Practise AP Calculus AB 2.1 questions by calculating average rates of change from intervals, tables, graphs and applied data.
Juice is sold in 20-centimeter-tall bottles with horizontal cross sections parallel to the base that are circles, as shown in the
figure. The radius of the circular cross section at height h above the base of the bottle is given by a differentiable function
r, where h and r(h) are measured in centimeters. Selected values of r(h) are given in the table shown.
Let f be the continuous function defined on the closed interval [-1,3] whose graph, consisting of three line segments, is
shown. Let g be the function given by g(x)=∫0xf(t)dt.
Approximate r′(3.1) using the average rate of change of r over the interval 0≤h≤6.2. Show the computations that
lead to your answer. Indicate units of measure.
On what intervals, if any, is g decreasing? Give a reason for your answer.
To earn the first point a response must provide a difference and a quotient. A response of only 6.22.7−2.5 or 6.2−00.2 earns the
first point.
To earn the second point a response must have cmcm (or the equivalent) attached to a numerical value. These units do not
reduce to 1 because the numerator, cm, refers to radius of the bottle and the denominator, cm, refers to height of the
bottle.
\begin{tabular}[t]{|l|l|l|}
\hline 0
1
The student response accurately includes both of the criteria below.
\begin{itemize}
\item[□]
Answer
\item[□]
Units
Solution:
r′(3.1)≈6.2−0r(6.2)−r(0)=6.22.7−2.5
0.032 centimeter per centimeter
Part B
Select a point value to view scoring criteria, solutions, and/or examples to score the response.