Question 1 f(x) = VTx Рcos (Tx). Prove that the equation f(x) = 0 has at least a solution p in the interval [0, 1]. By the Bisection method, find pn, n< 2 on [0, 1]. Use the table 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1 f(x) -1 -0.297 0.179 0.702 1.253 1.783 2.242 2.581 2.772 and write your answers in the next table. bn f(Pn) An Pn 1 1 2 How many iterations are necessary to solve VTx Рcos (Tx) = 0 with accuracy 10¼ on [0, 1]?
Question 1 f(x) = VTx Рcos (Tx). Prove that the equation f(x) = 0 has at least a solution p in the interval [0, 1]. By the Bisection method, find pn, n< 2 on [0, 1]. Use the table 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1 f(x) -1 -0.297 0.179 0.702 1.253 1.783 2.242 2.581 2.772 and write your answers in the next table. bn f(Pn) An Pn 1 1 2 How many iterations are necessary to solve VTx Рcos (Tx) = 0 with accuracy 10¼ on [0, 1]?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Question 1
\[ f(x) = \sqrt{\pi x} - \cos(\pi x) \]
**Prove that the equation \( f(x) = 0 \) has at least a solution \( p \) in the interval \([0, 1]\).**
**By the Bisection method, find \( p_n \), \( n \leq 2 \) on \([0, 1]\). Use the table:**
\[
\begin{array}{c|c|c|c|c|c|c|c|c|c}
x & 0 & 0.125 & 0.250 & 0.375 & 0.500 & 0.625 & 0.750 & 0.875 & 1 \\
\hline
f(x) & -1 & -0.297 & 0.179 & 0.702 & 1.253 & 1.783 & 2.242 & 2.581 & 2.772 \\
\end{array}
\]
**and write your answers in the next table.**
\[
\begin{array}{c|c|c|c|c}
n & a_n & b_n & p_n & f(p_n) \\
\hline
0 & 0 & 1 & & \\
1 & & & & \\
2 & & & & \\
\end{array}
\]
**How many iterations are necessary to solve \( \sqrt{\pi x} - \cos(\pi x) = 0 \) with accuracy \(10^{-5}\) on \([0, 1]\)?**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe25f92a2-1c4d-41a3-af19-0cdf00d27604%2F61b50ebb-a93b-4581-bc6c-f88522716087%2Fowyozu_processed.png&w=3840&q=75)
Transcribed Image Text:### Question 1
\[ f(x) = \sqrt{\pi x} - \cos(\pi x) \]
**Prove that the equation \( f(x) = 0 \) has at least a solution \( p \) in the interval \([0, 1]\).**
**By the Bisection method, find \( p_n \), \( n \leq 2 \) on \([0, 1]\). Use the table:**
\[
\begin{array}{c|c|c|c|c|c|c|c|c|c}
x & 0 & 0.125 & 0.250 & 0.375 & 0.500 & 0.625 & 0.750 & 0.875 & 1 \\
\hline
f(x) & -1 & -0.297 & 0.179 & 0.702 & 1.253 & 1.783 & 2.242 & 2.581 & 2.772 \\
\end{array}
\]
**and write your answers in the next table.**
\[
\begin{array}{c|c|c|c|c}
n & a_n & b_n & p_n & f(p_n) \\
\hline
0 & 0 & 1 & & \\
1 & & & & \\
2 & & & & \\
\end{array}
\]
**How many iterations are necessary to solve \( \sqrt{\pi x} - \cos(\pi x) = 0 \) with accuracy \(10^{-5}\) on \([0, 1]\)?**
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