Question: (a) Evaluate the sum S = \sqrt{3} + \sqrt{5} + \sqrt{7} to 4 significant digits and find its absolute and relative errors. 2mrks
(b) Evaluate f(1) using Taylor’s series for f(x) = x^3 - 3x^2 + 5x - 10 7mrks
(c) Given f(x) = \sin x, construct the Taylor series approximations of order 0 to 7 at x = \pi/3 and state their absolute errors. 8.5mrks
B
(a) By Bisection method, find a real root of the equation x^3 - 2x - 5 = 0 up to the 10th iteration. 7.5mrks
(b) Find the positive root, between 0 and 1, of the equation x = e^{-x} to a tolerance of 0.05%
10mrks
C
(a) The equation 2x = \log_{10} x + 7 has a root between 3 and 4. Find this root, correct to three decimal places, by regula-falsi method. 6mrks
(b) By iteration method, find a real root of the equation x^3 = 1 - x^2 on the interval [0,1] with an accuracy of 10^{-4}. 6mrks
(c) Use the Newton-Raphson method to find a root of the equation x^3 = 2x - 5 = 0 5.5mrks
D
(a) By Ramanujan’s method, find the smallest root of f(x) = x^3 - 9x^2 + 26x - 24 = 0. 6mrks
(b) Using the Secant method, find a real root of f(x = xe^x - 1 = 0 6mrks
(c) Using Graeffe’s method, find the real roots of x^3 - 6x^2 + 11x - 6 = 0 5.5mrks
E
(a) By iteration method up to 10th iterations, find a real root of the system of equations;
y^2 - 5y + 4 = 0
3yx^2 - 10x + 7 = 0 9mrks
(b) By Newton-Raphson method, solve the system;
sin x - y = 0.9793
cos y - x = 0.6703 8.5mrks
Graph/shape: no graph visible; text-only page. position_hint=centerUser: Explain
1. **Problem Statement:** Evaluate the sum $$S = \sqrt{3} + \sqrt{5} + \sqrt{7}$$ to 4 significant digits and find its absolute and relative errors.
2. **Step 1: Calculate each square root to high precision:**
$$\sqrt{3} \approx 1.73205$$
$$\sqrt{5} \approx 2.23607$$
$$\sqrt{7} \approx 2.64575$$
3. **Step 2: Sum the values:**
$$S = 1.73205 + 2.23607 + 2.64575 = 6.61387$$
4. **Step 3: Round to 4 significant digits:**
$$S \approx 6.614$$
5. **Step 4: Calculate absolute error:**
Absolute error = $$|\text{true value} - \text{approximate value}| = |6.61387 - 6.614| = 0.00013$$
6. **Step 5: Calculate relative error:**
Relative error = $$\frac{\text{absolute error}}{\text{true value}} = \frac{0.00013}{6.61387} \approx 1.97 \times 10^{-5}$$
**Final answers:**
- Sum to 4 significant digits: $$6.614$$
- Absolute error: $$0.00013$$
- Relative error: $$1.97 \times 10^{-5}$$