Find the average value f_("ave ") of the function f on the given interval. f(x)=3x^(2)+4x,quad[-1,2]

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Find the average value  f_("ave ") of the function  f on the given interval.  f(x)=3x^(2)+4x,quad[-1,2]

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Define Average Value Formula: To find the average value of a continuous function f(x) on the interval [a, b], we use the formula: f_(＂ave ＂) = (1/(b-a)) * ∫[a to b] f(x) dx In this case, f(x) = 3x^2 + 4x and the interval is [-1, 2].Calculate Definite Integral: First, we need to calculate the definite integral of f(x) from -1 to 2. ∫[-1 to 2] (3x^2 + 4x) dx This requires us to integrate the function term by term.Evaluate Antiderivative: The integral of 3x^2 with respect to x is x^3, and the integral of 4x with respect to x is 2x^2. So, ∫(3x^2) dx = x^3 and ∫(4x) dx = 2x^2.Subtract Lower Limit: Now we can write the antiderivative of f(x): ∫(3x^2 + 4x) dx = x^3 + 2x^2 + C, where C is the constant of integration. However, since we are calculating a definite integral, we do not need to include C.Simplify Expression: We evaluate the antiderivative at the upper and lower limits of the interval and subtract: (2^3 + 2*(2^2)) - ((-1)^3 + 2*(-1)^2) This simplifies to (8 + 8) - (-1 + 2).Find Average Value: Simplifying the expression gives us: (16) - (1) Which equals 15.Divide by Interval Length: Now we have the definite integral of f(x) from -1 to 2, which is 15. To find the average value, we divide this result by the length of the interval, which is 2 - (-1) = 3.Calculate Average Value: The average value of the function f(x) on the interval [-1, 2] is: f_(＂ave ＂) = (1/3) * 15Simplifying the expression gives us: f_(＂ave ＂) = 5

Find the average value $f_{\text{ave}}$ of the function $f$ on the given interval. $\newline$ $f(x)=3x^{2}+4x, \quad [-1,2]$

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Find the average value fave f_{\text{ave}} fave​ of the function f f f on the given interval.\newlinef(x)=3x2+4x,[−1,2] f(x)=3x^{2}+4x, \quad [-1,2] f(x)=3x2+4x,[−1,2]

Find the average value $f_{\text{ave}}$ of the function $f$ on the given interval. $\newline$ $f(x)=3x^{2}+4x, \quad [-1,2]$