Iteration and numerical solutions

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Notes de leçon Diapos Questions d'entraînement

Notes de leçon

What iteration is

Iteration repeatedly applies a formula. A suitable formula and starting value may converge to a root; convergence is not guaranteed.
The notation is x_n+1 = f(x_n), starting from x₀ or x₁.

Using iteration

Rearrange an equation into the form x = f(x). For x³ - 3x - 1 = 0: x = \frac{x³ - 1}{3} or x = \√[3]{3x + 1}.
For xₙ₊₁ = ∛(3xₙ + 1), starting x₁ = 2 gives x₂ ≈ 1.912931 and x₃ ≈ 1.888835. Continuing converges to about 1.879385; retain calculator precision between steps.
Use the calculator's ANS key to repeat quickly, and keep full calculator accuracy until you round at the end.

Decimal search and change of sign

If f(a) and f(b) have opposite signs, a root of f(x) = 0 lies between a and b (for a continuous function).
Narrow down by testing the midpoint (the decimal search or interval bisection).
To show a root is 1.88 to 2 d.p., show f(1.875) and f(1.885) have opposite signs.

Stating the answer

Give the number of decimal places asked, but check the rounding by looking at when successive iterations agree.
Agreement between a few rounded iterates supports an estimate but is not a proof of accuracy. For a continuous function, a sign-changing interval within the required rounding limits brackets a root.

Diapos

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Questions d'entraînement

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1.When using iteration to approximate a root, what process is repeated?
Easy
- Asolves exactly by factorising
- Brepeats a formula to get closer to a solution
- Cdraws a graph
- Dexpands brackets
2.If x_n+1 = x_n + 3 and x₁ = 2, what is x₂?
Easy
- A2
- B5
- C6
- D8
3.If x_n+1 = 2x_n and x₁ = 3, what is x₃?
Easy
- A6
- B12
- C9
- D18
4.If x_n+1 = \frac{x_n + 10}{2} and x₁ = 4, what is x₂?
Easy
- A5
- B7
- C14
- D2
5.If x_n+1 = \√{x_n + 6} and x₁ = 3, what is x₂?
Easy
- A3
- B2
- C9
- D\√{3}
6.If x_n+1 = 1 + \frac{1}{x_n} and x₁ = 1, what is x₂?
Easy
- A1
- B2
- C0.5
- D3
7.For f(x) = x² - 5, f(2) = -1 and f(3) = 4. What does this show?
Easy
- Aa root between 2 and 3
- Bno roots
- Cf(2.5) = 0
- Dthe root is 2
8.f(x) = x³ + x - 3. Which pair shows a root between 1 and 2?
Easy
- Af(1) = -1, f(2) = 7
- Bf(1) = 1, f(2) = 7
- Cf(1) = 3, f(2) = 7
- Df(1) = -1, f(2) = -7

Plus de questions

9.Which equation can be rearranged to x = \frac{x³ - 1}{3}?
Medium
- Ax³ - 3x - 1 = 0
- Bx³ + 3x - 1 = 0
- Cx³ - 3x + 1 = 0
- D3x³ - 1 = 0
10.When seeking the positive root of x² − 2x − 4 = 0, is x = √(2x + 4) a suitable rearrangement?
Medium
- Ayes
- Bno, it should be \√{2x - 4}
- Cno, it needs a cube root
- Dno, it should be -\√{2x}
11.Using x_n+1 = \√[3]{3x_n + 1} with x₁ = 2, what is x₂ to 3 d.p.?
Medium
- A1.913
- B2.000
- C1.879
- D2.080
12.Why do we keep full calculator accuracy until the end of an iteration?
Medium
- Ato avoid rounding errors building up
- Bto make the answer longer
- Cto find exact surds
- Dit makes no difference
13.If successive iterations are 1.8794, 1.8794 and 1.8794, the root to 3 d.p. is about:
Medium
- A1.879
- B1.880
- C1.88 to 1 d.p.
- D2
14.For a function continuous on the stated interval: f(1.87) < 0 and f(1.88) > 0. What can you conclude?
Medium
- Athe root is 1.87 to 2 d.p.
- Bthe root is between 1.87 and 1.88
- Cthere are no roots
- Dthe root is 1.9
15.For a continuous function with a single root near 1.88, which pair of values can bracket the root to show it rounds to 1.88 to 2 decimal places?
Medium
- Af(1.8) and f(1.9)
- Bf(1.875) and f(1.885)
- Cf(1.88) only
- Df(0)
16.If f(x) = x³ - 4x - 1, which interval contains a root: f(2) = -1, f(3) = 14?
Medium
- Abetween 2 and 3
- Bbetween 0 and 1
- Cbetween 3 and 4
- Dnone
17.Starting at x₁ = 1, which formula gives 1, 3, 7, 15?
Medium
- Ax_n+1 = 2x_n + 1
- Bx_n+1 = x_n + 2
- Cx_n+1 = 3x_n
- Dx_n+1 = x_n²
18.What is the ANS key used for in iteration?
Easy
- Ato store the previous answer for repeating
- Bto square a number
- Cto clear the screen
- Dto round
19.Which is a suitable decimal search to improve an estimate of a root in [1, 2]?
Medium
- Atest f(1.5)
- Btest f(5)
- Ctest f(-1)
- Dtest f(100)
20.For a function continuous on the stated interval: If f(1.5) > 0 and f(1) < 0, the root is between:
Easy
- A1 and 1.5
- B1.5 and 2
- C0 and 1
- D2 and 3
21.x_n+1 = 3 - \frac{1}{x_n}, x₁ = 2. What is x₂?
Medium
- A2.5
- B1
- C2
- D3.5
22.Which equation has an iteration x_n+1 = \frac{5}{x_n + 1}?
Medium
- Ax² + x - 5 = 0
- Bx² - x - 5 = 0
- Cx² + 5x = 1
- Dx² = 5x + 1
23.For an iteration that converges, the sequence of values:
Medium
- Agrows without limit
- Bgets closer to a fixed value
- Calternates between 0 and 1
- Dis always negative
24.Iteration is used when an equation:
Easy
- Acannot easily be solved algebraically
- Bis linear
- Chas no unknown
- Dis already solved
25.What is the first step for iteration on x² = 3x + 1?
Easy
- Arearrange into x = \ldots
- Bdraw a circle
- Cdivide by zero
- Dsquare root only
26.Iteration repeats a formula using each result as the next input.
Easy
True or false?
27.If f is continuous on [a, b] and f(a) and f(b) have opposite signs, there is at least one root strictly between a and b.
Easy
True or false?
28.Iteration always gives an exact answer.
Easy
True or false?
29.You should round at every step of an iteration.
Easy
True or false?
30.If f(1) > 0 and f(2) > 0 there must be a root between 1 and 2.
Medium
True or false?
31.For f continuous on [1, 2], which endpoint pairs guarantee at least one root between 1 and 2?
Medium
- Af(1) = -2, f(2) = 3
- Bf(1) = 4, f(2) = -1
- Cf(1) = 2, f(2) = 3
- Df(1) = -3, f(2) = -1
- Ef(1) = -1, f(2) = 1
32.Which practices are useful when approximating a nonlinear root by iteration? Select all that apply.
Medium
- Asolving x³ + x - 3 = 0 numerically
- Bsolving x + 2 = 5
- Cfinding a root to given accuracy
- Dexpanding brackets
- Eusing ANS repeatedly
33.Show that x³ - 3x - 1 = 0 has a root between 1 and 2.
Medium

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