AP Calculus AB 5.6: Concavity

Practice AP Calculus questions on using first and second derivatives to determine concavity, inflection points, and increasing rates.

[Maximum number: 1]

The temperature of coffee in a cup at time t minutes is modeled by a decreasing differentiable function C, where C(t) is measured in degrees Celsius. For $0 \leq t \leq 12$ , selected values of C(t) are given in the table shown.

For the model defined in part (c), it can be shown that $C^{\prime \prime}(t)=\frac{0.2455 e^{0.01 t}(100-t)}{t^{2}}$ . For 12<t<20, determine whether the temperature of the coffee is changing at a decreasing rate or at an increasing rate. Give a reason for your answer.

Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part.

Because $C^{\prime \prime}(t)>0$ on the interval 12<t<20, the rate of change
in the temperature of the coffee, $C^{\prime}(t)$ , is increasing on this
interval.

Answer with reason

1 point

That is, on the interval 12<t<20, the temperature of the coffee
is changing at an increasing rate.

Scoring notes:

- This point is earned only for a correct answer with a correct reason that references the sign of the

second derivative of C.

- A response that provides a reason based on the evaluation of $C^{\prime \prime}(t)$ at a single point does not earn

this point.

- A response that uses ambiguous pronouns (such as "It is positive, so increasing") does not earn this

point.

- A response does not need to reference the interval 12<t<20 to earn the point.

Total for part (d) 1 point

Total for question 19 points

Part A (AB): Graphing calculator required

9 points

General Scoring Notes

The model solution is presented using standard mathematical notation.

Answers (numeric or algebraic) need not be simplified. Answers given as a decimal approximation should be

correct to three places after the decimal point. Within each individual free-response question, at most one

point is not earned for inappropriate rounding.

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FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.6 question 1