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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 4.3.61
4장, 문제 4.3.61

Identifying Extrema


In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.


Graph of a blue curve labeled f prime showing changes in slope with x and y axes marked from -4 to 4.

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1
Recall that the graph given is of the derivative \(f'(x)\) of the function \(f(x)\). To find local maxima and minima of \(f(x)\), we need to analyze the behavior of \(f'(x)\) around its zeros.
Identify the points where \(f'(x) = 0\) by locating the x-values where the graph of \(f'(x)\) crosses the x-axis. These points are candidates for local extrema of \(f(x)\).
Determine the sign change of \(f'(x)\) around each zero: - If \(f'(x)\) changes from positive to negative at a zero, then \(f(x)\) has a local maximum there. - If \(f'(x)\) changes from negative to positive at a zero, then \(f(x)\) has a local minimum there.
Examine the graph near each zero of \(f'(x)\) and note the sign of \(f'(x)\) just before and just after the zero to classify the extrema accordingly.
Summarize the x-values where \(f'(x)\) crosses zero and the corresponding sign changes to list the local minima and maxima of \(f(x)\).

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
4m

주요 개념

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

Critical Points and Extrema

Critical points occur where the derivative f' is zero or undefined. These points are candidates for local maxima, minima, or saddle points. To identify local extrema, analyze the behavior of f' around these points, as they indicate where the slope of the original function f changes.
추천 영상:
04:50
Critical Points

First Derivative Test

The First Derivative Test uses the sign changes of f' to determine local extrema. If f' changes from positive to negative at a critical point, f has a local maximum there. If f' changes from negative to positive, f has a local minimum. No sign change means no local extremum at that point.
추천 영상:
07:09
The First Derivative Test: Finding Local Extrema

Interpreting the Graph of f'

The graph of f' shows the slope of f at each x-value. Points where f' crosses the x-axis (f' = 0) are critical points. By observing whether f' goes from positive to negative or vice versa at these points, one can identify local maxima and minima of f without needing the explicit form of f.
추천 영상:
03:56
Determining Concavity from the Graph of f' or f''
관련 실천
교과서 질문

Identifying Extrema


In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.


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교과서 질문

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

______

y = √𝓍² ― 1

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교과서 질문

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

2. y=x^4/4-2x^2+4

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교과서 질문

Absolute Extrema on Finite Closed Intervals


In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.


f(t) = 2 − |t|, −1 ≤ t ≤ 3

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교과서 질문

Identifying Extrema


In Exercises 15–18:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local and absolute extreme values, if any, saying where they occur.


153
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교과서 질문

Initial Value Problems


Find the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.

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