Define on the interval .

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Define s(x)=∫0xt2sin⁡t dts(x)=\displaystyle\int_{0}^{x} t^{2}\sin t\,dt on the interval [0,π][0,\pi]. (a) Without evaluating the integral explicitly, determine the average rate of change of ss over [0,π][0,\pi]. Express your answer in terms of definite integrals or known constants. (b) Prove that there exists at least one number cc in (0,π)(0,\pi) such that s′(c)s'(c) equals the average rate from part (a). Be explicit about any theorems or conditions you invoke. (c) Using sign changes or another justified numerical method, estimate one value of cc to within 0.10.1 and explain why your estimate satisfies the requirement from part (b).

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