Skip to main content
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 4.4.99
4장, 문제 4.4.99

99. In Exercises 99 and 100, the graph of f' is given. Determine x-values corresponding to inflection points for the graph of f.
Graph of f' showing a curve with inflection points, plotted on a Cartesian coordinate system with labeled axes.

검증된 단계별 안내
1
Step 1: Recall that inflection points occur where the second derivative of a function changes sign. This means we need to analyze the graph of f' to determine where the slope of f' (which represents f'') changes from positive to negative or vice versa.
Step 2: Observe the graph of f'. The slope of f' changes sign at points where the graph of f' transitions from increasing to decreasing or decreasing to increasing. These are the x-values where the graph of f' has local maxima or minima.
Step 3: Identify the x-values of local maxima and minima on the graph of f'. From the graph, it appears that there are local maxima at x ≈ -2 and local minima at x ≈ 2.
Step 4: Verify that these points correspond to changes in the sign of the slope of f'. At x ≈ -2, the slope of f' changes from positive to negative, and at x ≈ 2, the slope of f' changes from negative to positive.
Step 5: Conclude that the x-values corresponding to inflection points for the graph of f are approximately x = -2 and x = 2.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m
도움이 되었나요?

주요 개념

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

Derivative and Its Graph

The derivative of a function, denoted as f', represents the rate of change of the function f. The graph of f' provides insights into the behavior of f, such as where it is increasing or decreasing. Points where f' changes from positive to negative indicate local maxima, while points where it changes from negative to positive indicate local minima.
추천 영상:
가이드 코스
06:15
Graphing The Derivative

Inflection Points

Inflection points occur where the concavity of a function changes, which can be identified by analyzing the second derivative, f''. However, when given the graph of f', inflection points correspond to where f' changes its increasing or decreasing behavior. This means that the x-values where f' has local extrema (maxima or minima) are potential inflection points for f.
추천 영상:
04:50
Critical Points

Critical Points

Critical points are values of x where the derivative f' is either zero or undefined. These points are significant because they can indicate local maxima, minima, or inflection points. In the context of the given graph, identifying the x-values where f' equals zero helps in determining where the function f may have changes in concavity, leading to inflection points.
추천 영상:
04:50
Critical Points
관련 실천
교과서 질문

Root Finding

2. Use Newton's method to estimate the one real solution of x^3 +3x + 1 = 0. Start with x_0 = 0 and then find x_2.

240
views
교과서 질문

Checking the Mean Value Theorem


Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.


f(x) = {2x − 3, 0 ≤ x ≤ 2

6x − x² − 7, 2 < x ≤ 3

205
views
교과서 질문

Finding Extrema from Graphs


In Exercises 11–14, match the table with a graph.


171
views
교과서 질문

Finding Extrema from Graphs


In Exercises 7–10, find the absolute extreme values and where they occur.


321
views
교과서 질문

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

82. y' = sin t, for 0 ≤ t ≤ 2π

116
views
교과서 질문

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dr/dθ = −π sin (πθ), r(0) = 0

25
views