Use Newton's algorithm on f(x) = x² − 6 with a starting value x₀ = 2 to obtain two improved estimates of √6, then report the error after the second iteration.

Start with the thinking

• Newton's algorithm is x_{k+1} = x_k − f(x_k)/f′(x_k); for f = x² − N it collapses to the averaging form.
• Convergence is quadratic, so two iterations already give several correct digits.

Step-by-step solution

1. Formula

   $$x_{k+1} = x_k - f(x_k)/f'(x_k) = ½(x_k + N/x_k)$$

2. Iteration 1

   $$x_{1} = ½(2 + 6/2) = 2.50000$$

3. Iteration 2

   $$x_{2} = ½(2.50000 + 6/2.50000) = 2.45000$$

4. Exact value — √6 = 2.44949

5. Error — |x₂ − √6| = 0.000510

Why the other options are there

• 3.0000 (single division, no averaging)
• 4.0000 (forgot to divide by f′)

Reference: FE Reference Handbook — Mathematics → Euler's or Forward Rectangular Rule

Estimate ∫₀^3 (x² + 1) dx with the trapezoidal rule using n = 4 equal intervals, then compare with the exact value and report the percentage error.

Start with the thinking

• The trapezoidal rule weights interior ordinates by 2 and the two end ordinates by 1.
• For a convex function the trapezoidal rule over-estimates the true area.

Step-by-step solution

1. Step size

   $$h = (b - a)/n = (3 - 0)/4 = 0.7500$$

2. Formula — I ≈ (h/2)[f(a) + 2Σf(xᵢ) + f(b)]

3. Ordinates — f(0) = 1.000, interior sum Σf(xᵢ) = 10.8750, f(3) = 10.000

4. Substituting

   $$I \approx (0.7500/2)[32.7500] = 12.2813$$

5. Exact

   $$\int = b^{3}/3 + b = 12.0000$$

6. Error

   $$|I - exact|/exact = 2.344$$

Why the other options are there

• 24.563 (dropped the ½)
• 4.125 (single trapezoid)

Reference: FE Reference Handbook — Mathematics → Euler's or Forward Rectangular Rule

Use Newton's algorithm on f(x) = x² − 5 with a starting value x₀ = 4 to obtain two improved estimates of √5, then report the error after the second iteration.

Start with the thinking

• Newton's algorithm is x_{k+1} = x_k − f(x_k)/f′(x_k); for f = x² − N it collapses to the averaging form.
• Convergence is quadratic, so two iterations already give several correct digits.

Step-by-step solution

1. Formula

   $$x_{k+1} = x_k - f(x_k)/f'(x_k) = ½(x_k + N/x_k)$$

2. Iteration 1

   $$x_{1} = ½(4 + 5/4) = 2.62500$$

3. Iteration 2

   $$x_{2} = ½(2.62500 + 5/2.62500) = 2.26488$$

4. Exact value — √5 = 2.23607

5. Error — |x₂ − √5| = 0.028813

Why the other options are there

• 1.2500 (single division, no averaging)
• -7.0000 (forgot to divide by f′)

Reference: FE Reference Handbook — Mathematics → Euler's or Forward Rectangular Rule

Estimate ∫₀^2 (x² + 1) dx with the trapezoidal rule using n = 8 equal intervals, then compare with the exact value and report the percentage error.

Start with the thinking

• The trapezoidal rule weights interior ordinates by 2 and the two end ordinates by 1.
• For a convex function the trapezoidal rule over-estimates the true area.

Step-by-step solution

1. Step size

   $$h = (b - a)/n = (2 - 0)/8 = 0.2500$$

2. Formula — I ≈ (h/2)[f(a) + 2Σf(xᵢ) + f(b)]

3. Ordinates — f(0) = 1.000, interior sum Σf(xᵢ) = 15.7500, f(2) = 5.000

4. Substituting

   $$I \approx (0.2500/2)[37.5000] = 4.6875$$

5. Exact

   $$\int = b^{3}/3 + b = 4.6667$$

6. Error

   $$|I - exact|/exact = 0.446$$

Why the other options are there

• 9.375 (dropped the ½)
• 0.750 (single trapezoid)

Reference: FE Reference Handbook — Mathematics → Euler's or Forward Rectangular Rule

Use Newton's algorithm on f(x) = x² − 5 with a starting value x₀ = 4 to obtain two improved estimates of √5, then report the error after the second iteration.

Start with the thinking

• Newton's algorithm is x_{k+1} = x_k − f(x_k)/f′(x_k); for f = x² − N it collapses to the averaging form.
• Convergence is quadratic, so two iterations already give several correct digits.

Step-by-step solution

1. Formula

   $$x_{k+1} = x_k - f(x_k)/f'(x_k) = ½(x_k + N/x_k)$$

2. Iteration 1

   $$x_{1} = ½(4 + 5/4) = 2.62500$$

3. Iteration 2

   $$x_{2} = ½(2.62500 + 5/2.62500) = 2.26488$$

4. Exact value — √5 = 2.23607

5. Error — |x₂ − √5| = 0.028813

Why the other options are there

• 1.2500 (single division, no averaging)
• -7.0000 (forgot to divide by f′)

Reference: FE Reference Handbook — Mathematics → Euler's or Forward Rectangular Rule

Estimate ∫₀^4 (x² + 1) dx with the trapezoidal rule using n = 4 equal intervals, then compare with the exact value and report the percentage error.

Start with the thinking

• The trapezoidal rule weights interior ordinates by 2 and the two end ordinates by 1.
• For a convex function the trapezoidal rule over-estimates the true area.

Step-by-step solution

1. Step size

   $$h = (b - a)/n = (4 - 0)/4 = 1.0000$$

2. Formula — I ≈ (h/2)[f(a) + 2Σf(xᵢ) + f(b)]

3. Ordinates — f(0) = 1.000, interior sum Σf(xᵢ) = 17.0000, f(4) = 17.000

4. Substituting

   $$I \approx (1.0000/2)[52.0000] = 26.0000$$

5. Exact

   $$\int = b^{3}/3 + b = 25.3333$$

6. Error

   $$|I - exact|/exact = 2.632$$

Why the other options are there

• 52.000 (dropped the ½)
• 9.000 (single trapezoid)

Reference: FE Reference Handbook — Mathematics → Euler's or Forward Rectangular Rule

Use Newton's algorithm on f(x) = x² − 4 with a starting value x₀ = 4 to obtain two improved estimates of √4, then report the error after the second iteration.

Start with the thinking

• Newton's algorithm is x_{k+1} = x_k − f(x_k)/f′(x_k); for f = x² − N it collapses to the averaging form.
• Convergence is quadratic, so two iterations already give several correct digits.

Step-by-step solution

1. Formula

   $$x_{k+1} = x_k - f(x_k)/f'(x_k) = ½(x_k + N/x_k)$$

2. Iteration 1

   $$x_{1} = ½(4 + 4/4) = 2.50000$$

3. Iteration 2

   $$x_{2} = ½(2.50000 + 4/2.50000) = 2.05000$$

4. Exact value — √4 = 2.00000

5. Error — |x₂ − √4| = 0.050000

Why the other options are there

• 1.0000 (single division, no averaging)
• -8.0000 (forgot to divide by f′)

Reference: FE Reference Handbook — Mathematics → Euler's or Forward Rectangular Rule

Estimate ∫₀^4 (x² + 1) dx with the trapezoidal rule using n = 6 equal intervals, then compare with the exact value and report the percentage error.

Start with the thinking

• The trapezoidal rule weights interior ordinates by 2 and the two end ordinates by 1.
• For a convex function the trapezoidal rule over-estimates the true area.

Step-by-step solution

1. Step size

   $$h = (b - a)/n = (4 - 0)/6 = 0.6667$$

2. Formula — I ≈ (h/2)[f(a) + 2Σf(xᵢ) + f(b)]

3. Ordinates — f(0) = 1.000, interior sum Σf(xᵢ) = 29.4444, f(4) = 17.000

4. Substituting

   $$I \approx (0.6667/2)[76.8889] = 25.6296$$

5. Exact

   $$\int = b^{3}/3 + b = 25.3333$$

6. Error

   $$|I - exact|/exact = 1.170$$

Why the other options are there

• 51.259 (dropped the ½)
• 6.000 (single trapezoid)

Reference: FE Reference Handbook — Mathematics → Euler's or Forward Rectangular Rule

Use Newton's algorithm on f(x) = x² − 6 with a starting value x₀ = 5 to obtain two improved estimates of √6, then report the error after the second iteration.

Start with the thinking

• Newton's algorithm is x_{k+1} = x_k − f(x_k)/f′(x_k); for f = x² − N it collapses to the averaging form.
• Convergence is quadratic, so two iterations already give several correct digits.

Step-by-step solution

1. Formula

   $$x_{k+1} = x_k - f(x_k)/f'(x_k) = ½(x_k + N/x_k)$$

2. Iteration 1

   $$x_{1} = ½(5 + 6/5) = 3.10000$$

3. Iteration 2

   $$x_{2} = ½(3.10000 + 6/3.10000) = 2.51774$$

4. Exact value — √6 = 2.44949

5. Error — |x₂ − √6| = 0.068252

Why the other options are there

• 1.2000 (single division, no averaging)
• -14.0000 (forgot to divide by f′)

Reference: FE Reference Handbook — Mathematics → Euler's or Forward Rectangular Rule

Estimate ∫₀^8 (x² + 1) dx with the trapezoidal rule using n = 6 equal intervals, then compare with the exact value and report the percentage error.

Start with the thinking

• The trapezoidal rule weights interior ordinates by 2 and the two end ordinates by 1.
• For a convex function the trapezoidal rule over-estimates the true area.

Step-by-step solution

1. Step size

   $$h = (b - a)/n = (8 - 0)/6 = 1.3333$$

2. Formula — I ≈ (h/2)[f(a) + 2Σf(xᵢ) + f(b)]

3. Ordinates — f(0) = 1.000, interior sum Σf(xᵢ) = 102.8, f(8) = 65.000

4. Substituting

   $$I \approx (1.3333/2)[271.6] = 181.0$$

5. Exact

   $$\int = b^{3}/3 + b = 178.7$$

6. Error

   $$|I - exact|/exact = 1.327$$

Why the other options are there

• 362.1 (dropped the ½)
• 44.000 (single trapezoid)

Reference: FE Reference Handbook — Mathematics → Euler's or Forward Rectangular Rule