Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경

Hass 15th Edition

Ch. 4 - Applications of Derivatives

문제 125

4장, 문제 125

Graph f(x) = 2x^4 -4x^2 + 1 and its first two derivatives together. Comment on the behavior of f in relation to the signs and values of f' and f".

검증된 단계별 안내

Step 1: Identify the function f(x) = 2x^4 - 4x^2 + 1. This is a polynomial function of degree 4, which means it is a smooth and continuous curve.

Step 2: Find the first derivative f'(x) to determine the critical points and analyze the increasing or decreasing behavior of the function. Use the power rule to differentiate: f'(x) = d/dx (2x^4 - 4x^2 + 1) = 8x^3 - 8x.

Step 3: Find the second derivative f''(x) to analyze the concavity of the function and identify any inflection points. Differentiate f'(x): f''(x) = d/dx (8x^3 - 8x) = 24x^2 - 8.

Step 4: Graph f(x), f'(x), and f''(x) on the same set of axes. Observe where f'(x) = 0 to find critical points, and where f''(x) = 0 to find potential inflection points. These will help in understanding the behavior of f(x).

Step 5: Comment on the behavior of f(x) based on the signs of f'(x) and f''(x). Where f'(x) > 0, f(x) is increasing; where f'(x) < 0, f(x) is decreasing. Where f''(x) > 0, f(x) is concave up; where f''(x) < 0, f(x) is concave down. Use these observations to describe the overall shape and turning points of the graph.

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주요 개념

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Derivative

The derivative of a function, denoted as f'(x), represents the rate of change or the slope of the function at any given point. It provides critical information about the function's increasing or decreasing behavior. For the function f(x) = 2x^4 - 4x^2 + 1, calculating f'(x) helps identify intervals where the function is rising or falling.

Second Derivative

The second derivative, denoted as f''(x), indicates the concavity of the function and helps identify points of inflection. It shows whether the function is concave up (f''(x) > 0) or concave down (f''(x) < 0). For f(x) = 2x^4 - 4x^2 + 1, analyzing f''(x) helps understand the curvature and stability of the graph.

Critical Points and Inflection Points

Critical points occur where f'(x) = 0 or is undefined, indicating potential maxima, minima, or saddle points. Inflection points occur where f''(x) changes sign, indicating a change in concavity. Identifying these points for f(x) = 2x^4 - 4x^2 + 1 helps in sketching the graph and understanding the function's behavior.

관련 실천

교과서 질문

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".

251

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교과서 질문

In Exercises 121–124, find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function’s first and second derivatives. How are the values at which these graphs intersect the x-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function?

123. y=(4/5)x^5+16x^2-25

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교과서 질문

106. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (b) velocity equal to zero?

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241

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교과서 질문

103. A function f(x) has domain (-2, 2). The graph below is a plot of the derivative of f, not a plot of f itself. In other words, this is a graph of y = f'(x). Either use this graph to determine on which intervals the graph of f is concave up and on which intervals the graph of f is concave down, or explain why this information cannot be determined from the graph.

256

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