교과서 질문
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
iii. y = x³ − 3x² + 4 = (x + 1)(x − 2)²
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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 4.3.3a
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 − 1)²(x + 2)
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To find the critical points of the function f, we need to determine where its derivative f'(x) is equal to zero or undefined. In this case, f'(x) = (x - 1)²(x + 2).
Set the derivative equal to zero: (x - 1)²(x + 2) = 0. This equation will help us find the values of x where the derivative is zero, indicating potential critical points.
Solve the equation (x - 1)²(x + 2) = 0 by setting each factor equal to zero: (x - 1)² = 0 and (x + 2) = 0.
For (x - 1)² = 0, solve for x to get x = 1. This is a critical point because the derivative is zero at this value.
For (x + 2) = 0, solve for x to get x = -2. This is another critical point because the derivative is zero at this value.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
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Critical Points
Critical points of a function occur where its derivative is zero or undefined. These points are significant because they can indicate local maxima, minima, or points of inflection. To find critical points, set the derivative equal to zero and solve for the variable, or identify where the derivative does not exist.
추천 영상:
04:50
Critical Points
Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of its factors. This is crucial for solving equations where the polynomial is set to zero, as it allows us to find the roots or solutions. In the given derivative, f′(x) = (x − 1)²(x + 2), factoring helps identify the values of x that make the derivative zero.
추천 영상:
6:04Introduction to Polynomial Functions
Derivative Analysis
Analyzing the derivative of a function provides insights into the function's behavior, such as increasing or decreasing intervals and concavity. The sign and value of the derivative indicate the slope of the tangent line at any point on the function. For f′(x) = (x − 1)²(x + 2), examining the derivative helps determine where the function changes direction or has potential extrema.
추천 영상:
가이드 코스
06:15Derivatives Applied To Acceleration
관련 실천
교과서 질문
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
iv. y = x³ − 33x² + 216x = x(x - 9)(x − 24)
260
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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.
csc x cot x
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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.
(3/2)√x
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교과서 질문
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
g(x) = (x − 2) / (x²−1), 0 ≤ x < 1
208
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교과서 질문
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
i. y = x² − 4
231
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