62–65. {Use of Tech} Graphing f and f'
b. Compute and graph f'.
f(x)=e^−x tan^−1 x on [0,∞)

Verified step by step guidance

1. Step 1: Understand the function f(x) = e^(-x) * tan^(-1)(x). This function is a product of two functions: e^(-x) and tan^(-1)(x). To find the derivative f'(x), we will use the product rule.
2. Step 2: Recall the product rule for derivatives, which states that if you have two functions u(x) and v(x), then the derivative of their product is given by (uv)' = u'v + uv'. Here, let u(x) = e^(-x) and v(x) = tan^(-1)(x).
3. Step 3: Compute the derivative of u(x) = e^(-x). The derivative u'(x) is found using the chain rule: u'(x) = -e^(-x).
4. Step 4: Compute the derivative of v(x) = tan^(-1)(x). The derivative v'(x) is 1/(1 + x^2).
5. Step 5: Apply the product rule to find f'(x): f'(x) = u'(x)v(x) + u(x)v'(x) = (-e^(-x))tan^(-1)(x) + e^(-x)/(1 + x^2). Now, use a graphing tool to plot both f(x) and f'(x) over the interval [0, ∞).

Key Concepts

Derivative

The derivative of a function measures how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For the function f(x) = e^(-x) * arctan(x), the derivative f'(x) can be computed using the product rule and chain rule of differentiation.

Graphing Functions

Graphing a function involves plotting its output values against its input values on a coordinate system. For the function f(x) = e^(-x) * arctan(x), understanding its behavior as x approaches 0 and infinity is crucial. The graph of f' will provide insights into the function's increasing or decreasing behavior, as well as its critical points.

Behavior at Infinity

Analyzing the behavior of a function as x approaches infinity helps determine its long-term trends. For f(x) = e^(-x) * arctan(x), as x increases, e^(-x) approaches 0, while arctan(x) approaches π/2. This interplay affects the overall behavior of f and its derivative f', which is essential for understanding the function's limits and asymptotic behavior.