Exam-style Questions: Numerical Methods

1.

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Figure 1 shows the graphs of y = tan-l x and xy = 1 intersecting at the point A with x-coordinate.

a) (i) Show that 1.1

(ii) Use linear interpolation to find an approximation for A, giving your answer to two decimal places.

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Figure 2 shows the tangent to the curve xy = 1 at A. This tangent meets the x-axis at B. The region between the arc OA, the line AB and the x-axis is shaded.

b) Show that the x-coordinate of B is 2.

(Marks available: 7)

Answer

Answer outline and marking scheme for question: 1

Give yourself marks for mentioning any of the points below:

a) (i) The line intersect at a tan-l a =1.

Let f(x) = a tan-l a -1

Then f(1.1) = -0.0837206

and f(1.2) = +0.0512696 > 0

Therefore the change of sign (i.e. the x value of a is between 1.1 and 1.2)

(ii) Using linear interpolation to get a more accurate answer (to 2 decimal places).

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(4 marks)

b) Rearranging the equation of the straight line, gives:

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at point A, x = a, so:

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The equation to a tangent to the straight line is:

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At B, y = 0 therefore x = 2a.

(3 marks)

(Marks available: 7)

2.The variables x and y satisfy the differential equation

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and y = 2 when x = 1.

a) Use a local linear approximation to show that, when x = 1.02,

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b) By using the iterative equation

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find approximations for the values of y when x takes the values 1.02, 1.04 and 1.06, giving each value to three places of decimals.

(Marks available: 6)

Answer

Answer outline and marking scheme for question: 2

Give yourself marks for mentioning any of the points below:

a) Using the local linear approximation, gives:

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(2 marks)

b) Substituting values of x into the iterative equation given gives:

First value is 2.157

Second value is 2.316

Third value is 2.477

(4 marks)

(Marks available: 6)

3.

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Figure 1 above shows sketches of the graphs of

y = e-x and y = x

and their intersection at x = α, where α is approximately 0.57.

(i) Starting with x1 = 0.57, carry out the iteration

xn+1 = e-xn

up to and including x5, recording each value of xn to four decimal places as you proceed.

(ii) Write down an estimate of the value of α to three decimal places.

(Marks available: 4)

Answer

Answer outline and marking scheme for question: 3

Give yourself marks for mentioning any of the points below:

Performing the iteration on the equation given, gives:

x2 = 0.5655

x3 = 0.5681

x4 = 0.5666

x5 = 0.5675

Taking the results above, the closest approximation to α at 3d.p is 0.567.

(Marks available: 4)