Exam-style Questions: Numerical Methods
1.
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.
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 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).
(4 marks)
b) Rearranging the equation of the straight line, gives:
so 
at point A, x = a, so:
The equation to a tangent to the straight line is:
or 
At B, y = 0 therefore x = 2a.
(3 marks)
(Marks available: 7)
2.The variables x and y satisfy the differential equation
and y = 2 when x = 1.
a) Use a local linear approximation to show that, when x = 1.02,
b) By using the iterative equation
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 outline and marking scheme for question: 2
Give yourself marks for mentioning any of the points below:
a) Using the local linear approximation, gives:
(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.
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 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)