Textbook Question
Root Finding
2. Use Newton's method to estimate the one real solution of x^3 +3x + 1 = 0. Start with x_0 = 0 and then find x_2.
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 4.2.19
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x⁴ + 3x + 1, [−2, −1]
Verified step by step guidance1
First, understand that a zero of a function f(x) is a value x = c such that f(c) = 0. We need to show that the function f(x) = x⁴ + 3x + 1 has exactly one zero in the interval [-2, -1].
Apply the Intermediate Value Theorem, which states that if a function is continuous on a closed interval [a, b] and takes on different signs at the endpoints, then there is at least one c in (a, b) such that f(c) = 0. First, evaluate f(x) at the endpoints of the interval: f(-2) and f(-1).
Calculate f(-2): Substitute x = -2 into the function: f(-2) = (-2)⁴ + 3(-2) + 1. Simplify this expression to find the value of f(-2).
Calculate f(-1): Substitute x = -1 into the function: f(-1) = (-1)⁴ + 3(-1) + 1. Simplify this expression to find the value of f(-1).
Check the signs of f(-2) and f(-1). If f(-2) and f(-1) have opposite signs, then by the Intermediate Value Theorem, there is at least one zero in the interval [-2, -1]. To show that there is exactly one zero, consider the derivative f'(x) = 4x³ + 3. Analyze the sign of f'(x) over the interval to ensure that f(x) is either strictly increasing or decreasing, which would imply exactly one zero.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Intermediate Value Theorem
The Intermediate Value Theorem states that if a continuous function, f(x), takes on different signs at two points a and b, then it must cross zero at some point between a and b. This theorem is crucial for proving the existence of a root within a specific interval, as it ensures that the function transitions from positive to negative or vice versa.
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Fundamental Theorem of Calculus Part 1
Continuity of Polynomial Functions
Polynomial functions, like f(x) = x⁴ + 3x + 1, are continuous everywhere on the real number line. This property is essential when applying the Intermediate Value Theorem, as it guarantees that there are no breaks or jumps in the function's graph, allowing us to confidently assert the existence of a zero within a given interval.
Recommended video:
6:04Introduction to Polynomial Functions
Derivative and Monotonicity
The derivative of a function provides information about its monotonicity, indicating whether the function is increasing or decreasing. By analyzing the derivative of f(x) = x⁴ + 3x + 1, we can determine if the function is strictly increasing or decreasing in the interval [−2, −1], which helps establish that there is exactly one zero by ensuring no additional turning points exist within the interval.
Recommended video:
05:44Derivatives
Related Practice
Textbook Question
Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
82. y' = sin t, for 0 ≤ t ≤ 2π
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.
∫(2x³ − 5x + 7) dx
Textbook Question
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = x²ᐟ³, [−1, 8]
Textbook Question
109. Suppose the derivative of the function y = f(x) is
y'=(x-1)^2(x-2).
At what points, if any, does the graph of f have a local minimum, local maximum, or
point of inflection? (Hint: Draw the sign pattern for y'.)
Textbook Question
"Roots (Zeros) Show that the functions in Exercises 19–26 have exactly one zero