Assessment Task:

Answer the three questions given on the assignment sheet, further instructions are provided on the assignment sheet.

Learning Outcomes to be assessed:

1 Demonstrate an understanding of the mathematics needed for engineering

2 Apply appropriate techniques to solve problems

1. (a) Evaluate 6α⁵ - 3α² + 15α - 3.

(b) Evaluate sin^-1(3/(ln 2α)) in degrees and in radians

(c) Calculate A^TB where

(d) Find the roots, expressed to 4 decimal places, of the equation x⁴ - αx + 20 = 1.

(e) Let y = e^3x(5 cos √3x - 4 sin √3x + αe^-2x) and calculate in its simplest form

d²y/dx² - 6dy/dx + 12y.

(f) Let y = (5 sin (x - 2))/x + 1 then between x = 2 and x = α calculate the area under the curve and the volume of the solid formed when the curve is rotated about the x axis, each expressed to 4 decimal places.

2. An object of mass 5α is to be hung from the end of an uniform rigid 12 metre horizontal pole of mass 8, as shown in Figure 1. The pole is attached to a wall by a pivot and is supported by a 9 metre cable which is attached to the wall at a higher point.

Figure 1: A weight suspended from an uniform rigid bar supported by a cable

The tension on the cable is given by the following equation: T = a.b.(Mog + 1/2Mpg)/c√(a² - c²)

where T is the tension on the cable (N), a is the length of the cable in metres (m), b is the length of the pole in metres (m), M_o is the mass of the object (kg), g is the gravitational constant (9.80665m/s²), M_p is the mass of the pole (kg) and c is the distance from the wall to the attachment point of the cable in metres (m).

(a) Write an m-file script to determine the distance c at which the cable should be attached to the pole to minimize the tension in the cable. To do this, the algorithm should calculate the tension at 0.75 metre intervals from c = 1.25 metres to c = 8.5 metres. Expand this script so that it also plots the tension of the cable as a function of c, with appropriate titles and axis labels.

(b) Write an m-file function to determine the distance c that minimizes T , this time making use of differentiation and stationary points. This function should accept inputs for a, b, M_o and M_p.

3. Consider the following integral:

I = ₀∫^2.5 5/α.x.cos(x +5)dx

Use MATLAB to approximate I using the midpoint rule, trapezium rule and Simpson's rule. In each case use 4 and 8 intervals and compare the absolute error for each method. Write a summary of your findings looking at the relationship between the number of intervals and the absolute error in the approximation. Your summary should document your approach, results and findings.