교과서 질문
Theory and Examples
Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
4장, 문제 4.3.9a
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
a. What are the critical points of f?
f′(x) = 1− 4/x², x ≠ 0
검증된 단계별 안내1
To find the critical points of the function f, we need to determine where the derivative f'(x) is equal to zero or undefined. Critical points occur where the derivative changes sign or is undefined.
Given the derivative f'(x) = 1 - 4/x², we first set f'(x) equal to zero to find where the derivative changes sign: 1 - 4/x² = 0.
Solve the equation 1 - 4/x² = 0 for x. This involves isolating x² by adding 4/x² to both sides, resulting in 1 = 4/x².
Next, solve for x² by multiplying both sides by x², giving x² = 4. Then, take the square root of both sides to find the values of x: x = ±2.
Since x ≠ 0, the critical points are x = 2 and x = -2. These are the points where the derivative is zero, indicating potential maxima, minima, or points of inflection.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Critical Points
Critical points of a function occur where its derivative is zero or undefined. These points are significant because they can indicate potential local maxima, minima, or points of inflection. To find critical points, set the derivative equal to zero and solve for x, or identify where the derivative does not exist.
추천 영상:
04:50
Critical Points
Derivative Analysis
Analyzing the derivative of a function helps determine the behavior of the original function. The sign of the derivative indicates whether the function is increasing or decreasing. In this context, f′(x) = 1 - 4/x² must be analyzed to find where it equals zero or is undefined, revealing critical points.
추천 영상:
가이드 코스
06:15Derivatives Applied To Acceleration
Rational Functions
A rational function is a ratio of two polynomials. The derivative given, f′(x) = 1 - 4/x², is a rational function. Understanding how to manipulate and solve rational functions is crucial for finding where the derivative equals zero or is undefined, which helps identify critical points.
추천 영상:
6:04Intro to Rational Functions
관련 실천
교과서 질문
Finding displacement from an antiderivative of velocity
a. Suppose that the velocity of a body moving along the s-axis is
ds/dt = v = 9.8t − 3.
i. Find the body’s displacement over the time interval from t = 1 to t = 3 given that s = 5 when t = 0.
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교과서 질문
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫tanθ sec²θ dθ = sec³θ / 3 + C
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교과서 질문
Identifying Extrema
In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).
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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.
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교과서 질문
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
−πsin πx
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교과서 질문
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
a. What are the critical points of f?
f′(x) = x(x − 1)
149
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