Unit 7 Calculus to solve engineering Problems Assignment 1. In this assignment i got a distinction grade which can be seen from the quality of notes displayed in within. This assignment is about differentiation.
BTEC Assignment Brief
Pearson BTEC Level 3 National Extended Diploma in
Electrical/Electronic Engineering
Pearson BTEC Level 3 National Extended Diploma in
Qualification
Manufacturing Engineering
Pearson BTEC Level 3 National Extended Diploma in
Aeronautical Engineering
Unit number and title Unit 7: Calculus to solve engineering problems
Learning aim(s) A: Examine how differential calculus can be used to solve
(For NQF only) engineering problems
Assignment title Differential Calculus
Assessor S.Walsh
Issue date 01/11/21
Hand in deadline 23/11/21
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.
Vocational Scenario or Part of your apprenticeship is to spend time working in all
Context 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.
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.
Task 1 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:
, 𝟏
𝒔 = 𝒖𝒕 + 𝒂𝒕𝟐
𝟐
The tasks are to:
a) Plot a graph of distance (s) vs time (t) for the first 10s
of motion if 𝑢 = 𝑼 𝑚𝑠 −1 and 𝑎 = 5 𝑚𝑠 −2 .
b) Determine the gradient of the graph at 𝑡 = 2𝑠 and
𝑡 = 6𝑠.
c) Differentiate the equation to find the functions for
𝑑𝑠
i) Velocity (𝑣 = 𝑑𝑡 )
𝑑𝑣 𝑑2 𝑠
ii) Acceleration (𝑎 = = 𝑑𝑡 2 )
𝑑𝑡
d) Use your result from part c to calculate the velocity at
𝑡 = 2𝑠 and 𝑡 = 6𝑠.
e) Compare your results for part b and part d.
Task 1 2. The equation for the instantaneous voltage across a
𝒕
discharging capacitor is given by 𝒗 = 𝑽𝑶 𝒆−𝝉 , where 𝑉𝑂 is
the initial voltage and 𝜏 is the time constant of the circuit.
The tasks are to:
a) Draw a graph of voltage against time for 𝑉𝑂 = 12𝑉
and 𝜏 = 𝝉 𝑠, between 𝑡 = 0𝑠 and 𝑡 = 10𝑠.
b) Calculate the gradient at 𝑡 = 2𝑠 and 𝑡 = 4𝑠.
𝑡
𝑑𝑣
c) Differentiate 𝑣 = 12𝑒 −𝝉 and calculate the value of 𝑑𝑡
at 𝑡 = 2𝑠 and 𝑡 = 4𝑠.
d) Compare your answers for part b and part c.
e) Calculate the second derivative of the instantaneous
𝑑2 𝑣
voltage ( 𝑑𝑡 2 ).
The same capacitor circuit is now charged up to 12V and the
𝒕
instantaneous voltage is now given by 𝒗 = 𝟏𝟐 (𝟏 − 𝒆−𝝉 ).
, f) Differentiate 𝑣 with respect to 𝑡 to give an equation
𝑑𝑣
for 𝑑𝑡 .
𝑑𝑣
g) Calculate the value of 𝑑𝑡 at 𝑡 = 2𝑠 and 𝑡 = 4𝑠.
𝑑2 𝑣
h) Find the second derivative ( 𝑑𝑡 2 ).
The sales of a product (S) over time (t) can be represented
according to the following natural logarithm 𝑺 = 𝒍𝒏(𝟐𝒕) .
𝑑𝑆
i) Find the first derivative ( dt ).
𝑑2 𝑆
j) Find the second derivative ( dt2 ).
3. The displacement, 𝑦(m), of a body in damped oscillation is
given by the equation 𝒚 = 𝟐𝒆−𝒕 𝐬𝐢𝐧(𝟑𝒕) .
a) Use the Product Rule to find an equation for the
𝑑𝑦
velocity of the object if 𝑣 = .
𝑑𝑡
Task 1
The velocity of a moving vehicle is given by the equation
𝒗 = (𝟐𝒕 + 𝟑)𝟒 .
b) Use the Chain Rule to determine an equation for the
𝑑𝑣
acceleration when 𝑎 = .
𝑑𝑡
𝐬𝐢𝐧(𝒕)
A communication signal is given by the function 𝒚 = .
𝒕
𝑑𝑦
c) Derive an equation for 𝑑𝑡 using the Quotient Rule.
4. The displacement of an oscillating mass is given by the
function 𝒚 = 𝒔𝒊𝒏 𝟑𝒕 .
The tasks are to:
a) Draw a graph of the displacement y(m) against time t(s)
for the time 𝑡 = 0𝑠 to 𝑡 = 2𝑠.
, 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.
5. A company is required to mark out a rectangular area
around a production area to install a new coolant tank.
They have a perimeter L m of barriers available.
The task is to:
a) Find the maximum area that can be used.
Task 1 The cylindrical tank is designed to have a maximum coolant
volume of V m3. Determine the following:
b) Minimum surface area of the tank.
c) Dimensions of the cylindrical tank.
h
r
d) Comment on the result in part a and part c.
Checklist of evidence Your informal report should contain:
required ● written solutions and graphical printouts
● analysis
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.
,Criteria covered by tasks:
Unit/Criteria
To achieve the criteria you must show that you are able to:
reference
Evaluate, using technically correct language and a logical structure, the
correct graphical and analytical differential calculus solutions for each
7/A.D1
type of given routine and non-routine function, explaining how the
variables could be optimised in at least two functions.
Find accurately the graphical and analytical differential calculus solutions
7/A.M1 and, where appropriate, turning points for each type of given routine and
non-routine function and compare the results.
Find the first and second derivatives for each type of given routine
7/A.P1
function.
Find, graphically and analytically, at least two gradients for each type of
7/A.P2
given routine function.
Find the turning points for given routine polynomial and trigonometric
7/A.P3
functions.
Sources of information to http://www.mathsisfun.com/index.htm
support you with this
Assignment http://www.mathcentre.ac.uk/students/topics
Other assessment
materials attached to this Student Datasets
Assignment Brief
Q1 Q2 Q5
Initial Time
Student L V
Velocity Constant
(m) (m3)
U (m/s) (s)
1 1 1 60 30
2 2 2 80 40
3 3 3 100 50
4 4 4 120 60
5 5 5 140 70
6 6 6 160 80
7 7 7 180 90
8 8 8 200 100
9 9 9 220 110
10 10 10 240 120
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