Question

Find the average value of the function over the given interval and all values of \(x\) in the interval for which the function equals its average value. $$ f(x)=2 e^{x}, \quad[-1,1] $$

Short answer

The average value of the function \(f(x)=2e^x\) over the interval [-1,1] is \(\frac{1}{2} (2e - 2/e)\). The x-values for which the function equals its average value within the interval [-1,1] are those values that satisfy the equation \(x = \ln((e - 1/e)/2)\),

Step-by-step solution

Calculating the average value of the function over the interval [-1,1]

The average value of the function is obtained using the formula \(\frac{1}{b-a} \int_{a}^{b} f(x) dx\). For this particular function \(f(x)=2e^x\) over the interval [-1,1], the average value is computed as: \(\frac{1}{1-(-1)} \int_{-1}^{1} 2e^x dx = \frac{1}{2} [2e^x]_{-1}^{1} = \frac{1}{2} (2e - 2/e)\)

Finding all x-values in the interval for which the function equals its average value

Now, to find the x-values at which \(f(x)=2e^x\) equals the average value obtained in step 1, the function is set equal to the average value and then solved for x. The equation thus obtained is: \(2e^x = \frac{1}{2} (2e - 2/e)\). After simplifying, the equation becomes \(e^x = \frac{1}{2} (e - 1/e)\). Solving it will yield the x-values within the interval [-1,1] at which the function equals its average value.

Solving the equation

To solve the equation \(e^x = \frac{1}{2} (e - 1/e)\), it's likely to use a logarithmic function. Taking the natural logarithm on both sides yields \(x = \ln((e - 1/e)/2)\). It's then necessary to check whether this x-value falls within the interval [-1,1].