Textbook Question
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)
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.2.15aiii
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
Verified step by step guidance1
First, identify the zeros of the polynomial y = x³ − 3x² + 4. To do this, set y equal to zero and solve for x. The polynomial is factored as (x + 1)(x − 2)², so the zeros are x = -1 and x = 2.
Next, find the first derivative of the polynomial y = x³ − 3x² + 4. Use the power rule to differentiate each term: the derivative of x³ is 3x², the derivative of -3x² is -6x, and the derivative of 4 is 0. Therefore, the first derivative is y' = 3x² - 6x.
Set the first derivative equal to zero to find its zeros: 3x² - 6x = 0. Factor out the common term, which is 3x, resulting in 3x(x - 2) = 0. The zeros of the derivative are x = 0 and x = 2.
Plot the zeros of the original polynomial and its derivative on a number line. The zeros of the polynomial are x = -1 and x = 2, and the zeros of the derivative are x = 0 and x = 2. Note that x = 2 is a common zero.
Consider the behavior of the function and its derivative around these zeros. The zero at x = 2 is a repeated zero in the polynomial, indicating a change in concavity or a local extremum. The zero at x = 0 in the derivative suggests a potential inflection point or change in slope.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Roots (Zeros) 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 = (x + 1)(x − 2)², the zeros are x = -1 and x = 2. These points are where the graph of the polynomial intersects the x-axis.
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Introduction to Polynomial Functions
First Derivative and Critical Points
The first derivative of a function, denoted as y', represents the rate of change or slope of the function. Finding the zeros of the first derivative helps identify critical points, which are potential locations for local maxima, minima, or points of inflection. For y = x³ − 3x² + 4, the derivative is y' = 3x² - 6x, and its zeros are found by solving 3x(x - 2) = 0, giving x = 0 and x = 2.
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Graphical Representation of Zeros
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. The zeros of the polynomial indicate where the graph crosses the x-axis, while the zeros of the derivative indicate where the slope is zero, marking potential turning points or changes in concavity.
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Guided course
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Related Practice
Textbook Question
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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Textbook Question
Identifying Extrema
In Exercises 19–40:
b. Identify the function’s local extreme values, if any, saying where they occur.
f(r) = 3r³ + 16r
132
views
Textbook Question
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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Textbook Question
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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Textbook Question
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
views