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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.82.2
Chapter 2, Problem 2.82.2

b. Estimate a solution to the equation in the given interval using a root finder.


x^5+7x+5=0; (−1,0)

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1
First, understand the problem: We need to find a root of the equation x^5 + 7x + 5 = 0 within the interval (-1, 0). A root finder method, such as the bisection method, can be used to estimate the solution.
Start by evaluating the function at the endpoints of the interval. Calculate f(-1) and f(0) to determine if there is a sign change, which indicates a root exists between these points.
Apply the bisection method: Divide the interval into two halves and evaluate the function at the midpoint. If the function changes sign between the midpoint and one of the endpoints, the root lies in that subinterval.
Continue the bisection process: Repeatedly halve the interval and check for sign changes at the midpoints. This iterative process narrows down the interval where the root is located.
Stop the process when the interval is sufficiently small, or when the function value at the midpoint is close enough to zero, indicating an approximate root of the equation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Root Finding

Root finding is a numerical method used to determine the values of x for which a given function f(x) equals zero. Common techniques include the bisection method, Newton's method, and the secant method. These methods iteratively approximate the root by evaluating the function at specific points and refining the interval or estimate until a satisfactory level of accuracy is achieved.
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Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function takes on two values at two points, it must also take on any value between those two points. This theorem is crucial for establishing the existence of roots within a given interval. In the context of the equation x^5 + 7x + 5 = 0, verifying that the function changes sign over the interval (-1, 0) indicates that a root exists within that range.
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Function Behavior and Graphing

Understanding the behavior of the function, including its continuity, increasing or decreasing nature, and critical points, is essential for effective root finding. Graphing the function can provide visual insights into where the function crosses the x-axis, indicating potential roots. Analyzing the function's derivative can also help identify intervals where the function is increasing or decreasing, aiding in the selection of appropriate root-finding methods.
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Related Practice
Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

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Textbook Question

A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,…} is specified by the function f(n) = 2n, where n=1,2,3,….The limit of such a sequence is lim n→∞ f(n), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. 


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f(x)=1−cos(2x−2)(x−1)2;a=1f\(\left\)(x\(\right\))=\(\frac{1-\cos\left(2x-2\right)}{\left(x-1\right)^2}\);a=1

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Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. <IMAGE>

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Textbook Question

Suppose limx→a f(x)=L{\(\displaystyle\]\lim\)_{x\(\to\) a}}\(\text{ }\)f\(\left\)(x\(\right\))=L and limx→a g(x)=M{\(\displaystyle\]\lim\)_{x\(\to\) a}}\(\text{ }\)g\(\left\)(x\(\right\))=M. Prove that limx→a (f(x)−g(x))=L−M{\(\displaystyle\[\lim\)_{x\(\to\) a}}\(\text{ }\]\left\)(f\(\left\)(x\(\right\))-g\(\left\)(x\(\right\))\(\right\))=L-M.

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