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OCR A-level Maths Iterative methods

Iterative methods

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Suppose we have an equation such as xΒ² – 3x + 2 = 0. One approach of solving this is to rearrange the equation so that x is the subject of the formula, giving x = √(3x – 2). A value of x which satisfies this is also a solution of our original equation. We can find this value of x by considering this as an iterative formula xβ‚™β‚Šβ‚ = √(3xβ‚™ – 2). We then pick some starting value xβ‚€ and put it in the right hand side of our equation to give some new x₁. We then iteratively put the x₁ in the right side again to get xβ‚‚, and so on. The values we get from the equation should converge on an approximate solution to our original equation, but this can depend on the function and what starting xβ‚€ we use.

Iterative formulas

Suppose we are trying to find the root of some equation f(x). We can use the change of signs method and find 2 points a and b so that f(a) < 0 and f(b) > 0, and therefore we know the root will lie somewhere in between. We then consider the midpoint m of these 2 points (dashed line on graph). If f(m) < 0, then we now know the root is between m and b, and otherwise it's between a and m. Iterative bisection just involves repeadedly doing this process, finding a new smaller intervals until it converges to an approximate solution. After a certain number of iterations one obviously needs to stop, and then we just give the interval's midpoint as the approximate root. It doesn't strictly need to find a root either, we could check for values f(a) < 1 and f(b) > 1 when trying to solve f(x) = 1.

Iterative Bisection

Consider this graph of f(x), where we know f(x) = 0 between some f(a) < 0 and f(b) > 0. Linear Interpolation is similar to iterative bisection, but instead of finding the midpoint of a and b to get our new interval, we consider the straight line from f(a) to f(b). The x–intercept (c) of this straight line can be easy calculated as c = ( a|f(b)| + b |f(a)| ) / ( |f(a)| + |f(b)| ), and this will be closer to the root of f(x) (note that the '| |' means the absolute, positive value). Then just as we did for iterative bisection, we identify the smaller interval which must contain the root and iteratively repeat the process, converging onto the root.

Linear Interpolation

Iterative methods can be used to find the roots of f(x) = 0 after rearranging it to the form x = g(x) and using the iterative formula x_(n + 1) = g(x_n). One method of doing this is with successive iterations which alternate between being below and above the root. If the iterations converge, a cobweb diagram is formed. The interval in which a root lies can be found by plugging values for x in the formula and seeing where the value of f(x) changes sign.

Cobweb diagrams

One iterative method is one in which the iterations get progressively closer to the root from the same direction. When this process is plotted, a diagram called a staircase diagram is formed.

Staircase diagrams

Iterative methods can be used to model situations and find their solutions.

Iterative methods problems in context

1

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Construct the iterative formula for xΒ² + 2x – 8 = 0.

OCR A-level Maths Iterative methods Construct the iterative formula for xΒ² + 2x – 8 = 0.
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2

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Find a root of xΒ² + 2x – 8 = 0 to 1 decimal place using an iterative formula with xβ‚€ = 3.

OCR A-level Maths Iterative methods Find a root of xΒ² + 2x – 8 = 0 to 1 decimal place using an iterative formula with xβ‚€ = 3.
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3

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Find a solution to xΒ² + 2x – 3 = 0, using iterative bisection and knowing the root exists in the interval between x = 0 and x = 1.5.

OCR A-level Maths Iterative methods Find a solution to xΒ² + 2x – 3  = 0, using iterative bisection and knowing the root exists in the interval between x = 0 and x = 1.5.
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4

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Find a root of f(x) = sin(x) + x + 2 using iterative bisection, knowing the root is between x = –1 and x = –2.

OCR A-level Maths Iterative methods Find a root of f(x) = sin(x) + x + 2 using iterative bisection, knowing the root is between x = –1 and x = –2.
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5

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Find a solution to f(x) = cos(x) + sin(x) + x, using iterative bisection. Suppose you initially know a root exists between x = –1 and x = 0. .

OCR A-level Maths Iterative methods Find a solution to f(x) = cos(x) + sin(x) + x, using iterative bisection. Suppose you initially know a root exists between x =  –1 and x = 0.  .
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6

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Find xΒ² + 5x + 6, using linear interpolation and knowing a root exists in the interval between x = – 3.5 and x = –2.5.

OCR A-level Maths Iterative methods Find xΒ² + 5x + 6, using linear interpolation and knowing a root exists in the interval between x = – 3.5 and x = –2.5.
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7

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Show that there is a root of f(x) = 2x + tan(x) + 1 at around x = –0.3 using linear interpolation between x = 0 and x = 0.5.

OCR A-level Maths Iterative methods Show that there is a root of f(x) = 2x + tan(x) + 1 at around x = –0.3 using linear interpolation between x = 0 and x = 0.5.
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8

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Find the root of f(x) = xΒ³ + √(x) – 3/2, using linear interpolation (to 1 decimal place).

OCR A-level Maths Iterative methods Find the root of f(x) = xΒ³ + √(x) – 3/2, using linear interpolation (to 1 decimal place).
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9

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Show that the curve y = xΒ³ + 4x βˆ’ 1 has a root between x = 0 and x = 1.

OCR A-level Maths Iterative methods Show that the curve y = xΒ³ + 4x βˆ’ 1 has a root between x = 0 and x = 1.
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10

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Draw a cobweb diagram starting at x = 0.5 to show how the intersection point of y = cosx and y = x is iteratively reached.

OCR A-level Maths Iterative methods Draw a cobweb diagram starting at x = 0.5 to show how the intersection point of y = cosx and y = x is iteratively reached.
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11

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Describe the iterative process which graphically forms a staircase diagram.

OCR A-level Maths Iterative methods Describe the iterative process which graphically forms a staircase diagram.
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12

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Let f(x) = xΒ³ βˆ’ 2xΒ² βˆ’ 3x + 2. Show that an iterative formula is x_(n+1) = √((x_nΒ³ βˆ’ 3x_n + 2)/2).

OCR A-level Maths Iterative methods Let f(x) = xΒ³ βˆ’ 2xΒ² βˆ’ 3x + 2. Show that an iterative formula is x_(n+1) = √((x_nΒ³ βˆ’ 3x_n + 2)/2).
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13

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Sketch a staircase diagram for locating the root of the equation 2 + lnx = x, starting at x = 1.2.

OCR A-level Maths Iterative methods Sketch a staircase diagram for locating the root of the equation 2 + lnx = x, starting at x = 1.2.
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14

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The price of a car after x years is modelled by f(x) = 2200(0.7)Λ£ βˆ’ 1200sinx. Show that the function has a root between x = 12 and x = 13 and using this, criticise the model.

OCR A-level Maths Iterative methods The price of a car after x years is modelled by f(x) = 2200(0.7)Λ£ βˆ’ 1200sinx. Show that the function has a root between x = 12 and x = 13 and using this, criticise the model.
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15

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The motion of an object is modelled by the function f(x) = βˆ’xΒ² + 5x + 30, x > 0, where x represents the horizontal distance travelled. Show that the ball will hit the ground between x = 8 and x = 9.

OCR A-level Maths Iterative methods The motion of an object is modelled by the function f(x) = βˆ’xΒ² + 5x + 30, x > 0, where x represents the horizontal distance travelled. Show that the ball will hit the ground between x = 8 and x = 9.
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16

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The journeys of two cars are modelled by the functions f(x) = √x + 7 and g(x) = x. Sketch a staircase diagram starting at x = 5 to show where the journeys of the two cars will intersect.

OCR A-level Maths Iterative methods The journeys of two cars are modelled by the functions f(x) = √x + 7 and g(x) = x. Sketch a staircase diagram starting at x = 5 to show where the journeys of the two cars will intersect.
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