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MathCalculus

Fundamental Theorem of Calculus: Find F′(2) and the Average Value of F′

Use the Fundamental Theorem of Calculus to find F′(2) and the average value of F′ for an integral-defined function.

Question

Let F(x)=∫1x(1−t) dtF(x)=\int_1^x(1-t)\,dt. Find F′(2)F'(2) and the average value of F′F' on [1,2][1,2].

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Evidence boundary

The prompt is a faithful paraphrase of OpenStax Exercise 147. The numerical results follow from the displayed integral and the Fundamental Theorem of Calculus; no measured or student supplied data are involved.