1. **Problem 1: Find the approximate root of $f(x) = x^3 + 2x^2 - 1$ between 0 and 1 using the bisection method after four iterations.** The bisection method formula is: $$C = \frac{a+b}{2}$$ where $a$ and $b$ are interval endpoints and $C$ is the midpoint. We evaluate $f(a)$, $f(b)$, and $f(C)$ to decide which subinterval contains the root. **Iteration 1:** - $a=0$, $b=1$ - $f(0) = 0^3 + 2\cdot0^2 - 1 = -1$ - $f(1) = 1 + 2 - 1 = 2$ - $C = \frac{0+1}{2} = 0.5$ - $f(0.5) = 0.5^3 + 2\cdot0.5^2 - 1 = 0.125 + 0.5 - 1 = -0.375$ - Since $f(a)\cdot f(C) < 0$, root lies between $0.5$ and $1$, update $a=0.5$ **Iteration 2:** - $a=0.5$, $b=1$ - $f(0.5) = -0.375$ - $f(1) = 2$ - $C = \frac{0.5+1}{2} = 0.75$ - $f(0.75) = 0.75^3 + 2\cdot0.75^2 - 1 = 0.422 + 1.125 - 1 = 0.547$ - Since $f(a)\cdot f(C) < 0$ is false, root lies between $0.5$ and $0.75$, update $b=0.75$ **Iteration 3:** - $a=0.5$, $b=0.75$ - $f(0.5) = -0.375$ - $f(0.75) = 0.547$ - $C = \frac{0.5+0.75}{2} = 0.625$ - $f(0.625) = 0.625^3 + 2\cdot0.625^2 - 1 = 0.244 + 0.781 - 1 = 0.025$ - Since $f(a)\cdot f(C) < 0$ is true, root lies between $0.5$ and $0.625$, update $b=0.625$ **Iteration 4:** - $a=0.5$, $b=0.625$ - $f(0.5) = -0.375$ - $f(0.625) = 0.025$ - $C = \frac{0.5+0.625}{2} = 0.5625$ - $f(0.5625) = 0.5625^3 + 2\cdot0.5625^2 - 1 = 0.178 + 0.633 - 1 = -0.189$ - Since $f(a)\cdot f(C) < 0$ is true, root lies between $0.5625$ and $0.625$, update $a=0.5625$ **Approximate root after 4 iterations:** $x \approx 0.5625$ --- 2. **Problem 2: Calculate the relative error when approximating $5x^2 - 1$ using 4-digit chopping for $x=11110.3338$.** - First, calculate the exact value: $$5x^2 - 1 = 5 \times (11110.3338)^2 - 1$$ - Compute $x^2 = 11110.3338^2 \approx 123456790.7$ - Then exact value $= 5 \times 123456790.7 - 1 = 617283953.5$ - 4-digit chopping of $x$ is $11110$ (keep first 4 digits only, chopping rest). - Approximate $x^2$ with chopping: $11110^2 = 123432100$ - Approximate value $= 5 \times 123432100 - 1 = 617160499$ - Relative error formula: $$\text{Relative error} = \frac{|\text{exact} - \text{approx}|}{|\text{exact}|}$$ - Calculate: $$= \frac{|617283953.5 - 617160499|}{617283953.5} = \frac{123454.5}{617283953.5} \approx 0.0002$$ - Among given options, closest is $0.00048215$. --- 3. **Problem 3: Evaluate $x^3 - 3x^2 + 4x + 0.21$ at $x=2.15$ using nested arithmetic and 3-digit rounding.** Nested form: $$x(x(x-3) + 4) + 0.21$$ Stepwise evaluation with 3-digit rounding: 1. Compute $x-3 = 2.15 - 3 = -0.85$ 2. Multiply $x(x-3) = 2.15 \times (-0.85) = -1.8275 \approx -1.83$ 3. Add 4: $-1.83 + 4 = 2.17$ 4. Multiply by $x$: $2.15 \times 2.17 = 4.6655 \approx 4.67$ 5. Add 0.21: $4.67 + 0.21 = 4.88$ **Final approximate value:** $4.88$