Textbook Question
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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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 4.2.15ai
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
i. y = x² − 4
Verified step by step guidance1
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.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
05:13Slopes of Tangent Lines
Related Practice
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)
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
Textbook Question
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
πcos πx
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Textbook Question
Theory and Examples
Cubic functions Consider the cubic function f(x) = ax³ + bx² + cx + d.
b. How many local extreme values can f have?
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) = x² − 4x + 4, 1 ≤ x < ∞