Final answer:
The equation that could be used to find the value of b in the given scenario is (f(b)-f(1))/(b-1) = 20 (Option D).
Step-by-step explanation:
In order to find the value of b in the equation for the average rate of change of the function f(x)=e^(2x) over the interval [1,b] being equal to 20, we need to use the given options and analyze which one satisfies this condition.
To find the correct equation to determine the value of b, given that the average rate of change of the function \( f(x) = e^{2x} \) over the interval \([1,b]\) is 20 and \( b > 1 \), we can use the definition of the average rate of change of a function.
The average rate of change of a function \( f(x) \) over an interval \([a,b]\) is given by the formula: \[ \frac{f(b) - f(a)}{b - a} \] In this case, the function is \( f(x) = e^{2x} \), our interval is \([1,b]\), and the given average rate of change is 20.
Thus, we have \( a = 1 \) and the formula for average rate of change becomes: \[ \frac{f(b) - f(1)}{b - 1} \] Setting this equal to 20 gives us the equation: \[ \frac{f(b) - f(1)}{b - 1} = 20 \]
We can now substitute \( f(x) = e^{2x} \) into the equation: \[ \frac{e^{2b} - e^{2(1)}}{b - 1} = 20 \] This equation will allow us to solve for b and best matches with the given options. Hence, the correct equation from the options provided is: \[ C) \quad \frac{f(b) - f(1)}{b - 1} = 20 \]
The correct equation would be (f(b)-f(1))/(b-1) = 20 (Option D) because it represents the average rate of change of the function over the interval. By plugging in the function values, we get (e^(2b) - e^2)/(b-1) = 20. Solving this equation will give us the value of b.