Task 1
Produce a report that contains written descriptions, analysis and mathematics that shows how calculus can be used to solve engineering problems as set out below.
1 The equation for a distance, s(m), travelled in time t(s) by an object starting with an initial velocity u(ms-1) and uniform acceleration a(ms-2) is:
s=ut+1/2 at^2
The tasks are to:
Plot a graph of distance (s) vs time (t) for the first 10s of motion if u=10ms^(-1) and a=5ms^(-2).
Determine the gradient of the graph at t=2s and t=6s.
Differentiate the equation to find the functions for
Velocity (v=ds/dt)
Acceleration (a=dv/dt=(d^2 s)/?dt?^2 )
Use your result from part c to calculate the velocity at t=2s and t=6s.
Compare your results for part b and part d.
2 The displacement of a mass is given by the function
y=sin 3t .
The tasks are to:
Draw a graph of the displacement y(m) against time t(s) for the time t=0s to t=2s.
Identify the position of any turning points and whether they are maxima, minima or points of inflexion.
Calculate the turning points of the function using differential calculus and show which are maxima, minima or points of inflexion by using the second derivative.
Compare your results from parts b and c.
3 The equation for the instantaneous voltage across a discharging capacitor is given by v=V_O e^(-t/t), where V_O is the initial voltage and t is the time constant of the circuit.
The tasks are to:
Draw a graph of voltage against time for V_O=12V and t=2s, between t=0s and t=10s.
Calculate the gradient at t=2
s and t=4s.
Differentiate v=12e^(-t/2) and calculate the value of dv/dt at t=2s and t=4s.
Compare your answers for part b and part c.
Calculate the second derivative of the instantaneous voltage ((d^2 v)/?dt?^2 ).
4 The same capacitor circuit is now charged up to 12V and the instantaneous voltage is v=12(1-e^(-t/2) ).
The tasks are to:
Differentiate v with respect to t to give an equation for dv/dt.
Calculate the value of dv/dt at t=2s and t=4s.
Find the second derivative ((d^2 v)/?dt?^2 ).
5 The gain of an amplifier is found to be G=20 log?(10V_out ),:
The tasks are to find equations for:
dG/(dV_Out )
(d^2 G)/?dV_Out?^2
6 The displacement, y(m), of a body in damped oscillation is y=2e^(-t) sin?3t.
The task is to:
Use the Product Rule to find an equation for the velocity of the object if v=dy/dt.
7 The velocity of a moving vehicle is given by the equation v=(2t+3)^4
The task is to:
Use the Chain Rule to determine an equation for the acceleration when a=dv/dt.
8 A communication signal is given by the function y=sin?t/t
The task is to:
Derive an equation for dy/dt using the Quotient Rule.
9 A company is required to fence off a square/rectangular area around a robot arm to comply with health and safety law. They have 750m of fencing available.
The task is to:
Find the maximum square/rectangular area they can fence off?
10 You plan to make a simple, open topped box from a piece of sheet metal by cutting a square – of equal size – from each corner and folding up the sides as shown in the diagram: