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,∞)

검증된 단계별 안내

Step 1: Understand the function f(x) = e^(-x) * tan^(-1)(x). This function is a product of the exponential function e^(-x) and the inverse tangent function tan^(-1)(x).

Step 2: Use a graphing utility or software that supports graphing, such as Desmos, GeoGebra, or a graphing calculator, to input the function f(x) = e^(-x) * tan^(-1)(x).

Step 3: Set the domain for the graph. Since the problem specifies the interval [0, ∞), ensure that the graphing utility is set to display the graph starting from x = 0 and extending towards positive infinity.

Step 4: Observe the behavior of the graph. Note how the exponential decay of e^(-x) affects the overall shape of the graph, and how the inverse tangent function, which approaches π/2 as x approaches infinity, influences the graph.

Step 5: Analyze the graph for key features such as intercepts, asymptotic behavior, and any points of interest. This will help in understanding the overall behavior of the function on the given interval.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a * e^(bx), where 'e' is Euler's number (approximately 2.718). In the given function f(x) = e^(-x) * tan^(-1)(x), the term e^(-x) represents a decaying exponential function, which approaches zero as x increases. Understanding the behavior of exponential functions is crucial for analyzing the overall shape and limits of the graph.

Inverse Tangent Function

The inverse tangent function, denoted as tan^(-1)(x) or arctan(x), is the function that returns the angle whose tangent is x. It has a range of (-π/2, π/2) and approaches these limits as x approaches ±∞. In the context of the function f(x) = e^(-x) * tan^(-1)(x), this function influences the growth of f(x) as x increases, particularly since tan^(-1)(x) approaches π/2.

Graphing Utilities

Graphing utilities are software or tools that allow users to visualize mathematical functions and their derivatives. They can plot complex functions, helping to analyze their behavior over specified intervals. For the function f(x) = e^(-x) * tan^(-1)(x) on the interval [0, ∞), using a graphing utility will provide insights into the function's growth, decay, and asymptotic behavior, which is essential for understanding its overall characteristics.

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,∞)