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 4.2.15ai
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
i. y = x² − 4
검증된 단계별 안내1
First, identify the zeros of the polynomial y = x² − 4. To find the zeros, set the equation equal to zero: x² − 4 = 0.
Solve the equation x² − 4 = 0 by factoring it as (x - 2)(x + 2) = 0. This gives the zeros x = 2 and x = -2.
Next, find the first derivative of the polynomial y = x² − 4. The derivative, using basic differentiation rules, is y' = 2x.
Set the first derivative equal to zero to find its zeros: 2x = 0. Solving this gives x = 0.
Plot the zeros of the original polynomial (x = 2 and x = -2) and the zero of its first derivative (x = 0) on a number line. This visual representation helps in understanding the behavior of the function and its slope at these points.
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.
Roots of a Polynomial
The roots or zeros of a polynomial are the values of x for which the polynomial equals zero. For the polynomial y = x² − 4, the roots can be found by setting the equation to zero and solving for x, resulting in x = ±2. These roots represent the points where the graph of the polynomial intersects the x-axis.
추천 영상:
6:04
Introduction to Polynomial Functions
First Derivative of a Polynomial
The first derivative of a polynomial provides the rate of change of the function and is used to find critical points, such as maxima, minima, and points of inflection. For y = x² − 4, the first derivative is y' = 2x. Setting the derivative to zero helps identify the x-values where the slope of the tangent is zero, indicating potential extrema.
추천 영상:
07:09The First Derivative Test: Finding Local Extrema
Plotting on a Number Line
Plotting the zeros of a polynomial and its derivative on a number line helps visualize the relationship between the function and its rate of change. For y = x² − 4, plot the roots x = ±2 and the zero of the derivative x = 0. This visualization aids in understanding how the function behaves around these critical points.
추천 영상:
가이드 코스
05:13Slopes of Tangent Lines
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
views
교과서 질문
Identifying Extrema
In Exercises 19–40:
b. Identify the function’s local extreme values, if any, saying where they occur.
f(θ) = 3θ² − 4θ³
139
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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
28
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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 − 1)²(x + 2)
172
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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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