Consider the continuous function on defined by

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the continuous function ff on [0,6][0,6] defined by f(t)=sin⁡t−t4f(t)=\sin t-\frac{t}{4}. Let g(x)=∫πxf(t) dt,0≤x≤6g(x)=\int_{\pi}^{x} f(t)\,dt, \qquad 0\le x\le 6. (a) Determine all critical points of gg in the interval (0,6)(0,6). Classify each as a location of a relative maximum, a relative minimum, or neither. Provide complete justification. (b) Find the intervals of concavity for gg on (0,6)(0,6). Support your answer with appropriate calculations. (c) Evaluate g′′(π)g''(\pi). Interpret the sign of this value in terms of the shape of the graph of gg near x=πx=\pi. (d) Using a second-degree Taylor polynomial centered at x=πx=\pi, approximate g(π+0.2)g(\pi+0.2). Include all steps in your construction of the polynomial.

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