Math Problem Statement
- Consider the following equation f(x) = x * sin(3x) whose roots we seek, answer the following question: with the initial bracket [2.5 , 3.5] how many bisections will it take to achieve an error tolerance of 10^-5? a) 18 b) 16 c) 13 d) 17 e) 5
Solution
To determine the number of bisections needed to achieve a given error tolerance in the interval using the Bisection Method, we use the following formula:
where:
- is the number of bisections,
- is the initial bracket,
- is the error tolerance.
Step 1: Calculate the width of the initial bracket
Step 2: Plug values into the formula
Simplify the logs:
Thus, the formula becomes:
Since , we get:
Step 3: Round up
Since must be an integer, round up to the next whole number. Thus, .
Conclusion:
The number of bisections needed is , so the correct answer is d) 17.
Would you like further details on how bisection works or any other questions on this?
Here are 5 related questions:
- How does the Bisection Method guarantee convergence?
- What is the formula for error reduction in the Bisection Method?
- Can you apply the Bisection Method to discontinuous functions?
- What are other root-finding methods that might be faster than bisection?
- How can you determine whether the Bisection Method will work for a given function?
Tip: Always check the signs of and in the Bisection Method to ensure the function crosses the x-axis in the interval.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Numerical Methods
Root-Finding Methods
Bisection Method
Logarithms
Formulas
n ≥ (log(b - a) - log(ε)) / log(2)
Theorems
Bisection Method Convergence Theorem
Suitable Grade Level
University Level or Advanced High School
Related Recommendation
Finding Roots of f(x) = e^x - cos(x) - 2 Using the Bisection Method
Finding Roots Using Bisection Method for f(x) = sin(x) + x^3 + 1
Numerical Solutions: Bisection, Newton, and Secant Methods for Root Finding
Solving Root-finding Algorithms with Bisection, Newton, and Secant Methods
Solving f(x) = sqrt(x) - cos(x) Using the Bisection Method