AP Calculus AB 2.5 Applying the Power Rule Question Bank

Practise AP Calculus AB 2.5 questions by rewriting powers and radicals, then applying the power rule to calculate derivatives.

[Maximum number: 2]

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 $g(x)=\int_{0}^{x} f(t) d t$ .

The radius of the circular cross section at height h above the base of the bottle can also be modeled by the function f,

where h and f(h) are measured in centimeters. Let $f(h)=3 \sqrt{2-1.25 \cos \left(\frac{\pi}{100}(h-20)^{2}\right)}$ for $0 \leq h \leq 20$ . Based

on the model, at a height of h=12 centimeters, is the radius increasing or decreasing as h increases? Give a reason

for your answer.

What is the absolute minimum value of g on the closed interval [-1, 3]? Justify your answer.

To earn the second point, a response does not need to provide a value for $f^{\prime}(12)$ ; the sign of $f^{\prime}(12)$ is sufficient.

Any presented value of $f^{\prime}(12)$ must be correct for the digits presented, up to three decimal places.

If a response presents an incorrect sign or value for $f^{\prime}(12)$ , the response does not earn the second point.

Degree mode: A response that presents answers obtained by using a calculator in degree mode does not earn the first point

it would have otherwise earned. The response is eligible for all subsequent points (unless no answer is possible in degree

mode or the question is made simpler by using degree mode). In degree mode, $f^{\prime}(12)=-0.0006660598$ . If a

response presents this value, it earns the first point but is not eligible to earn the second point.

2022 Form I FRQ

\begin{tabular}[t]{|l|l|l|}
\hline 0

1

The student response accurately includes both of the criteria below.
\begin{itemize}
\item[□]

Considers $f^{\prime}(12)$
\item[□]

Answer with reason

Solution:

$f^{\prime}(12)=-0.535902$

At a height of 12 centimeters, the radius is decreasing as h increases because $f^{\prime}(12)<0$ .

Part D

Select a point value to view scoring criteria, solutions, and/or examples to score the response.

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FUN-3.A—Calculate derivatives of familiar functions question 1