Find the average value of f(x) = e⁴^x over the interval [1, 3]. Approximate value to two decimal places. A) 20,337.52 B) 40,675.05 C) 20,344.35 D) 20,351.17 E) 6.82

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Quizplus · Accepted Answer

The average value of a function f(x) over an interval [a, b] is given by the formula:$$f_{avg} = \frac{1}{b-a} \int_a^b f(x) dx$$In this case, we have f(x) = e^4x and [a, b] = [1, 3]. So, the average value of f(x) over [1, 3] is:$$f_{avg} = \frac{1}{3-1} \int_1^3 e^4x dx$$$$= \frac{1}{2} \left[ \frac{1}{4} e^4x \right]_1^3$$$$= \frac{1}{8} (e^4(3) - e^4(1))$$$$= \frac{1}{8} (e^12 - e^4)$$$$= \frac{1}{8} (20,337.52 - 54.5982)$$$$= 20,344.35$$Therefore, the average value of f(x) over [1, 3] is 20,344.35, which is closest to choice C.

Find the Average Value of F(x) = E4x Over the Interval

Find the Average Value of F(x) = E⁴^x Over the Interval