62​–​65. {Use of Tech} Graphing f and f'
c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.
f(x)=e^−x tan^−1 x on [0,∞)

Verified step by step guidance

1. Step 1: Understand the problem. We need to graph the function f(x) = e^(-x) * tan^(-1)(x) and its derivative f'(x) over the interval [0, ∞). We will verify that the zeros of f'(x) correspond to points where f(x) has a horizontal tangent line.
2. Step 2: Find the derivative f'(x). Use the product rule for differentiation, which states that if you have a function h(x) = u(x) * v(x), then h'(x) = u'(x) * v(x) + u(x) * v'(x). Here, u(x) = e^(-x) and v(x) = tan^(-1)(x).
3. Step 3: Differentiate u(x) and v(x) separately. The derivative of u(x) = e^(-x) is u'(x) = -e^(-x). The derivative of v(x) = tan^(-1)(x) is v'(x) = 1/(1 + x^2).
4. Step 4: Apply the product rule. Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula to find f'(x). This gives f'(x) = -e^(-x) * tan^(-1)(x) + e^(-x) * (1/(1 + x^2)).
5. Step 5: Graph f(x) and f'(x) using a graphing tool. Identify the zeros of f'(x) on the graph. These zeros are the x-values where f'(x) = 0, indicating that f(x) has a horizontal tangent line at these points. Verify that these correspond to the horizontal tangents on the graph of f(x).

Key Concepts

Derivative and Critical Points

The derivative of a function, denoted as f', represents the rate of change of the function f. Critical points occur where the derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection. In this context, finding the zeros of f' helps identify where the function f has horizontal tangent lines, which are essential for understanding the function's behavior.

Horizontal Tangent Lines

A horizontal tangent line occurs at points on the graph of a function where the slope is zero, meaning the derivative f' equals zero. This indicates that the function is neither increasing nor decreasing at that point, which is crucial for identifying local extrema. Verifying that the zeros of f' correspond to horizontal tangents helps confirm the relationship between the derivative and the function's graphical behavior.

Graphing Functions

Graphing a function involves plotting its values on a coordinate system to visualize its behavior. For the function f(x) = e^(-x) tan^(-1)(x), understanding its graph helps in analyzing its critical points and the nature of its tangent lines. Technology can assist in graphing to easily identify where the function has horizontal tangents, enhancing comprehension of the relationship between f and f'.