AP Calculus BC 3.6 Higher-Order Derivatives Overview

Review higher-order derivatives by differentiating repeatedly and interpreting second-derivative information in context.

[Maximum number: 2]

An ice sculpture melts in such a way that it can be modeled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function r, where r(t) is measured in centimeters and t is measured in days. The table above gives selected values of $r^{\prime}(t)$ , the rate of change of the radius, over the time interval $0 \leq t \leq 12$ .

Approximate $r^{\prime \prime}(8.5)$ using the average rate of change of $r^{\prime}$ over the interval $7 \leq t \leq 10$ . Show the computations that lead to your answer, and indicate units of measure.

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$r^{\prime \prime}(8.5) \approx \frac{r^{\prime}(10)-r^{\prime}(7)}{10-7}=\frac{-3.8-(-4.4)}{10-7}$ $r^{\prime \prime}(8.5)$ with
supporting work

$=\frac{0.6}{3}=0.2$ centimeter per day per day

Units

Scoring notes:

- To earn the first point the supporting work must include at least a difference and a quotient.

- Simplification is not required to earn the first point. If the numerical value is simplified, it must be

correct.

- The second point can be earned with an incorrect approximation for $r^{\prime \prime}(8.5)$ but cannot be earned

without some value for $r^{\prime \prime}(8.5)$ presented.

- Units may be written in any equivalent form (such as cm / day $^{2}$ ).

Total for part (a)

2 points

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Covered syllabus points

3.6 Calculating Higher-Order Derivatives question 1