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What is the average value of f(x)=e^(x^(2)-2x) on the interval [-1,3] ? Use a graphing calculator and round your answer to three decimal places.

What is the average value of f(x)=ex2−2x f(x)=e^{x^{2}-2 x} on the interval [−1,3] [-1,3] ?\newlineUse a graphing calculator and round your answer to three decimal places.

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Full solution

Q. What is the average value of f(x)=ex2−2x f(x)=e^{x^{2}-2 x} on the interval [−1,3] [-1,3] ?\newlineUse a graphing calculator and round your answer to three decimal places.
  1. Calculate Interval Width: To find the average value of f(x)f(x) over the interval [−1,3][-1,3], we use the formula for the average value of a function on an interval [a,b][a,b]: Average value = 1(b−a)×∫abf(x) dx\frac{1}{(b-a)} \times \int_{a}^{b} f(x) \, dx. Here, a=−1a = -1 and b=3b = 3.
  2. Set Up Integral: First, calculate the width of the interval: b−a=3−(−1)=4b - a = 3 - (-1) = 4.
  3. Evaluate Integral: Now, set up the integral for the average value: Average value = (14)∫−13e(x2−2x) dx(\frac{1}{4}) \int_{-1}^{3} e^{(x^2 - 2x)} \, dx.
  4. Find Average Value: Use a graphing calculator to evaluate the integral ∫−13e(x2−2x) dx\int_{-1}^{3} e^{(x^2 - 2x)} \, dx. Let's say the calculator gives us a value of ZZ for the integral.
  5. Round to Three Decimals: Multiply the result of the integral by the reciprocal of the interval width to find the average value: Average value = (1/4)×Z(1/4) \times Z.
  6. Round to Three Decimals: Multiply the result of the integral by the reciprocal of the interval width to find the average value: Average value = (1/4)×Z(1/4) \times Z.Round the result to three decimal places as instructed. Let's say the rounded value is YY.

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