Problem

Question Find the value of $c$ in $[-5,-2]$ such that $f(c)$ equals the average value of $f(t)=-6 t-2$ over the interval $[-5,-2]$. Enter your answer as an exact fraction if necessary.

Solution

Step 1 :The average value of a function \(f(t)\) over an interval \([a, b]\) is given by the formula \(\frac{1}{b-a}\int_{a}^{b}f(t)dt\). We need to find the value of \(c\) in \([-5,-2]\) such that \(f(c)\) equals this average value.

Step 2 :First, we calculate the average value of \(f(t)\) over the interval \([-5,-2]\).

Step 3 :The average value of the function over the interval \([-5,-2]\) is found to be 19.

Step 4 :Next, we solve the equation \(f(c) = 19\) to find the value of \(c\) that makes this true.

Step 5 :The solution to the equation is \(c = -\frac{7}{2}\).

Step 6 :Final Answer: The value of \(c\) in \([-5,-2]\) such that \(f(c)\) equals the average value of \(f(t)\) over the interval \([-5,-2]\) is \(\boxed{-\frac{7}{2}}\).

From Solvely APP
Source: https://solvelyapp.com/problems/8237/

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