Questions and Answers
In Example 3, what is the interval used to find the approximate root of f(x) = x^2 - 2?
[1, 2]
What is the approximate root of the function f(x) = xlog(x) - 1?
1.76322284
What is the value of x1 in the Secant method when a=1, b=2, f(a)=-1, and f(b)=2?
1.33333
What is the value of x3 in the Secant method for finding the root of f(x) = x^2 - 2?
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What is the function f(x) in Example 3?
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What is the root of the function $f(x)=x^2-2$ found using the FalsePosition method with error $\epsilon=10^{-3}$?
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What is the approximate root of the function $f(x)=x\log(x)-1$ in the interval [1,2] using the FalsePosition method with error $\epsilon=10^{-3}$?
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What is the value of x3 in the FalsePosition method for finding the root of $f(x)=x\log(x)-1$?
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In the FalsePosition method for $f(x)=x^2-2$, what is the interval used to find the root?
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What is the value of x7 in the Secant method for finding the root of $f(x)=x^2-2$?
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What is the root of the function $f(x)=x^2-2$ found using the Secant method?
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What is the value of x4 in the Secant method when finding the root of f(x) = xlog(x) - 1?
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What is the value of x2 in the Secant method for finding the root of f(x) = x^2 - 2?
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What is the approximate root of the function f(x) = x^2 - 2 obtained using the Secant method?
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What is the value of x3 in the Secant method when finding the root of f(x) = x^2 - 2?
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What is the value of x1 in the Secant method when a=1, b=2, f(a)=-1, and f(b)=2 for the function f(x) = x^2 - 2?
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What is the absolute difference between x1 and x2 in the Secant method when finding the root of f(x) = x^2 - 2?
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In the False-Position method, if f(a) and f(b) have opposite signs, what does this indicate?
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In the False-Position method, how is the next approximation x3 calculated after finding x2 and x1?
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What condition is checked in the False-Position method to determine the existence of a root between x3 and x2?
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How is the interval narrowed down in the False-Position method to find a more accurate root approximation?
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What is the termination condition in the False-Position method to stop the iteration process?
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In the False-Position method, what does it mean when f(x_{n+1}) * f(x_n) > 0?
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Study Notes
False-Position Method (Regula Falsi Method)
- The method is used to find an approximate root of a continuous function f(x) defined on the interval [a, b].
- The function f(x) has opposite signs at a and b (i.e., f(a) × f(b) < 0), indicating a root between x3 and x2.
Example 1: f(x) = x^2 - 2
- The function has a root between x3 and x2, and the first iteration yields x4 = 1.3999.
- The process is repeated until the error |x_i - x_(i-1)| < ε, where ε is the desired error tolerance.
- The approximate root of f(x) = x^2 - 2 is x = 1.4142.
Example 2: f(x) = x log(x) - 1
- The function has a root between x2 and x1, and the first iteration yields x3 = 1.76154.
- The process is repeated until the error |x_i - x_(i-1)| < ε, where ε is the desired error tolerance.
- The approximate root of f(x) = x log(x) - 1 is x = 1.76322284.
Secant Method
- The method is used to find an approximate root of a function f(x) using the secant line approximation.
- The iterative formula for the secant method is x_(n+1) = x_n - f(x_n) × (x_n - x_(n-1)) / (f(x_n) - f(x_(n-1))).
- The process is repeated until the error |x_i - x_(i-1)| < ε, where ε is the desired error tolerance.
Example 3: f(x) = x^2 - 2
- The function has a root between x1 and x2, and the first iteration yields x3 = 1.41466898.
- The process is repeated until the error |x_i - x_(i-1)| < ε, where ε is the desired error tolerance.
- The approximate root of f(x) = x^2 - 2 is x = 1.4142115.
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