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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.R.83
Chapter 2, Problem 2.R.83

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


x=cos x; (0,π/2)

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1
Identify the function for which we need to find the root. In this case, the function is f(x) = x - cos(x). We are looking for a value of x where f(x) = 0.
Choose a numerical method to estimate the root. A common method is the bisection method, which is suitable for continuous functions on a closed interval where the function changes sign.
Set the initial interval as given: [a, b] = [0, π/2]. Check the values of f(a) and f(b) to ensure that they have opposite signs, confirming that a root exists in the interval.
Apply the bisection method: Calculate the midpoint c = (a + b) / 2. Evaluate f(c). If f(c) is close to zero (within a desired tolerance), c is an approximate root. Otherwise, determine the subinterval [a, c] or [c, b] where the sign change occurs and repeat the process.
Continue iterating the bisection method, narrowing down the interval until the midpoint c is sufficiently close to the actual root, as determined by your tolerance level.

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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 a variable that make a function equal to zero. In this context, we are looking for the value of x that satisfies the equation x = cos(x) within the interval (0, π/2). Common methods for root finding include the bisection method, Newton's method, and the secant method, each with its own advantages and limitations.
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Finding Limits by Direct Substitution

Fixed Point Iteration

Fixed point iteration is a technique used to find solutions to equations of the form x = g(x). In this case, we can rearrange the equation x = cos(x) to g(x) = cos(x). By iteratively substituting values into g(x), we can converge to a fixed point, which represents the solution to the original equation. This method is particularly useful when the function g(x) is continuous and the interval is chosen appropriately.
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Critical Points

Interval and Convergence

The interval (0, π/2) is crucial for ensuring that the root finding method converges to a solution. Within this interval, the function f(x) = x - cos(x) is continuous and changes sign, indicating the presence of a root according to the Intermediate Value Theorem. Understanding the behavior of the function within the specified interval helps in selecting the appropriate numerical method and ensuring that the iterations lead to a valid solution.
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Derivatives
Related Practice
Textbook Question

Suppose the rental cost for a snowboard is \$25 for the first day (or any part of the first day) plus \$15 for each additional day (or any part of a day).

e. For what values of t is f continuous? Explain.

Textbook Question

Evaluate limx→∞f(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}} andlimx→−∞f(x){\(\displaystyle\)\(\lim\)_{x\(\to\)-\(\infty\)}{f(x)}}.


f(x)=1−e−2xf\(\left\)(x\(\right\))=1-e^{-2x}

Textbook Question

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→1^− f(x)

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Textbook Question

Determine the following limits.


a. limx→2+1x(x−2){\(\displaystyle\)\(\lim\)_{x\(\to\)2^{+}}}\(\frac{1}{\sqrt{x\left(x-2\right)}\)}

Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=3x2−2x+53x+4f\(\left\)(x\(\right\))=\(\frac{3x^2-2x+5}{3x+4}\)

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Textbook Question

Let g(x)={5x−2if x<1aif x=1ax2+bxif x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)5x-2 & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ ax^2+bx & \(\text{if }\)x>1\(\end{cases}\).


Determine values of the constants aa and bb , if possible, for which gg is continuous at x=1x=1.