Textbook Question
Theory and Examples
Maximum height of a vertically moving body The height of a body moving vertically is given by s = −12gt² + υ₀t + s₀, g > 0, with s in meters and t in seconds. Find the body’s maximum height.
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 4.6.26
[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).
26. Factoring a quartic Find the approximate values of r_1 through r_4 in the factorization
8x^4-14x^3-9x^2+11x-1=8(x-r_1)(x-r_2)(x-r_3)(x-r_4)
Verified step by step guidance1
Identify the polynomial equation given: \( y = 8x^4 - 14x^3 - 9x^2 + 11x - 1 \). This is a quartic equation, which means it is a polynomial of degree 4.
To factor the quartic polynomial, we need to find its roots, \( r_1, r_2, r_3, \) and \( r_4 \). These roots are the values of \( x \) where the polynomial equals zero.
Use the graph provided to estimate the x-intercepts, which are the approximate roots of the polynomial. The graph shows where the curve crosses the x-axis.
From the graph, observe the approximate x-values where the curve crosses the x-axis. These points are the approximate roots of the polynomial.
Use a calculator or computer software to find more precise values of the roots, if necessary, by solving the equation \( 8x^4 - 14x^3 - 9x^2 + 11x - 1 = 0 \) using numerical methods such as the Newton-Raphson method or other root-finding algorithms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quartic Functions
A quartic function is a polynomial of degree four, which can be expressed in the form f(x) = ax^4 + bx^3 + cx^2 + dx + e, where a, b, c, d, and e are constants and a ≠ 0. The graph of a quartic function can have up to four real roots, and its shape can vary significantly, exhibiting behaviors such as turning points and inflection points.
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Properties of Functions
Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of simpler polynomials or linear factors. For a quartic polynomial, this means finding values r_1, r_2, r_3, and r_4 such that the polynomial can be written as a product of the form a(x - r_1)(x - r_2)(x - r_3)(x - r_4). This process is essential for solving polynomial equations and understanding their roots.
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Roots and Intercepts
The roots of a polynomial are the values of x for which the polynomial equals zero, corresponding to the x-intercepts of its graph. In the context of the quartic function given, finding the approximate values of r_1 through r_4 involves determining where the graph intersects the x-axis. These roots are crucial for understanding the behavior of the polynomial and its factorization.
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Related Practice
Textbook Question
Finding Position from Velocity or Acceleration
Exercises 45–48 give the acceleration a=d²s/dt², initial velocity, and initial position of an object moving on a coordinate line. Find the object’s position at time t.
a = 9.8, v(0) = −3, s(0) = 0
Textbook Question
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(cscθ cotθ) / 2 dθ
Textbook Question
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x − 3x²ᐟ³
Textbook Question
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x³ + 4x² + 7, (−∞, 0)
Textbook Question
Identifying Extrema
In Exercises 15–18:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local and absolute extreme values, if any, saying where they occur.