62​–​65. {Use of Tech} Graphing f and f'
a. Graph f with a graphing utility.
f(x)=e^−x tan^−1 x on [0,∞)

Verified step by step guidance

1. Start by defining the function f(x) = e^{-x} an^{-1}(x). This function combines an exponential decay component and the arctangent function, which will influence its behavior on the interval [0, ∞).
2. Use a graphing utility to plot the function f(x) over the interval [0, ∞). Make sure to set appropriate limits for the x-axis and y-axis to capture the behavior of the function as x approaches infinity.
3. Next, calculate the derivative f'(x) using the product and chain rules. The derivative will help you understand the rate of change of the function and identify critical points.
4. Once you have f'(x), graph this derivative function alongside f(x) on the same interval. This will allow you to compare the original function with its rate of change.
5. Analyze the graphs of f(x) and f'(x) to identify key features such as local maxima, minima, and points of inflection, which will provide insights into the behavior of the function.