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AP Calculus AB 2.5 Derivative Rules Question Bank

Practise AP Calculus AB 2.5 questions by rewriting powers and radicals, applying the power rule and evaluating derivatives in context.

Syllabus
Effective Fall 2025
Course
AP Calculus AB

Exam points

  • rewrite radicals and negative exponents as powers before differentiating
  • apply the power rule and evaluate derivatives at specified inputs

Covered syllabus points

FUN-3.A—Calculate derivatives of familiar functions

2.5 Applying the Power Rule question 1

[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)=∫0xf(t)dtg(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)=32−1.25cos⁡(π100(h−20)2)f(h)=3 \sqrt{2-1.25 \cos \left(\frac{\pi}{100}(h-20)^{2}\right)} for 0≤h≤200 \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.

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