Understanding Numerical Methods for Root Finding and Iteration

Explore the concepts of root finding, locating roots, stationary points, and iteration in numerical methods. Learn to determine roots, stationary points, and convergence/divergence types, as well as apply the Newton-Raphson method for function approximation.

• Numerical Methods
• Root Finding
• Stationary Points
• Convergence
• Iteration

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1. Chapter 10 Chapter 10- - Numerical methods Numerical methods

2. Locating roots (10.1) Locating roots (10.1)

3. 1 Where does the function ? ? = 4 ?+ 3 have at least one root? 1 a) 3.9 < ? < 4.1 b) 4.1 < ? < 4.4 d) 3.6 < ? < 3.9 c) 4.1 < ? < 4.3

4. 2 Where does the function ? ? = cos2(?) ? 2? have a stationary point? a) 1.2 < ? < 1.4 b) 1.3 < ? < 1.4 d) 3.2 < ? < 3.3 c) 3.1 < ? < 3.2

5. 3 Where does the function ? ? = sin 2? + ?4? have a stationary point? a) 0.3 < ? < 0.1 b) 0.7 < ? < 0.5 d) 0.9 < ? < 0.8 c) 1.0 < ? < 0.9

6. Iteration (10.2) Iteration (10.2)

7. Which of the following is not a rearrangement of ?5 ?2 20 = 0? 1 31 +20 5?2+ 20 b) ? = a) ? = ?2 c) ? = 20? 4+ ? 2 d) ? = 20 ?5

8. Describe the type of convergence/ divergence produced in the table below: 2 b) Staircase convergence a) Cobweb convergence d) Staircase divergence c) Cobweb divergence

9. Describe the type of convergence/ divergence produced in the table below: 3 b) Staircase convergence a) Cobweb convergence d) Staircase divergence c) Cobweb divergence

10. Describe the type of convergence/ divergence produced in the table below: 4 b) Staircase convergence a) Cobweb convergence d) Staircase divergence c) Cobweb divergence

11. The Newton The Newton- -Raphson method (10.3) Raphson method (10.3)

12. 1 Using ?0= 3 as a first approximation, use the Newton s Raphson method to compute ?1 of the following: ? ? = ?4 10?2 11. a) 3.5001 b) 3.4101 d) 3.4167 c) 3.4892

13. 2 Using ?0= 1 as a first approximation, use the Newton s Raphson method to compute ?1 of the following: ? ? = 4?3+ ? 7. a) 1.2109 b) 1.1555 d) 1.2001 c) 1.1538

14. 3 Using ?0= 1 as a first approximation, use the Newton s Raphson method to compute ?1 of the following: ? ? = 2?3 cos(?). a) 0.7901 b) 0.7866 d) 0.7861 c) 0.7822

15. Applications to modelling (10.4) Applications to modelling (10.4)

16. 1 A boomerang is thrown from a point on level ground. The height in metres of the boomerang above the ground ? seconds after it is thrown can be modelled by the function ?2 36. When does the boomerang hit the ground? (?) = 50sin a) 10 < ? < 11 b) 11 < ? < 12 d) 12 < ? < 13 c) 9 < ? < 10

17. 2 The value of a mobile phone, t years After purchase , is modelled by the function ? ? = 450e 0.5t 40cos t ,t > 0. When does the function have a value of nothing? a) 1 < ? < 3 b) 3 < ? < 4 d) 5 < ? < 6 c) 4 < ? < 5