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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 4 - Applications of Derivatives문제 126
4장, 문제 126

Graph f(x) = x cos x and its second derivative together for 0 ≤ x ≤ 2pi. Comment on the behavior of the graph of f in relation to the signs and values of f".

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Step 1: Begin by finding the first derivative of the function f(x) = x cos(x). Use the product rule, 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) = x and v(x) = cos(x).
Step 2: Calculate the first derivative f'(x). Using the product rule, differentiate u(x) = x to get u'(x) = 1, and differentiate v(x) = cos(x) to get v'(x) = -sin(x). Therefore, f'(x) = 1 * cos(x) + x * (-sin(x)) = cos(x) - x sin(x).
Step 3: Find the second derivative f''(x) by differentiating f'(x) = cos(x) - x sin(x). Differentiate cos(x) to get -sin(x) and use the product rule on -x sin(x) to get -sin(x) - x cos(x). Thus, f''(x) = -sin(x) - sin(x) - x cos(x) = -2sin(x) - x cos(x).
Step 4: Graph the function f(x) = x cos(x) and its second derivative f''(x) = -2sin(x) - x cos(x) over the interval 0 ≤ x ≤ 2π. Use a graphing tool or software to visualize these functions. Observe the points where the second derivative changes sign, as these indicate potential inflection points where the concavity of f(x) changes.
Step 5: Analyze the behavior of the graph of f(x) in relation to the signs and values of f''(x). When f''(x) > 0, the graph of f(x) is concave up, and when f''(x) < 0, it is concave down. Note how these concavity changes correspond to the shape and turning points of the graph of f(x).

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주요 개념

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

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate plane to visualize its behavior over a specified interval. For f(x) = x cos x, this means plotting points for x between 0 and 2π and connecting them smoothly. Understanding the shape and key features of the graph, such as intercepts and turning points, is crucial for analysis.
추천 영상:
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Graph of Sine and Cosine Function

Second Derivative

The second derivative of a function, denoted as f''(x), provides information about the concavity of the function's graph. If f''(x) > 0, the graph is concave up, indicating a local minimum, while f''(x) < 0 suggests concave down, indicating a local maximum. Analyzing the second derivative helps in understanding the acceleration of the function's rate of change.
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The Second Derivative Test: Finding Local Extrema

Behavior of Functions

The behavior of a function refers to how it changes over its domain, including increasing or decreasing trends, and concavity. By examining the signs and values of f(x) and its derivatives, one can infer critical points, inflection points, and overall trends. This analysis is essential for interpreting the relationship between f(x) and its derivatives, especially in the context of graphing.
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Graphs of Exponential Functions
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