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BTEC Level 3 Unit 7 Engineering Calculus Assignment: Differentiation Applications, Graphs & Optimisation Problems
| University | Milton Keynes College |
| Subject | Unit 7: Calculus to solve engineering problems |
Learning aim(s)
A: Examine how differential calculus can be used to solve engineering problems
Assignment title: Solving engineering problems that involve differentiation
Vocational Scenario
You are working as an apprentice engineer at a company involved in the research, design production and maintenance of bespoke engineering solutions for larger customers.
Part of your apprenticeship is to spend time working in all departments, however a certain level of understanding needs to be shown before the managing director allows apprentices into the design team and so she has developed a series of questions on differentiation to determine if you are suitable.
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. Distance-Time Relationship
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 + ½at2
a) Plot a graph of distance (s) vs time (t) for the first 10s of motion if u = 5ms-1 and a = 2ms-2.
b) Determine the gradient of the graph at t = 3s and t = 6s.
c) Differentiate the equation to find the functions for:
i) Velocity
ii) Acceleration
d) Use your result from part c to calculate the velocity at t = 3s and t = 6s.
e) Compare your results for part b and part d.
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2. Displacement Function
The displacement of a mass is given by the function y = t3 – 6t2 + 9t.
a) Draw a graph of the displacement y(m) against time t(s) for the time t = 0s to t = 5s.
b) Identify the position of any turning points and whether they are maxima, minima or points of inflexion.
c) 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. Discharging Capacitor
The equation for the instantaneous voltage across a discharging capacitor is given by v = V0e-t/τ, where V0 is the initial voltage and τ is the time constant of the circuit.
a) Draw a graph of voltage against time for V0 = 10V and τ = 2s, between t = 0s and t = 10s.
b) Calculate the gradient at t = 1s and t = 4s.
c) Differentiate and calculate the value of dv/dt at t = 1s and t = 4s.
d) Compare your answers for part b and part c.
e) Calculate the second derivative of the instantaneous voltage.
4. Charging Capacitor
The same capacitor circuit is now charged up to 12V and the instantaneous voltage is v = 12(1 – e-t/τ).
a) Differentiate with respect to t to give an equation for dv/dt.
b) Calculate the value of dv/dt at t = 0.5s and t = 2s.
c) Find the second derivative d2v/dt2.
5. Amplifier Gain
The gain of an amplifier is found to be G = 10/(1 + 0.1f2).
The tasks are to find equations for:
a) dG/df
b) d2G/df2
6. Damped Oscillation
The displacement, y(m), of a body in damped oscillation is y = te-t.
The task is to:
a) Use the Product Rule to find an equation for the velocity of the object if v = dy/dt.
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7. Vehicle Motion
The velocity of a moving vehicle is given by the equation v = (3t + 2)4.
The task is to:
a) Use the Chain Rule to determine an equation for the acceleration when a = dv/dt.
8. Communication Signal
A communication signal is given by the function f(t) = sin(t)/(1 + t2).
The task is to:
a) Derive an equation for df/dt using the Quotient Rule.
9. Fencing Problem
A company is required to fence off the area around a robot arm to comply with health and safety law. They have 750m of fencing available.
The task is to:
a) Find the maximum area they can fence off?
10. Box Volume Problem
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.
If the sheet is 1m × 0.5m and the squares have side x, calculate:
a) The value of x which will give the maximum volume
b) The maximum volume of the box
c) Comment on the value obtained in part b.
Checklist of evidence required
Your informal report should contain:
- Analysis
- Worked solutions to the problems
Each worked solution should be laid out clearly and contain brief explanations of the stages of the calculation to indicate your understanding of how calculus can be used to solve an engineering problem. Your explanation should be detailed in response to questions 9 and 10 to show how the variables are optimised in each case. Graphs should be well presented and clearly labelled and comparisons between methods should be accurate and well presented.
Criteria covered by this task:
7/A.D1 Evaluate, using technically correct language and a logical structure, the correct graphical and analytical differential calculus solutions for each type of given routine and non-routine function, explaining how the variables could be optimised in at least two functions.
7/A.M1 Find accurately the graphical and analytical differential calculus solutions and, where appropriate, turning points for each type of given routine and non-routine function and compare the results.
7/A.P1 Find the first and second derivatives for each type of given routine function.
7/A.P2 Find, graphically and analytically, at least two gradients for each type of given routine function.
7/A.P3 Find the turning points for given routine polynomial and trigonometric functions.
Sources of information to support you with this Assignment
- mathsisfun.com/index.htm
- mathcentre.ac.uk/students/topics
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