BTEC Level 3 Unit 7 Calculus to Solve Engineering Problems Assignment Brief

Assignment Brief A

Vocational Scenario or Context

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)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:

The tasks are to:

(a) Plot a graph of distance (s) vs time (t) for the first 10s of motion if and
(b) Determine the gradient of the graph at  and .
(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 and .
(e) Compare your results for part b and part d. 2 The displacement of a mass is given by the function

(2)

The tasks are to:

(a) Draw a graph of the displacement y(m) against time t(s) for the time to .

(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)The equation for the instantaneous voltage across a discharging capacitor is given by , where is the initial voltage and  is the time constant of the circuit.

The tasks are to:

(a) Draw a graph of voltage against time for and , between  and .
(b) Calculate the gradient at and .
(c) Differentiate and calculate the value of  at  and .
(d) Compare your answers for part b and part c.
(e) Calculate the second derivative of the instantaneous voltage .

(4)The same capacitor circuit is now charged up to 12V and the instantaneous voltage is .

The tasks are to:

(1) Differentiate with respect to  to give an equation for
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Calculate the value of at  and .
Find the second derivative .

(5) The gain of an amplifier is found to be ,:

The tasks are to find equations for:

(6) The displacement, (m), of a body in damped oscillation is

The task is to:

(a) Use the Product Rule to find an equation for the velocity of the object if .

(7) The velocity of a moving vehicle is given by the equation The task is to:

(a)Use the Chain Rule to determine an equation for the acceleration when .

(8) A communication signal is given by the function

The task is to:

(a) Derive and equation for using the Quotient Rule.

(9) 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) 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:

If and calculate:

(a) The value of x which will give the maximum volume

(b) The maximum volume of the box

(c) Comment of 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. You 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.

BTEC Assignment Brief B

Vocational Scenario or Context

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 integration to determine if you are suitable.

Task 1

1 The tasks are to:

a) Find the indefinite integral of the function
    * y = 3t² + 2e³ᵗ + 1/t + 2 cos 3t

b) Calculate the definite integral
    ∫₁² [3t² + 2e³ᵗ + 1/t + 2 cos 3t] dt

2 An object is moving with a uniform acceleration a = 2 ms⁻²,
determine the functions for:

a) Velocity v (ms⁻¹) – given v(0) = 10 ms⁻¹
b) Displacement s (m) – given s(0) = 5 m
c) Calculate the values of v and s for:
  i) t = 2 s
  ii) t = 5 s

3 The extension, y, of a material with an applied force, F, is
given by y = e⁽ᶠ×1×10⁻³⁾.

a) Calculate the work done if the force increases from
    100 N to 500N using:

i) An analytical integration technique
ii) A numerical integration technique

[Note: the work done is given by the area under the curve]

b) Compare the two answers
c) Using a computer spreadsheet increase the number of values used for your numerical method
d) Analyse any affect the size of numerical step has on the result.

4. For the function y = 12 sin 4θ, calculate the:
 a) Mean
 b) Root mean square (RMS)

Over a range of 0 ≤ θ ≤ π/4 radians.

[Note the trigonometric identity cos 2θ = 1 – 2 sin² θ]

5. A complex function can be modelled by the equation:

   y = cos(x³) 3x²

Find the indefinite integral of the function
∫ cos(x³) 3x² dx using a substitution method.

6. The acceleration of an object moving in a strange way has been modelled as a = e⁽ᶠˣ⁾.
a) Use integration by parts to find an equation to model the velocity if v = ∫ x e⁽ᶠˣ⁾ dx.
b) Is the problem any different if you find v = ∫ e⁽ᶠˣ⁾ dx?

7. Newton’s laws of cooling proposes that the rate of change of temperature is proportional to the temperature difference to the ambient (room) temperature. And can be modelled using the equation:

   dT/dt = –k(T – Tₐ)

This can also be written as:

   dT / (T – Tₐ) = –k dt

Where:
T = Temperature of material
Tₐ = Ambient (room) temperature
k = A cooling constant

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. Graphs should be well presented and clearly labelled and comparisons between methods should be accurate and well presented.

Criteria Covered by this Task:

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Assignment Brief C

Vocational Scenario or Context

You are working as a Junior Engineer for a small motor racing team. You have been given a proposed mathematical model to calculate the velocity of a car accelerating from rest in a straight line. The equation is:

is the instantaneous velocity of the car (m/s) t is the time in seconds  is the time to reach the maximum speed in seconds A is a constant.

The Team Manager has asked you to carry out an analysis of the model and produce a written report. As part of the analysis you have been given some data on a known model.

Task 1

Produce a report that contains written descriptions, analysis and mathematics that shows how calculus can be used to solve an engineering problem. The tasks are to:

• Use thinking methods to analyse the given engineering problem, e.g. break the problem down into a series of manageable elements, and produce a specification
• Prepare a valid proposal for solving the problem and present it
• Produce mathematical models for the identified elements of the problem
• Apply calculus methods to produce answers for each of the elements
• Bring the elements together in a formal presentation

In your proposal you need to outline the problem and the methods needed to solve it. You need to include how to

(1) Identify the

1. • units of the coefficient A
   • physical meaning of A
   • velocity of the car at t = 0
   • asymptote of this function as t → ∞?
3. (2) Sketch a graph of velocity vs. time.

(3) Derive an equation x(t) for the instantaneous position of the car as a function of time. Identify the

• • value x when t = 0 s
  • asymptote of this function as t → ∞

(4) Sketch a graph of position vs. time.

(5) Derive an equation for the instantaneous acceleration of the car as a function of time.

Identify the

• acceleration of the car at t = 0 s
• asymptote of this function as t → ∞

(6) Sketch a graph of acceleration vs. time.

(7) Apply your mathematical models to your allocated car. Use the given data for the 0 – 28 m/s and 400m times to calculate the:

• • value of the coefficient A
  • maximum velocity

maximum acceleration.

Checklist of evidence required

Your informal report should contain:

• written descriptions,
• 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. Diagrams and sketches should be well presented and clearly labelled.

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