# 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 x_{1} = 0.57, carry out the iteration

x^{n+1} = e^{-xn}

up to and including x_{5}, recording each value of x_{n} 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:

x_{2} = 0.5655

x_{3} = 0.5681

x_{4} = 0.5666

x_{5} = 0.5675

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

**(Marks available: 4)**