A recurrence relationship can be used to generate all the terms of a sequence. It describes each term as a function of the previous term.

The Newton-Raphson method uses tangent lines to find approximations of roots of equations in the form f(x) = 0. A value for x₀ is chosen and a tangent at that point is drawn. The next x value is taken to be the point at which the tangent intersects the x-axis. This process is continued to find increasingly accurate approximations of the root.

The Newton-Raphson formula can be used to calculate increasingly accurate approximations of a root, given a starting value. If the starting value is chosen to be a turning point, the formula cannot be used, since its derivative will be 0 and division by 0 in the formula will not be possible. If the starting value is chosen to be near a turning point, the gradient will be small, therefore the tangent will intersect the x-axis a long way away from the starting value, and therefore the Newton-Raphson method may fail.

The Newton-Raphson method can be used to model situations and find their solutions.

1

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The recurrence relationship of a sequence is given by u_(n+1) = 100 − u_n, where u₁ = 18. Calculate the next three terms of the sequence.

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2

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Find the first term of the sequence given by the recurrence relationship u_(n + 1) = 15u_n + 3, where u₂ = 33.

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3

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Explain how the Newton-Raphson method is used graphically to approximate a root.

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4

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Taking x = −2 as the first approximation, apply the Newton-Raphson method graphically to find the root of the graph of y = (0.5x + 5)².

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5

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Show that the equation y = x² − 1/x has a root in the interval [0.8, 1.2].

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6

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Let f(x) = ln(3x + 5). Taking the first approximation to be x = −1.5, apply the Newton-Raphson method to calculate a second approximation.

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7

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The cross-section of a river bank is given by the equation y = e⁻ˣ(x² − 5x). Using x = 3.5 as a first approximation, calculate a second approximation for the point at which the river levels off.

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8

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The journey of a skier is modelled by the equation y = −x³ + 5x, 0 < x < a. Using x = 2 as a first approximation, calculate a second approximation for x = a.

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9

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The depth of a stream is given by the given function. Using the first approximation of x = 6, calculate a second approximation for the point at which the graph crosses the x-axis

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