Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on the following concepts:

• Bisection Method (Interval Halving): Iterative numerical method for finding roots by repeatedly dividing the interval in half

• Intermediate Value Theorem (IVT): Ensures existence of a root when $f(a)$ and $f(b)$ have opposite signs

• Root Finding: Locating values $x$ where $f(x)=0$ for polynomial and transcendental equations

• Convergence Criteria: Determining when the approximation is accurate to two decimal places

• Error Bounds: Understanding maximum possible error after $n$ iterations

• Transcendental Equations: Solving equations involving exponential, logarithmic, and trigonometric functions

• Bracketing Methods: Maintaining an interval that contains the root throughout iterations

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Important Formulas

Below are the key formulas used in this exercise:

Midpoint Calculation: $c=\frac{a+b}{2}$

Error Bound after $n$ iterations: $|x_n-x^{\ast }|\le \frac{b-a}{2^n}$

Stopping Criterion (for 2 decimal places accuracy): $|b-a|<0.01\text{or}|f(c)|<0.005$

Number of iterations needed for error $ϵ$: $n\ge \frac{\ln (b-a)-\ln (ϵ)}{\ln (2)}$

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Step-by-Step Procedure

1. Verify bracketing: Check that $f(a)\cdot f(b)<0$ (IVT guarantees a root exists in $[a,b]$).
2. Compute midpoint: $c=\frac{a+b}{2}$.
3. Evaluate $f(c)$:
   • If $f(c)=0$: $c$ is the exact root. Stop.
   • If $f(a)\cdot f(c)<0$: root is in $[a,c]$; set $b=c$.
   • If $f(c)\cdot f(b)<0$: root is in $[c,b]$; set $a=c$.
4. Check stopping criterion: If $|b-a|<0.005$ (for 2 d.p. accuracy), stop. Otherwise repeat from step 2.

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Summary

This exercise provides comprehensive practice in applying the bisection method to locate roots of various equations with an accuracy of two decimal places. The problems span polynomial equations (cubic and quartic), transcendental equations involving trigonometric and exponential functions, and practical application problems.

Key strategies include: verifying the initial interval $[a,b]$ satisfies $f(a)\cdot f(b)<0$, systematically computing midpoints, determining which subinterval contains the root based on sign changes, and iterating until the desired precision is achieved. Common patterns involve recognizing that transcendental equations often require the method due to lack of algebraic solutions, while polynomial equations may have multiple roots requiring careful interval selection.