is:
To determine when the graph of y=M(d)y = M(d) has a point of inflection, we must identify when the concavity of the graph changes. This occurs when the second derivative of M(d)M(d), denoted M′′(d)M”(d), changes sign.
Given that R(d)R(d) is the rate of change of M(d)M(d), we know: R(d)=M′(d).R(d) = M'(d).
To find M′′(d)M”(d), we compute the derivative of R(d)R(d): R′(d)=M′′(d).R'(d) = M”(d).
The function R(d)R(d) is given as: R(d)=−d+35d3−411d2+1845d−2686.5.R(d) = -d + 35d^3 – 411d^2 + 1845d – 2686.5.
Steps to Solve:
- Compute R′(d)R'(d):
R′(d)=ddd[−d+35d3−411d2+1845d−2686.5]=105d2−822d+1845−1.R'(d) = \frac{d}{dd}[-d + 35d^3 – 411d^2 + 1845d – 2686.5] = 105d^2 – 822d + 1845 – 1.
- Simplify R′(d)R'(d):
R′(d)=105d2−822d+1844.R'(d) = 105d^2 – 822d + 1844.
- Find where R′(d)=0R'(d) = 0 to identify possible points of inflection:
105d2−822d+1844=0.105d^2 – 822d + 1844 = 0.
This is a quadratic equation that can be solved using the quadratic formula: d=−b±b2−4ac2a,d = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a},
where a=105a = 105, b=−822b = -822, and c=1844c = 1844.
- Solve for dd:
d=−(−822)±(−822)2−4(105)(1844)2(105).d = \frac{-(-822) \pm \sqrt{(-822)^2 – 4(105)(1844)}}{2(105)}. d=822±675684−773640210.d = \frac{822 \pm \sqrt{675684 – 773640}}{210}. d=822±−97956210.d = \frac{822 \pm \sqrt{-97956}}{210}.
Since the discriminant (−97956-97956) is negative, there are no real roots, meaning R′(d)≠0R'(d) \neq 0 for any real dd. Thus, there is no change in concavity, and y=M(d)y = M(d) does not have a point of inflection.
Explanation:
The key to identifying points of inflection is finding where R′(d)=M′′(d)=0R'(d) = M”(d) = 0 and checking for sign changes in M′′(d)M”(d). In this case, the quadratic equation for R′(d)R'(d) has no real solutions, so the concavity of M(d)M(d) does not change. Hence, the graph of y=M(d)y = M(d) does not have a point of inflection for any value of dd.