LO$3$ Approximate solutions of contextualised examples with graphical and numerical

methods

Assignment Brief and Guidance

Task $1$

The engineering department has developed the following equation for the bending moment of a

beam and you have asked to investigate its behaviour

$M(x)=x^3-3x^2-4$

The Beam is $4m$ long and the design team suspect the is problem if the bending moment is zero

in the range between $3-4m$ and you have been asked to

a$)$ Plot the bending moment at $0\mathrm{.}5m$ interval for the range $0x4m,0x4$ and

determine if the bending moment is zero in range $3mx4m$

b$)$ Use the graph to estimate where the bending moment is zero

c$)$ Use the bisection method to numerically estimate the exact location where the bending

moment is zero

d$)$ Newton$-$Raphson method to obtain the required location

e$)$ Compare the results of the above method to determine which gives a best solution

Task $2$

The following offsets are taken from a chain line to an irregular boundary towards right side

of the chain line.

Common distance $d=25m$

You have been asked to estimate the area using the following methods and compare and

comment on their difference and accuracy.

a$)$ Trapezium Rule

b$)$ Simpson's Rule

Task $3$

The equation governing a body travelling in a water channel is given by the following equation

$\frac{dv}{dt}=1-v^2$

Plot the velocity time graph for the object and determine the final velocity and the time taken to

reach this velocity