Practice AP Calculus BC questions on calculating second and higher derivatives and interpreting higher-order derivative notation.
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′(t), the rate of change of the radius, over the time interval 0≤t≤12.
Approximate r′′(8.5) using the average rate of change of r′ over the interval 7≤t≤10. Show the computations that lead to your answer, and indicate units of measure.
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r′′(8.5)≈10−7r′(10)−r′(7)=10−7−3.8−(−4.4)r′′(8.5) with
supporting work
=30.6=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′′(8.5) but cannot be earned
without some value for r′′(8.5) presented.
- Units may be written in any equivalent form (such as cm / day 2 ).
Total for part (a)
2 points