Question

In Exercises \(1-6,\) find the average rate of change of the function over each interval. \(f(x)=e^{x}\) (a) $$[-2,0] \quad$$ (b) $$[1,3]$$

Short answer

The average rate of change over the interval \([-2,0]\) is \(\frac{1 - e^{-2}}{2}\) and over the interval \([1,3]\) is \(\frac{e^{3} - e^{1}}{2}\)

Step-by-step solution

Average rate of change on interval [-2,0]

Apply the formula for the average rate of change to the first interval \([-2,0]\) in the function \(f(x)=e^{x}\). Here, \(f(-2) = e^{-2}\) and \(f(0) = e^{0}\). Therefore, the average rate of change for this interval is \(\frac{f(0) - f(-2)}{0-(-2)} = \frac{e^{0} - e^{-2}}{0-(-2)}\)

Simplify

Simplify the expression from step 1 to get a number. Since \(e^{0}\) equals 1 and \(0-(-2)\) equals 2, the average rate of change for this interval is \(\frac{1 - e^{-2}}{2}\)

Average rate of change on interval [1,3]

Apply the formula for the average rate of change to the second interval \([1,3]\) in the function \(f(x)=e^{x}\). Here, \(f(1) = e^{1}\) and \(f(3) = e^{3}\). Therefore, the average rate of change for this interval is \(\frac{f(3) - f(1)}{3-1} = \frac{e^{3} - e^{1}}{3-1}\)

Simplify

Simplify the expression from step 3 to get a number. Since \(3-1\) equals 2, the average rate of change for this interval is \(\frac{e^{3} - e^{1}}{2}\)